Communications In Mathematical Physics - Volume 267

Commun. Math. Phys. 267, 1–12 (2006) Digital Object Identifier (DOI) 10.1007/s00220-006-0033-1 Communications in Mathe...

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Commun. Math. Phys. 267, 1–12 (2006) Digital Object Identifier (DOI) 10.1007/s00220-006-0033-1

Communications in

Mathematical Physics

Simple Waves and a Characteristic Decomposition of the Two Dimensional Compressible Euler Equations Jiequan Li1, , Tong Zhang2 , Yuxi Zheng3, 1 Department of Mathematics, Capital Normal University, Beijing, 100037, P.R. China 2 Institute of Mathematics, Chinese Academy of Sciences, Beijing, 100080, P.R. China 3 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.

E-mail: [email protected] Received: 15 September 2005 / Accepted: 27 December 2005 Published online: 5 May 2006 – © Springer-Verlag 2006

Abstract: We present a characteristic decomposition of the potential flow equation in the self-similar plane. The decomposition allows for a proof that any wave adjacent to a constant state is a simple wave for the adiabatic Euler system. This result is a generalization of the well-known result on 2-d steady potential flow and a recent similar result on the pressure gradient system. 1. Introduction The one-dimensional wave equation u tt − c2 u x x = 0

(1)

with constant speed c has an interesting decomposition (∂t + c∂x )(∂t − c∂x )u = 0,

(2)

(∂t − c∂x )(∂t + c∂x )u = 0,

(3)

or

known from elementary text books. One can rewrite them as ∂+ ∂− u = 0, or ∂− ∂+ u = 0,

(4)

where ∂± = ∂t ± c∂x . Sometimes, the same fact is written in Riemann invariants ∂t R + c∂x R = 0, ∂t S − c∂x S = 0

(5)

Research partially supported by NSF of China with No. 10301022, NSF from Beijing Municipality, Fok Ying Tong Educational Foundation, and the Key Program from Beijing Educational Commission with no. KZ200510028018. Research partially supported by NSF-DMS-0305497, 0305114.

2

J. Li, T. Zhang, Y. Zheng

for the Riemann invariants R := (∂t − c∂x )u, S := (∂t + c∂x )u. For a pair of a system of hyperbolic conservation laws u f (u, v) 0 , + = v t g(u, v) x 0

(6)

(7)

it is known that a pair of Riemann invariants exist so that the system can be rewritten as ∂t R + λ1 (u, v)∂x R = 0, (8) ∂t S + λ2 (u, v)∂x S = 0, where (R, S) are the Riemann invariants and the λ’s are the two eigenvalues of the system. These decompositions and Riemann invariants are useful in the construction of solutions, for example, the construction of the D’Alembert formula, and proof of development of singularities ([5]). An example of the system is the system of isentropic irrotational steady two-dimensional Euler equations for compressible ideal gases 2 (c − u 2 )u x − uv(u y + vx ) + (c2 − v 2 )v y = 0, (9) u y − vx = 0, supplemented by Bernoulli’s law c2 u 2 + v2 + = k2, γ −1 2

(10)

where γ > 1 is the gas constant while k > 0 is an integration constant. This system has two unknowns (u, v), and by the existence of Riemann invariants, any solution adjacent to a constant state is a simple wave. A simple wave means a solution (u, v) that depends on a single parameter rather than the pair parameters (x, y). Since there is the lack of explicit expressions, the concept of Riemann invariants plays a limited role in a much broader sense, e. g., to treat the full Euler equations. In recent years, the pressure gradient system  u t + px = 0,  vt + p y = 0, (11)  E + (up) + (vp) = 0, t x y where E = p + (u 2 + v 2 )/2, has been known to have “simple waves” adjacent to a constant state (u, v, p) in the self-similar variable plane (ξ, η) = (x/t, y/t). This system has three equations and no Riemann invariants have been found. But the equation for the unknown variable p in the (ξ, η) plane ( p − ξ 2 ) pξ ξ − 2ξ ηpξ η + ( p − η2 ) pηη +

(ξ pξ + ηpη )2 − 2(ξ pξ + ηpη ) = 0 (12) p

allows for a decomposition ∂+ ∂− p = m + ∂− p, m + :=

r 4 λ+ pr , 2 p2

(13)

Simple Waves and Decomposition of 2D Compressible Euler Equations

3

where (r, θ ) denotes the polar coordinates of the (ξ, η) plane and ∂+ =

∂θ + λ−1 + ∂r ;

∂− =

∂θ + λ−1 − ∂r ;

λ± = ±

p . − p)

r 2 (r 2

(14)

For convenience of verification we state that the p equation in polar coordinates takes the form ( p − r 2 ) prr +

p p 1 pθθ + pr + (r pr )2 − 2r pr = 0. 2 r r p

(15)

The characteristics are defined by dθ = λ± . dr

(16)

∂± λ ± = n ± ∂ ± p

(17)

In addition, we know that

for some nice factors n ± . These facts allow for expressions ∂∓ (∂± λ± ) = (∂± λ± ) f ±

(18)

for some nice factors f ± . This decomposition leads directly to the fact that Proposition 1. A state adjacent to a constant state for the pressure gradient system must be a simple wave in which p is constant along characteristics of a plus (or minus) family. These lead to the desire to consider the pseudo-steady isentropic irrotational Euler system which has three equations with source terms,   (ρU )ξ + (ρV )η = −2ρ, (ρU 2 + p(ρ))ξ + (ρU V )η = −3ρU,  (ρU V )ξ + (ρV 2 + p(ρ))η = −3ρV,

(19)

where (ξ, η) = (x/t, y/t), (U, V ) = (u − ξ, v − η) is the pseudo-velocity, and the pressure p = p(ρ) is the function of the density ρ. It turns out that we are unable to find explicit forms of the Riemann invariants, but decompositions similar to ∂+ ∂− λ− = m∂− λ− hold for some m, presented in Sect. 4. We use the characteristic decomposition of Sect. 4 to establish in Sect. 5 that adjacent to a constant state a wave must be a simple wave for the pseudo-steady irrotational isentropic Euler system. A simple wave for this case is such that one family of wave characteristics are straight lines and the physical quantities velocity, speed of sound, pressure, and density are constant along the wave characteristics. Further, using the fact that entropy and vorticity are constant along the pseudo-flow characteristics (the pseudo-flow lines), our irrotational result extends to the adiabatic full Euler system, see Sect. 5.

4


2. Two-by-Two System Consider a 2 × 2 hyperbolic system in the Riemann invariants ∂t R + λ1 ∂x R = 0, ∂t S + λ2 ∂x S = 0.

(20)

So we find that ∂2 λ2 := (∂t + λ2 ∂x )λ2 = λ2,R ∂2 R,

(21)

where λ2,R := ∂ R λ2 . We go on to find ∂ 1 ∂2 R = and so

∂1

∂1 λ2 − ∂2 λ1 ∂2 R, λ2 − λ1

1 λ2,R

∂2 λ 2

=

(22)

∂1 λ2 − ∂2 λ1 ∂2 λ2 , λ2 − λ1 λ2,R

(23)

which implies ∂1 ∂2 λ2 =

∂1 λ2 − ∂2 λ1 ∂1 λ2,R + λ2 − λ1 λ2,R

∂2 λ 2 .

(24)

The elegant form is undermined by the dependence on the Riemann invariant R via the term λ2,R . It is not very useful when the explicit form of the Riemann invariants are not known. But we think it is worth mentioning. For example, (24) can be applied directly to show that all characteristics in a wave adjacent to a constant state are straight and thus such a wave is a simple wave. 3. Steady Euler System Let us build explicitly the characteristic decomposition for the steady Euler system for isentropic irrotational flow (9)(10) in the absence of the explicit form of the Riemann invariants. The same technique can be extended to the case of pseudo-steady flows in Sect. 4. We write the system in the form −2uv c2 −v 2 u u + c2 −u 2 c2 −u 2 = 0. (25) v x v y −1 0 The matrix has eigenvalues λ± =

uv ±

c2 (u 2 + v 2 − c2 ) u 2 − c2

dy = , dx

(26)

which are solutions to the characteristic equation λ2 +

2uv c2 − v 2 λ + = 0. c2 − u 2 c2 − u 2

(27)


5

We have the left eigenvectors ± = [1, λ∓ ],

(28)

where we have used the relation λ± λ∓ =

c2 − v 2 . c2 − u 2

(29)

The characteristic form of the system is therefore u u + λ± ± = 0, ± v x v y

(30)

which is equivalent to ∂± u + λ∓ ∂± v = 0.

(31)

We then have ∂− λ− = ∂x λ− + λ− ∂ y λ− = ∂u λ− ∂− u + ∂v λ− ∂− v = (∂u λ− − ∂v λ− /λ+ ) ∂− u.

(32)

We shall ignore the similar calculation for ∂+ λ+ for simplicity of notation. Now that the term ∂− λ− differs from ∂− u by a lower-order factor, we shall focus our attention on ∂− u. First we see that we can derive a second-order equation for u, i. e., 2uv c2 − u 2 u yy − 2 u − u = 0, (33) y x c − v2 c2 − v 2 x or equivalently uxx

2uv c2 − v 2 c2 − v 2 − 2 u + u = x y yy c − u2 c2 − u 2 c2 − u 2

2uv c2 − v 2

uy − x

c2 − u 2 c2 − v 2

ux . x

(34)

We now compute the ordered derivative ∂+ ∂− u to find ∂+ ∂− u = u x x + (λ+ + λ− )u x y + λ+ λ− u yy + ∂+ λ− u y .

(35)

∂+ λ− = ∂+ u(∂u λ− − ∂v λ− /λ− ).

(36)

2 2uv c2 − v 2 c − u2 uy − 2 ux ∂+ ∂− u = 2 c − u2 c2 − v 2 x c − v2 x +(u x + λ+ u y )u y (∂u λ− − ∂v λ− /λ− ).

(37)

We find that

Thus we obtain

We notice that the above right-hand side is a quadratic form in (u x , u y ), once we substitute vx by u y . So we compute further. We use the Bernoulli’s law to find (c2 )x = −(γ − 1)(uu x + vu y ).

(38)

6


So we find 2uv 2 = 2 [vu x (c2 − u 2 − v 2 + γ u 2 ) + uu y (c2 + γ v 2 )], c2 − v 2 x (c − v 2 )2 2 c − u2 −1 = 2 [uu x (2c2 − v 2 − u 2 + γ u 2 − γ v 2 ) 2 2 c −v x (c − v 2 )2 −vu y (2c2 − v 2 + γ v 2 − u 2 − γ u 2 )]. We now compute the factor

∂u λ− − λ−1 − ∂v λ − .

We use Bernoulli’s law to find

(c )u = −(γ − 1)u, (c )v = −(γ − 1)v. 2

(39)

2

(40)

We use the characteristic equation (c2 − u 2 )λ2 + 2uvλ + c2 − v 2 = 0

(41)

to obtain ∂u λ− =

λ2− (γ + 1)u − 2vλ− + (γ − 1)u , 2λ− (c2 − u 2 ) + 2uv

λ2 (γ − 1)v − 2uλ− + (γ + 1)v ∂v λ− = − . 2λ− (c2 − u 2 ) + 2uv

(42)

We then simply compute to find ∂u λ− − λ−1 − ∂v λ − =

2λ−

(c2

(uλ− − v)3 γ +1 . 2 − u ) + 2uv c 2 λ−

(43)

Coming back to our equation for ∂+ ∂− u, we have (c2 − u 2 )(c2 − v 2 )∂+ ∂− u = u 2x u(2c2 − u 2 − v 2 + γ u 2 − γ v 2 ) +u x u y (−vu 2 − v 3 + 3γ vu 2 − γ v 3 + Q) +u 2y [2u(c2 + γ v 2 ) + λ+ Q],

(44)

where we have introduced the notation Q :=

(γ + 1)(c2 − v 2 )(u H − vc)3 (c2 − u 2 )(c2 − v 2 ) γ + 1 3 , (uλ − v) = − 2λ− (c2 − u 2 ) + 2uv c2 λ− 2(c2 − u 2 )H (cH − uv)

where H :=

u 2 + v 2 − c2 .

(45)

(46)

We then factorize the quadratic form to find finally ∂+ ∂− u =

u(2c2 − u 2 − v 2 + γ u 2 − γ v 2 ) (∂x u + α∂ y u)∂− u, (c2 − u 2 )(c2 − v 2 )

(47)

where α=

2u(c2 + γ v 2 ) + λ+ Q . λ− u(2c2 − u 2 − v 2 + γ u 2 − γ v 2 )

(48)


7

Proposition 2. There holds the identity ∂+ ∂− u = m(∂x u + α∂ y u)∂− u

(49)

for α(u, v) given in (48) and m given by m=

u(2c2 − u 2 − v 2 + γ u 2 − γ v 2 ) . (c2 − u 2 )(c2 − v 2 )

(50)

We use the relation ∂− u = ∂− λ− /(∂u λ− − ∂v λ− λ−1 + )

(51)

to go back to ∂+ ∂− λ− . We find ∂u λ− − ∂v λ− λ−1 + =−

[4c2 + (γ − 3)(u 2 + v 2 )][vλ− (c2 + u 2 ) + (c2 − v 2 )u] =: G. (c2 − v 2 )(c2 − u 2 )[2λ− (c2 − u 2 ) + 2uv] (52)

So we have

∂+

1 ∂− λ − G

= m ∂α u

∂− λ − , G

(53)

or ∂+ ∂− λ− = (m∂α u + ∂+ (ln |G|))∂− λ− .

(54)

Proposition 3. There holds the identity ∂+ ∂− λ− = m∂− λ−

(55)

for some m = m(u, v)(∂x u + β(u, v)∂ y u). A similar identity holds for ∂− ∂+ λ+ . We remark that in the application on simple waves, the equation for u is sufficient and the equations for λ± are not needed. 4. Pseudo-Steady Euler We consider the two-dimensional isentropic irrotational ideal flow in the self-similar plane (ξ, η) = (x/t, y/t). Bernoulli’s law holds: U2 + V 2 c2 + = −ϕ, γ −1 2

(56)

where c is the speed of sound, (U, V ) = (u − ξ, v − η) are the pseudo-velocity, while (u, v) is the physical velocity, and ϕ is the pseudo-potential such that ϕξ = U,

ϕη = V.

The equations of motion can be written as 2 (c − U 2 )Uξ − U V (Uη + Vξ ) + (c2 − V 2 )Vη = −2c2 + U 2 + V 2 , Vξ − Uη = 0.

(57)

(58)

8


We can rewrite the equations of motion in a new form −2U V c2 −V 2 u u + c2 −U 2 c2 −U 2 =0 v ξ v η −1 0

(59)

to draw as much parallelism to the steady case as possible. We emphasize the mixed use of the variables (U, V ) and (u, v), i. e., (U, V ) is used in the coefficients while (u, v) is used in differentiation. This way we obtain zero on the right-hand side for the system. The eigenvalues are similar as before: dη U V ± c2 (U 2 + V 2 − c2 ) = ± = . (60) dξ U 2 − c2 The left eigenvectors are ± = [1, ∓ ].

(61)

∂± u + ∓ ∂± v = 0.

(62)

And we have similarly

Our ± now depend on more than (U, V ). But, let us regard ± as a simple function of three variables ± = ± (U, V, c2 ) as given in (60). Thus we need to build differentiation laws for c2 . We can directly obtain 2 2 c c + U u ξ + V vξ = 0, + U u η + V vη = 0. (63) γ −1 ξ γ −1 η We have ∂± c2 = −(γ − 1)(U ∂± u + V ∂± v).

(64)

So we move on to compute ∂± ± = ∂U ± ∂± U + ∂V ± ∂± V + ∂c2 ± ∂± c2 = ∂U ± (∂± u − 1) + ∂V ± (∂± v − ± ) + ∂c2 ± ∂± c2 = ∂U ± ∂± u + ∂V ± ∂± v + ∂c2 ± ∂± c2 − ∂U ± − ∂V ± ± .

(65)

We need to handle the term ∂U ± + ∂V ± ± . We show it is zero. Recalling that (c2 − U 2 ) 2 + 2U V + c2 − V 2 = 0,

(66)

and regarding that depends on the three quantities (U, V, c2 ) independently, we can easily find

U =

(U − V ) U − V , V = − .

(c2 − U 2 ) + U V

(c2 − U 2 ) + U V

(67)

Thus

U +

V = 0.

(68)


Therefore we end up with

−1 ∂± ± = ∂U ± − −1 ∂

− (γ − 1)∂

(U −

V ) ∂± u. 2 V ± ± c ∓ ∓

9

(69)

Thus, if one of the quantities (u, v, c2 , − ) is a constant along − , so are all the rest. So far the properties are very similar to the steady case. We derive an equation for ∂− u. We have a similar second-order equation for u, 2U V c2 − U 2 uη − 2 uξ . (70) u ηη = c2 − V 2 c − V2 ξ We have similarly ∂+ ∂− u = u ξ ξ + ( + + − )u ξ η + − + u ηη + ∂+ − u η

2 2U V c2 − V 2 c − U2 = 2 uη − 2 u ξ + ∂+ − u η . c − U2 c2 − V 2 ξ c − V2 ξ

(71)

We compute ∂+ − = ∂U − ∂+ U + ∂V − ∂+ V + ∂c2 − ∂+ c2 1 1 = ∂U − − ∂V − − (γ − 1)∂c2 − U − V ∂+ u

−

− −(∂U − + + ∂V − ).

(72)

We continue to find uη c2 − V 2 ∂+ ∂− u = 2 c − U 2 (c2 − V 2 )2

× 2Vξ U (c2 − V 2 ) + 2V Uξ (c2 − V 2 ) − 2V U ((c2 )ξ − 2V Vξ )

uξ 2 2 2 2 2 2 ((c ) − 2UU )(c − V ) − (c − U )((c ) − 2V V ) − 2 ξ ξ ξ ξ (c − V 2 )2 1 1 ∂V − − (γ − 1)∂c2 − (U − V) +∂+ u u η ∂U − −

−

− (73) −u η (∂U − + + ∂V − ). We apply the rule Uξ = u ξ − 1, Vξ = vξ = u η to find

uη c2 − V 2 2u η U (c2 − V 2 ) + 2V u ξ (c2 − V 2 ) ∂+ ∂ − u = 2 c − U 2 (c2 − V 2 )2 +2V U ((γ − 1)U u ξ + (γ + 1)V u η )

uξ − ((γ + 1)U u ξ + (γ − 1)V u η )(c2 − V 2 ) − 2 2 2 (c − V ) +(c2 − U 2 )((γ − 1)U u ξ + (γ + 1)V u η ) 1 1 +∂+ u u η ∂U − − ∂V − − (γ − 1)∂c2 − (U − V)

−

− −2V vξ − 2U u ξ − u η (∂U − + + ∂V − ). c2 − U 2

(74)

10


We note that terms appear which are linear in the derivatives of (u, v) in addition to the pure quadratic form as in the steady case. The pure quadratic form is identical to the steady case, so we do not need to handle it further. The linear form can be handled as follows. First we use the derivatives ( U , V ) to compute ∂U − + + ∂V − = ∂U − +

1 c2 − V 2 2(U − − V ) ∂V − = .

− c2 − U 2 c2 − U 2

(75)

Then we have −2V vξ − 2U u ξ 2U − u η (∂U − + + ∂V − ) = − 2 ∂− u. 2 2 c −U c − U2

(76)

Thus the linear form is also in the direction of − . Combining the steps we end up with Theorem 4. There holds ∂+ ∂− u =

U (2c2 − U 2 − V 2 + γ U 2 − γ V 2 ) 2U (∂ξ u + A∂η u)∂− u − 2 ∂− u, (c2 − U 2 )(c2 − V 2 ) c − U2 (77)

where A :=

2U (c2 + γ V 2 ) + + Q˜ ,

− U (2c2 − U 2 − V 2 + γ U 2 − γ V 2 )

(78)

and (c2 − U 2 )(c2 − V 2 ) γ + 1 (U − − V )3 . Q˜ := 2 − (c2 − U 2 ) + 2U V c2 −

(79)

We then have Theorem 5. There holds ∂+ (∂− − ) = m ∂− − .

(80)

Similarly ∂− (∂+ + ) = n ∂+ + holds. 5. Application: Simple Waves For a system of hyperbolic conservation laws in one-space dimension, a centered rarefaction wave is a simple wave, in which one family of characteristics are straight lines and the dependent variables are constant along a characteristic. See any text book on systems of conservation laws, e. g., Courant and Friedrichs [2], p.59, and others’ [9, 3]. Simple waves for the two-dimensional steady Euler system are similar, i. e., one family of characteristics are straight and the velocity are constant along the characteristics. For the two-dimensional self-similar pressure gradient system, see [4], simple waves can be defined similarly, i. e., one nonlinear family of characteristics are straight and the pressure is constant along them. We note that we do not require the velocity to be constant. This way, by the characteristic decomposition, we find that a wave adjacent to a constant state is a simple wave.


11

In the construction of solutions to the two-dimensional Riemann problem for the Euler system, see any of the sources [7–9], it is important to know how to construct solutions adjacent to a constant state in addition to the constructions of the interaction of rarefaction waves ([6]), subsonic solutions, and transonic shock waves. “Simple waves play a fundamental role in describing and building up solutions of flow problems.” ( See ˘ c and Keyfitz explored simple waves further and generalized pp.59–60, [2]). In [1], Cani´ Courant and Friedrichs’ theorem by allowing the coefficients of a 2 × 2 system (say (7)) to depend on the independent variables (t, x) linearly as well as the dependent variables (u, v). Here the characteristic decomposition ∂1 ∂2 λ2 = m∂2 λ2 allows us to conclude that Theorem 6. Adjacent to a constant state in the self-similar plane of the potential flow system is a simple wave in which the physical variables (u, v, c) are constant along a family of characteristics which are straight lines.

5.1. Simple waves for full Euler. Consider the adiabatic Euler system for an ideal fluid   ρt + ∇ · (ρu) = 0, (ρu)t + ∇ · (ρu ⊗ u + p I ) = 0, (81)  (ρ E) + ∇ · (ρ Eu + pu) = 0, t where E :=

1 2 |u| + e, 2

where e is the internal energy. For a polytropic gas, there holds e=

1 p , γ −1ρ

where γ > 1. In the self-similar plane and for smooth solutions, the system takes the form  1   ρ ∂s ρ + u ξ + vη = 0,      ∂s u + 1 pξ = 0,  ρ (82)   ∂s v + ρ1 pη = 0,       1 ∂s p + u ξ + vη = 0, γp where ∂s := (u − ξ )∂ξ + (v − η)∂η , which we call pseudo-flow directions, as opposed to the other two characteristic directions, called (pseudo-)wave characteristics. We easily derive ∂s ( pρ −γ ) = 0,

(83)

12


and

ωt + (uω)x + (vω) y +

py ρ

− x

px ρ

=0

(84)

y

for the vorticity ω := vx − u y . So entropy pρ −γ is constant along the pseudo-flow lines. For a region whose pseudo-flow lines come from a constant state, we see that the entropy is constant in the region. For the isentropic region, vorticity has zero source of p production since ( ρy )x − ( pρx ) y = 0. Thus vorticity tω = vξ − u η (setting t = 1 then) satisfies ∂s (ω/ρ) = 0.

(85)

Hence, for a region whose pseudo-flow lines come from a constant state, the vorticity must be zero everywhere. So the region is irrotational and isentropic. Thus our formulas for the potential flow apply. We have Theorem 7. Adjacent to a constant state in the self-similar plane of the adiabatic Euler system is a simple wave in which the physical variables (u, v, c, p, ρ) are constant along a family of wave characteristics which are straight lines, provided that the region is such that its pseudo-flow characteristics extend into the state of constant. Note added in proof: Lax has a concept of Riemann invariants for large systems, see “hyperbolic systems of conservation laws. II.” Comm. Pure Appl. Math. 10(1957), pp. 537–566. For the steady irrotational Euler system (9), a pair of Riemann invariants are pointed out to us by Marshall Slemrod as W± (θ, q) = θ ± qc(q 2 − c2 )−1/2 dq for u = q cos θ, u = q sin θ and c2 depends on q via Bernoulli’s law (10). Acknowledgements. Y. Zheng would like to thank the mathematics department at Capital Normal University for its hospitality during his visit when this work was done. J. Li thanks Matania Ben-Artzi for his interest.

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Communications in


The Cohomology Algebra of the Semi-Infinite Weil Complex Andrew R. Linshaw Department of Mathematics, Brandeis University, Waltham, MA 02454, USA. E-mail: [email protected] Received: 17 September 2005 / Accepted: 28 March 2006 Published online: 11 July 2006 – © Springer-Verlag 2006

Abstract: In 1993, Lian-Zuckerman constructed two cohomology operations on the BRST complex of a conformal vertex algebra with central charge 26. They gave explicit generators and relations for the cohomology algebra equipped with these operations in the case of the c = 1 model. In this paper, we describe another such example, namely, the semi-infinite Weil complex of the Virasoro algebra. The semi-infinite Weil complex of a tame Z-graded Lie algebra was defined in 1991 by Feigin-Frenkel, and they computed the linear structure of its cohomology in the case of the Virasoro algebra. We build on this result by giving an explicit generator for each non-zero cohomology class, and describing all algebraic relations in the sense of Lian-Zuckerman, among these generators. 1. Introduction The BRST cohomology of a conformal vertex algebra of central charge 26 is a special case of the semi-infinite cohomology of a tame Z-graded Lie algebra g (in this case the Virasoro algebra) with coefficients in a g-module M. The theory of semi-infinite cohomology was developed by Feigin and Frenkel-Garland-Zuckerman [3, 7], and is an analogue of classical Lie algebra cohomology. In general, there is an obstruction to the semi-infinite differential being square-zero which arises as a certain cohomology class in H 2 (g, C). The semi-infinite Weil complex of g is obtained by taking M to be the module of “adjoint semi-infinite symmetric powers” of g [4, 1]. In this case, an anomaly cancellation ensures that the differential is always square-zero. The semi-infinite Weil complex is a vertex algebra and its differential arises as the zeroth Fourier mode of a vertex operator, a fact which is useful for doing computations. This paper is organized as follows. First, we define vertex algebras and their modules, which have been discussed from various different points of view in the literature [2, 6, 8, 9, 15, 10, 13, 16]. We will follow the formalism introduced in [13]. We describe the main examples we need, and then define the BRST complex of a conformal vertex algebra

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A. R. Linshaw

A with central charge 26. We then recall the two cohomology operations introduced in [12] on the BRST cohomology H ∗ (A), namely, the dot product and the bracket. We examine in detail the case where A = S, i.e., the βγ -ghost system associated to a one-dimensional vector space. This coincides with the module of adjoint semi-infinite symmetric powers of the Virasoro algebra, so the BRST complex of S is exactly the semi-infinite Weil complex of the Virasoro algebra. Finally, we prove our main result, which is a complete description of the algebraic structure of H ∗ (S) in the sense of Lian-Zuckerman. Theorem 1.1. Let V ir+ denote the Lie subalgebra of the Virasoro algebra generated by L n , n ≥ 0. As a Lie superalgebra with respect to the bracket, H ∗ (S) is isomorphic to the semi-direct product of V ir+ with its adjoint module. As an associative algebra with respect to the dot product, H ∗ (S) is a polynomial algebra on one even variable and one odd variable. 2. Vertex Algebras Let V = V0 ⊕ V1 be a super vector space over C, and let z, w be formal variables. By Q O(V ), we mean the space of all linear maps V → V ((z)) =

v(n)z −n−1 |v(n) ∈ V ; v(n) = 0 f or n >> 0 .

(2.1)

n∈Z

Each a ∈ Q O(V ) can be uniquely represented as a power series a(z) = element −n−1 ∈ (End V )[[z, z −1 ]], although the latter space is clearly much larger n∈Z a(n)z than Q O(V ). We refer to a(n) as the n th Fourier mode of a(z). Each a ∈ Q O(V ) is assumed to be of the shape a = a0 + a1 , where ai : V j → Vi+ j ((z)) for i, j ∈ Z/2, and we write |ai | = i. On Q O(V ) there is a set of non-associative bilinear operations, ◦n , indexed by n ∈ Z, which we call the n th circle products. They are defined by a(w) ◦n b(w) = Resz a(z)b(w) ι|z|>|w| (z −w)n −(−1)|a||b| b(w)a(z)ι|w|>|z| (z −w)n . Here ι|z|>|w| f (z, w) ∈ C[[z, z −1 , w, w −1 ]] denotes the power series expansion of a rational function f in the region |z| > |w|. Note that ι|z|>|w| (z − w)n = ι|w|>|z| (z − w)n for n < 0. We usually omit the symbol ι|z|>|w| and just write (z − w)n to mean the expansion in the region |z| > |w|, and write (−1)n (w − z)n to mean the expansion in |w| > |z|. It is easy to check that a(w) ◦n b(w) above is a well-defined element of Q O(V ). The non-negative circle products are connected through the operator product expansion (OPE) formula ([13], Prop. 2.3). For a, b ∈ Q O(V ), we have a(z)b(w) =

a(w) ◦n b(w) (z − w)−n−1 + : a(z)b(w) :

n≥0

as formal power series in z, w. Here : a(z)b(w) : = a(z)− b(w) + (−1)|a||b| b(w)a(z)+ ,

(2.2)

Cohomology Algebra of the Semi-Infinite Weil Complex

15

−n−1 and a(z) = −n−1 . Equation (2.2) is where a(z)− = + n 0, and let C be the one-dimensional b-module on which (x, 0)t n and (0, x )t m act trivially and the central element τ acts by the identity. Denote the linear operators representing (x, 0)t n , (0, x )t n on Uh ⊗Ub C by β x (n), γ x (n − 1), respectively, for n ∈ Z. The power series β x (z) = β x (n)z −n−1 , γ x (z) = γ x (n)z −n−1 ∈ Q O(Uh ⊗Ub C) n∈Z

n∈Z

generate a vertex algebra S(V ) inside Q O(Uh ⊗Ub C), and the generators satisfy the OPE relations

β x (z)γ x (w) ∼ x , x(z − w)−1 , β x (z)β y (w) ∼ 0, γ x (z)γ y (w) ∼ 0. This algebra was introduced in [9] and is known as a βγ -ghost system, or a semi-infinite symmetric algebra. The creation map χ : S(V ) → Uh ⊗Ub C, which sends a(z) → a(−1)(1 ⊗ 1), is easily seen to be a linear isomorphism. By the Poincaré-Birkhoff-Witt theorem, the vector space Uh ⊗Ub C has the structure of a polynomial algebra with generators given by the negative Fourier modes β x (n), γ x (n), n < 0, which are linear in x ∈ V and x ∈ V ∗ . It follows from (2.4) that S(V ) is spanned by the collection of iterated Wick products of the form

µ = : ∂ n 1 β x1 · · · ∂ n s β xs ∂ m 1 γ x1 · · · ∂ m t γ xt : . S(V ) has a natural Z-grading which we call the βγ -ghost number. Fix a basis x1 , . . . , xn for V and a corresponding dual basis x1 , . . . , xn for V ∗ . Define the βγ -ghost number to be the eigenvalue of the diagonalizable operator [B, −], where B is the zeroth Fourier mode of the vertex operator n i=1

: β xi γ xi : .

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Clearly B is independent of our chosen basis of V , and β x , γ x have βγ -ghost numbers −1, 1 respectively. We can also regard V ⊕ V ∗ as an odd abelian Lie (super) algebra, and consider its loop algebra and a one-dimensional central extension by Cτ with bracket [(x, x )t n , (y, y )t m ] = (y , x + x , y)δn+m,0 τ. Call this Lie algebra j = j(V ), and form the induced module Uj ⊗Ua C. Here a is the subalgebra of j generated by τ , (x, 0)t n , and (0, x )t m , for n ≥ 0 and m > 0, and C is the one-dimensional a-module on which (x, 0)t n and (0, x )t m act trivially and τ acts by 1. There is a vertex algebra E(V ), analogous to S(V ), which is generated by the odd vertex operators b x (n)z −n−1 , c x (z) = c x (n)z −n−1 ∈ Q O(Uj ⊗Ua C), b x (z) = n∈Z

n∈Z

which satisfy the OPE relations

b x (z)c x (w) ∼ x , x(z − w)−1 , b x (z)b y (w) ∼ 0, c x (z)c y (w) ∼ 0. This vertex algebra is known as a bc-ghost system, or a semi-infinite exterior algebra. Again the creation map E(V ) → Uj ⊗Ua C, a(z) → a(−1)(1 ⊗ 1), is a linear isomorphism. As in the symmetric case, the vector space Uj ⊗Ua C has the structure of an odd polynomial algebra with generators given by the negative Fourier modes b x (n), c x (n), n < 0, which are linear in x ∈ V and x ∈ V ∗ . As above, it follows that E(V ) is spanned by the collection of all iterated Wick products of the vertex operators ∂ k b x and ∂ k c x , for k ≥ 0. E(V ) has a Z-grading which we call the bc-ghost number (or fermion number). It is given by the eigenvalue of the diagonalizable operator [F, −], where F is the zeroth Fourier mode of the vertex operator −

n

: b xi c xi : .

i=1

F is independent of our choice of basis for V , and b x , c x have bc-ghost numbers −1, 1 respectively. We will denote the bc-ghost number of a homogeneous element a ∈ E(V ) by |a|. Note that this coincides with our earlier notation for the Z/2-grading on E(V ) coming from its vertex superalgebra structure. This causes no difficulty; since b x , c x are odd vertex operators, the mod 2 reduction of the bc-ghost number coincides with this Z/2-grading. Let us specialize to the case where V is a one-dimensional vector space. In this case, S(V ) coincides with the module of adjoint semi-infinite symmetric powers of the Virasoro algebra [4]. Fix a basis element x of V and a dual basis element x of V ∗ . We denote S(V ) by S, and we denote the generators β x , γ x by β, γ , respectively. Similarly, we denote E(V ) by E, and we denote the generators b x , c x by b, c, respectively. For a fixed scalar λ ∈ C, define LS λ = (λ − 1) : ∂βγ : +λ : β∂γ : ∈ S. An OPE calculation shows that −2 + ∂β(w)(z − w)−1 , LS λ (z)β(w) ∼ λβ(w)(z − w)

(2.7)


19

−2 LS + ∂γ (w)(z − w)−1 , λ (z)γ (w) ∼ (1 − λ)γ (w)(z − w)

S LS λ (z)L λ (w) ∼

k −2 −1 (z − w)−4 + 2L S + ∂ LS λ (w)(z − w) λ (w)(z − w) , 2

where k = 12λ2 − 12λ + 2. Hence L S λ is a Virasoro element of central charge k, and ) is a conformal vertex algebra in which β, γ have conformal weights λ, 1 − λ (S, L S λ respectively. Similarly, define L Eλ = (1 − λ) : ∂bc : −λ : b∂c : ∈ E.

(2.8)

A calculation shows that L Eλ is a Virasoro element with central charge k = −12λ2 + 12λ − 2, (E, L Eλ ) is a conformal vertex algebra, and b, c have conformal weights λ, 1 − λ respectively. 3. BRST Cohomology

Observe that if A, A are conformal vertex algebras with Virasoro elements L A , L A of central charges k, k , respectively, then A ⊗ A is a conformal vertex algebra with Virasoro element L A⊗A = L A + L A (i.e., L A ⊗ 1 + 1 ⊗ L A ) of central charge k + k . To simplify notation, the ordered product ab = a(z)b(z) of two vertex operators a, b in the same formal variable z will always denote the Wick product. Fix λ = 2 in (2.8), and denote the corresponding Virasoro element L E2 = −∂bc − 2b∂c ∈ E, by L E . With this choice, (E, L E ) is a conformal vertex algebra of central charge -26. For any conformal vertex algebra (A, L A ) of central charge k, let C ∗ (A) = E ⊗ A. Denote the Virasoro element L E + L A , by L C . The conformal weight and bcghost number are given, respectively, by the eigenvalues of the operators [L C 0 , −] and [F ⊗ 1, −] on C ∗ (A). Definition 3.1. Let JA be the following element of C ∗ (A): 1 3 JA = (L A + L E )c + ∂ 2 c. 2 4 A calculation shows that JA (z) ◦0 b(z) = L C (z).

(3.1)

(3.2)

We will denote the zeroth Fourier mode JA (0) by Q, so we may rewrite this equation as [Q, b(z)] = L C (z) by (2.3). Note that the operator [Q, −] preserves conformal weight and raises bc-ghost number by 1. Lemma 3.2. Q 2 = 0 iff k = 26. In this case, we can consider C ∗ (A) to be a cochain complex graded by bc-ghost number, with differential [Q, −]. Its cohomology is called the BRST cohomology associated to A, and will be denoted by H ∗ (A). Proof. First, note that Q 2 = 21 [Q, Q] = 21 Resw JA (w)◦0 JA (w). Computing the OPE of JA (z)JA (w) and extracting the coefficient of (z −w)−1 , we find that JA (w)◦0 JA (w) = 3 k−26 3 2 2 ∂(∂ c(w)c(w)) + 12 ∂ c(w)c(w). Since the residue of a total derivative is zero, only the second term contributes, and it follows that Resw JA (w)◦0 JA (w) = 0 iff k = 26.

From now on, we will only consider the case where k = 26.

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4. Algebraic Structure of H ∗(A) In this section, we recall without proof some facts from [12] on the algebraic structure of the BRST cohomology. We first note that any cohomology class can be represented by an element u(z) of conformal weight 0, since (3.2) implies that [Q, b(1)] = L C 0 . Since [Q, −] acts by derivation on each of the products ◦n on C ∗ (A), each ◦n descends to a product H ∗ (A). Since ◦n lowers conformal weight by n + 1, all these products are trivial except for the one induced by ◦−1 (the Wick product), which we call the dot product. We write the dot product of u and v as uv. The cohomology H ∗ (A) has another bilinear operation known as the bracket. First, we define the bracket on the space C ∗ (A). Definition 4.1. Given u(z), v(z) ∈ C ∗ (A), let {u(z), v(z)} = (−1)|u| (b(z) ◦0 u(z)) ◦0 v(z).

(4.1)

The equivalence between this definition and the one given in [12] is shown in [13]. From this description, it is easy to see that the bracket descends to H ∗ (A), inducing a well-defined bilinear operation. Theorem 4.2. The following algebraic identities hold on H ∗ (A): uv = (−1)|u||v| vu,

(4.2)

(uv)t = u(vt),

(4.3)

{u, v} = −(−1)(|u|−1)(|v|−1) {v, u},

(4.4)

(−1)(|u|−1)(|t|−1) {u, {v, t}} + (−1)(|t|−1)(|v|−1) {t, {u, v}} +(−1)(|v|−1)(|u|−1) {v, {t, u}} = 0,

(4.5)

{u, vt} = {u, v}t + (−1)(|u|−1)(|v|) v{u, t},

(4.6)

b ◦1 {u, v} = {b ◦1 u, v} + (−1)|u|−1 {u, b ◦1 v},

(4.7)

{, } : H p × H q → H p+q−1 .

(4.8)

Equations (4.2)–(4.3) say that H ∗ (A) is an associative, graded commutative algebra with respect to the dot product. Equations (4.4)–(4.5) say that under the bracket, H ∗ (A) is a Lie superalgebra with respect to the grading (bc-ghost number –1). Also, note that H 1 (A) is an ordinary Lie algebra under the bracket. Taking p = 1 in (4.8), we see that for every q, H q (A) is a module over H 1 (A).


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5. The Semi-infinite Weil Complex of the Virasoro Algebra In this section, we give a complete description of the algebraic structure of H ∗ (A) in the case A = S, with the choice λ = 2 in (2.7). In this case, the Virasoro element LS = LS 2 = ∂βγ + 2β∂γ ∈ S has central charge 26. Definition 5.1. C ∗ (S) = E ⊗ S, equipped with the differential [Q, −] and the Virasoro element L W = L E + L S , is called the semi-infinite Weil complex associated to V ir , and will be denoted by W. Note that W is naturally triply graded. In addition to the conformal weight and bc-ghost number, W is graded by the βγ -ghost number, which is the eigenvalue of [1 ⊗ B, −]. Note that [Q, −] preserves the βγ -ghost number. Let W i, j ⊆ W denote the conformal weight zero subspace of bc-ghost number i, βγ -ghost number j, and let Z i, j , B i, j ⊆ W i, j denote the cocycles and coboundaries, respectively, with respect to [Q, −]. Let H i, j = Z i, j /B i, j . Note that H i (S) decomposes as the direct sum j∈Z H i, j . In [4] and [5], Feigin-Frenkel computed the linear structure of H ∗ (S), namely, the dimension of each of the spaces H i, j . Theorem 5.2. For all j ≥ 0, dim H 0, j = dim H 1, j = 1. For all other values of i, j, dim H i, j = 0. This was proved by using the Friedan-Martinec-Shenker bosonization [9] to express S as a submodule of a direct sum of Feigin-Fuchs modules over V ir , and then using known results on the structure of these modules. We will assume the results in [4, 5], and use them to describe the algebraic structure of H ∗ (S). Our first step is to find a canonical generator for each non-zero cohomology class. Recall that W has a basis consisting of the monomials: ∂ n 1 b · · · ∂ n i b ∂ m 1 c · · · ∂ m j c ∂ s1 β · · · ∂ sk β ∂ t1 γ · · · ∂ tl γ

(5.1)

with n 1 > · · · > n i ≥ 0, m 1 > · · · > m j ≥ 0 and s1 ≥ · · · ≥ sk ≥ 0, t1 ≥ · · · ≥ tl ≥ 0. Let D ⊂ W be the subspace spanned by monomials which contain at least one derivative, i.e., at least one of the numbers n 1 , ..., n i , m 1 , ..., m j , s1 , ..., sk , t1 , ..., tl above is positive. Lemma 5.3. The image of [Q, −] is contained in D. Proof. A straightforward calculation shows that [Q, b] = L W , [Q, c] = c∂c, [Q, β] = c∂β + 2∂cβ, [Q, γ ] = c∂γ − ∂cγ . The claim follows by applying the graded derivation [Q, −] to a monomial of the form (5.1), and then using (2.5) to express the result as a linear combination of standard monomials of the form (5.1). Note that for any vertex operators a, b, c ∈ W, the expression : (ab :)c : − : abc : , which measures the non-associativity of the Wick product, always lies in D.

Lemma 5.4. Let x = βγ 2 − bcγ + 23 ∂γ . Then x ∈ Z 0,1 and x ∈ / B 0,1 , so x represents a non-zero cohomology class. Since H 0,1 is 1-dimensional, x generates H 0,1 . Similarly, let y = cβγ + 23 ∂c. Then y ∈ Z 1,0 and y ∈ / B 1,0 , so y generates H 1,0 .

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Proof. The proof that x ∈ Z 0,1 and y ∈ Z 1,0 is a straightforward calculation. Since x contains a monomial with no derivatives, x ∈ / D. By Lemma 5.3, B 0,1 ⊂ D, so x ∈ / B 0,1 . 1,0 Similarly, y ∈ / B .

Our main result is the following Theorem 5.5. For each integer k ≥ 0, x k represents a non-zero cohomology class in H 0,k , and yx k represents a non-zero class in H 1,k . By Theorem 5.2, these are all the non-zero classes in H ∗ (S). Proof. It is clear from the derivation property of [Q, −] that x k ∈ Z 0,k and yx k ∈ Z 1,k , so it suffices to show that x k ∈ / B 0,k and yx k ∈ / B 1,k . For each integer k ≥ 0, define: xk = β k γ 2k − kbcβ k−1 γ 2k−1 ,

yk = cβ k+1 γ 2k+1 .

We claim that: x k = x k + Dk ,

(5.2)

yx k = yk + Dk ,

(5.3)

for some Dk ∈ D and Dk ∈ D. Since xk and yk have no derivatives, it follows that x k and yx k do not lie in D. Now we can apply Lemma 5.3 to conclude that x k ∈ / B 0,k and yx k ∈ / B 1,k . We begin with (5.2) and proceed by induction. The cases k = 0 and k = 1 are obvious, so assume the statement true for k − 1, 3 x k = (βγ 2 − bcγ + ∂γ )x k−1 2

(5.4)

= (βγ 2 − bcγ )(β k−1 γ 2k−2 − (k − 1)bcβ k−2 γ 2k−3 + Dk−1 ) + E 0 , where E 0 = 23 ∂γ x k−1 . We expand this product and apply Lemma 2.4 repeatedly: (bcγ )((k − 1)bcβ k−2 γ 2k−3 ) = (k − 1)b2 c2 β k−2 γ 2k−2 + E 1 = E 1 ,

(5.5)

since b, c are anti-commuting variables, (bcγ )(β k−1 γ 2k−2 ) = bcβ k−1 γ 2k−1 + E 2 ,

(5.6)

(βγ 2 )((k − 1)bcβ k−2 γ 2k−3 ) = (k − 1)bcβ k−1 γ 2k−1 + E 3 ,

(5.7)

(βγ 2 )(β k−1 γ 2k−2 ) = β k γ 2k + E 4 ,

(5.8)

where E i ∈ D for i = 0, 1, 2, 3, 4. It is easy to see that (bcγ )Dk−1 ∈ D and (βγ 2 )Dk−1 ∈ D. Equation (5.2) follows by collecting terms from (5.5)–(5.8). Finally, the same argument proves (5.3).

Using Theorem 5.5, we can now describe the algebraic structure of H ∗ (S). Let V ir+ ⊆ V ir be the Lie subalgebra generated by L n , n ≥ 0. The Cartan subalgebra h of V ir+ is generated by L 0 .


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Corollary 5.6. As an associative algebra with respect to the dot product, H ∗ (S) is a polynomial algebra in one even variable, x, and one odd variable, y. In other words, H ∗ (S) is isomorphic to the classical Weil algebra associated to h. It is easy to check from the definition of the bracket (4.1) that {y, x} = −x. Using the graded derivation property of the bracket with respect to the dot product, we can write down all the bracket relations in H ∗ (S). For any n, m ≥ 0, {x n , x m }=0, {yx n , x m }=(n − m)x n+m , {yx n , yx m }=(n − m)yx n+m .

(5.9)

It follows that as a Lie algebra, H 1,∗ is isomorphic to V ir+ under the isomorphism yx k → L k , k ≥ 0. As an H 1,∗ -module, H 0,∗ is isomorphic to the adjoint representation of V ir+ . Finally, we obtain Corollary 5.7. As a Lie superalgebra with respect to the bracket, H ∗ (S) is isomorphic to the semi-direct product of V ir+ with its adjoint module. Acknowledgement. I would like to thank Bong H. Lian for many helpful conversations we have had during the course of this work.

References 1. Akman, F.: A characterization of the differential in semi-infinite cohomology. J. Alg. 162(1), 194–209 (1993) 2. Borcherds, R.: Vertex operator algebras, Kac-Moody algebras and the Monster. Proc. Nat. Acad. Sci. USA 83, 3068–3071 (1986) 3. Feigin, B.: The semi-infinite homology of Lie, Kac-Moody and Virasoro algebras. Russ. Math. Surv. 39(2), 195–196 (1984) 4. Feigin, B., Frenkel, E.: Semi-Infinite Weil complex and the Virasoro algebra. Commun. Math. Phys. 137, 617–639 (1991) 5. Feigin, B., Frenkel, E.: Determinant formula for the free field representations of the Virasoro and KacMoody algebras. Phys. Lett. B 286, 71–77 (1992) 6. Frenkel, I.B., Huang, Y.Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104(494), viii+64 (1993) 7. Frenkel, I.B., Garland, H., Zuckerman, G.J.: Semi-infinite cohomology and string theory. Proc. Natl. Acad. Sci. USA 83(22), 8442–8446 (1986) 8. Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. New York: Academic Press, 1988 9. Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B271, 93–165 (1986) 10. Li, H.: Local systems of vertex operators, vertex superalgebras and modules. http://arxiv.org/list// hep-th/9406185, 1994 11. Lian, B., Linshaw, A.: Chiral equivariant cohomology I. http://arxiv.org/list// math.DG/0501084, 2005, to appear in Adv. Math. 12. Lian, B., Zuckerman, G.J.: New perspectives on the BRST-algebraic structure of string theory. Commun. Math. Phys. 154, 613–646 (1993) 13. Lian, B., Zuckerman, G.J.: Commutative quantum operator algebras. J. Pure Appl. Alg. 100(1–3), 117– 139 (1995) 14. Lian, B.: Lecture notes on circle algebras. Preprint, 1994 15. Kac, V. G.: Vertex Algebras for Beginners. AMS Univ. Lecture Series, 10, 2nd corrected ed., Prividence, RI: Amer. Math. Soc., 2001 16. Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177– 254 (1989) Communicated by L. Takhtajan


Communications in


Quasi-Linear Quantum Field Theories for Maps to Groups and Their Quotients Clifford H. Taubes Department of Mathematics, Harvard University, Cambridge, MA 02138, USA. E-mail: [email protected] Received: 2 October 2005 / Accepted: 14 February 2006 Published online: 26 July 2006 – © Springer-Verlag 2006

Abstract: I describe a functional integral for maps from R × Rn to a Lie group or its quotient which has a simple renormalization that leads to a quantum field theory for maps from Rn into the Lie group or its quotient whose Hamiltonian is the time translation generator for a unitary action of the n+1 dimensional Poincaré group on the quantum Hilbert space. I also explain how the renormalization provides a functional integral for maps from a Riemann surface to a compact Lie group or its quotient that exhibits many conformal field theoretic properties. Contents 1. 2. 3. 4. 5. 6. 7. 8.

The Construction . . . . . . . . . . . . Renormalization . . . . . . . . . . . . Quantum field theories . . . . . . . . . The action of the Poincaré group . . . Field theories for quotient spaces . . . When the domain is a Riemann surface Remarks on conformal field theories . Free field theories . . . . . . . . . . .

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Introduction I described in a previous paper [T] a construction of a measure on spaces of maps from a topological space to a smooth manifold and my purpose here is to explore variants of the construction from [T] in the case when the range space is a compact Lie group or its quotient by a compact subgroup. In particular, I describe how a simple renormalization of [T] ’s measure when the domain is R × Rn leads to a quantum field theory of maps Support in part by a grant from the National Science Foundation

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to the Lie group or its quotient whose Hamiltonian is the time translation generator for a unitary action of the n + 1 dimensional Poincaré group on the quantum Hilbert space. I also explain how the renormalization provides a functional integral for maps from a Riemann surface to a compact Lie group or its quotient that exhibits many conformal field theoretic properties. The second to last section here describes what is lacking for a complete conformal field theory. The final subsection describes a Gaussian field theory for maps to a compact Lie group or its quotient. 1. The Construction This section constitutes a digression of sorts to summarize the measure that is constructed in [T]. The summary begins by setting notation. To this end, let M denote the domain space, and let a : M × M → [0, ∞) denote a continuous function that defines a non-negative definite kernel in the following sense: Fix any positive integer N ; and choose any N distinct points {z 1 , . . . , z N } and N real numbers {η1 , . . . , η N }. Then a(z i , z j )ηi η j ≥ 0. (1.1) 1≤i≤ j≤N

Let X denote the range space, a smooth manifold with a given Riemannian metric. In the cases under consideration, X is assumed to be both compact and oriented. I use d to denote the dimension of X . Let π : P → X denote a compact, fiber bundle with the following additional data: • A set, {∂1 , . . . , ∂d }, of d vector fields that generate T P/kernel(π∗ ) at each point. • A volume form, dp, with total mass 1 and such that each vector field from this set has zero divergence. (1.2) Note that the symbol dp is also used in what follows to indicate the product volume form on products of P. A bundle such as P always exists; as a last resort, one can take P to be the bundle of oriented, orthonormal frames over X . Suppose next that a positive integer N has been chosen. In what follows, δ N is used to denote the Dirac delta function on × N P with support along the full diagonal. To be precise, δ N dp is the measure on × N P that sends a continuous function to Fδ N dp ≡ F( p, . . . , p)dp. (1.3) ×N P

P

Here and below, measures, even those singular with respect to Lebesgue measure, are written as if they were honest functions times the volume form. Thus, the volume form is always present in the notation for integration. To continue with the notation, for each i ∈ {1, . . . , N } and a ∈ {1, . . . , d}, the symbol ∂ai denotes the vector field on × N P that differentiates according to the basis vector ∂a along the i th factor of P. Suppose now that z = (z 1 , . . . , z N ) ∈ × N M is a chosen point. This point defines the differential operator j Az ≡ a(z i , z j ) ∂ai ∂a (1.4) 1≤i, j≤N

on

C ∞ (×

N P).

1≤a≤d

Quasi-Linear Quantum Field Theories for Maps to Groups and Their Quotients

27

By virtue of (1.1) and (1.2), this operator is negative semi-definite and symmetric. As a consequence, there exists a measure valued solution to the heat equation on × N P that is characterized as follows: This solution, K z , defines a continuous map from [0, ∞) to the space of Borel measures on × N P whose value at 0 is the measure δ N . Moreover, K z is such that when F is twice differentiable, then the pairing F, K z is differentiable on (0, ∞) where it obeys d F K z |s dp = (A z F)K z |s dp. (1.5) ds × N P ×N P A theorem of Hörmander [H] guarantees that K z is smooth for s > 0 if the bilinear form in (1.1) is non-degenerate, and if the set of higher order Lie brackets of the vector fields in the set {∂a } span T P at each point. In general, • K z ≥ 0. • P K (z 1, z 2 ,...,z N ) ( p1 , . . . , p N −1 , p)dp = K (z 1 ,z 2 ,...,z N ) ( p1 , . . . , p N −1 ). • Let N and N be positive integers with N ≤ N . If the final N − N + 1 entries

of z ∈ × N M are identical, then K z ( p1, . . . , p N ) = δ N −N +1 K (z 1, ...,z N −1 ,z N ) ( p1 , . . . , p N ). • Let σ simultaneously denote an element in the group of permutations of {1, . . . , N } and the action of this element on both × N M and × N P. Then K z = σ ∗ (K σ (z) ). (1.6) Let P M denote the space of all maps (continuous or not) from M to P. The collection of all such K z can be used to define a measure on P M as follows: The measure in question is defined on the σ -algebra that is generated by ‘cylinder’ sets that are jointly ladled by a positive integer, N , together with a collection of N pairs {(z i , Ui )}1≤i≤N such that z = (z 1 , . . . , z N ) ∈ × N M has distinct entries and each U j is an open subset of P. The set labelled by the data (N , {(z j , U j )}) consists of the maps that send each z j to its partnered set U j . The measure of this set is deemed equal to K z |s=1 dp. (1.7) ×1≤ j≤N U j

A theorem of Kolmogorov (see Theorem 1.10 in [SV]) guarantees that the just asserted rules define a bonafide measure on P M . A push-forward measure is induced on X M from the measure just described on P M . This push-forward measure is defined by its values on certain cylinder sets of maps from M to X . Such a set is jointly labelled by a positive integer, N , together with a collection of N pairs {(z j , V j )}1≤ j≤N , where z is as above and where V j ⊂ X is an open set. The set with this label consists of those φ ∈ X M such that φ(z j ) ∈ V j for all 1 ≤ j ≤ N . The measure of this set is given by the version of (1.7), where each U j is taken to be the inverse image in P of the corresponding V j . 2. Renormalization The task here is that of salvaging something from the preceding construction in the case where M is a smooth manifold, but where the function a that appears in (1.1) diverges towards ∞ as the distance between its arguments limits to zero. In this case, the operator

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C. H. Taubes

that appears in (1.4) has no meaning. A strategy is described below for proceeding in certain instances. Note in this regard that the quantum field theory story that follows uses a divergent version of the function a. To motivate the focus in the quantum field theory on divergent versions of a, first recall from [T] that the measure in the case where M = R × Y is ‘reflection positive’ when the function a in (1.1) is the Green’s function for a differential operator of the form 2 d − + L ; (2.1) dt 2 here L is a non-positive, self-adjoint elliptic differential operator on Y whose order is greater than twice the dimension of Y . This lower bound on the degree of L guarantees the continuity of the Green’s function on the whole of M × M. The reflection positivity property is used in [T] to construct a Hamiltonian quantum field theory using ideas of Osterwalder-Schrader [OS]. These quantum field theories are of little interest to physics in part because L is not a second order operator. For example, a reasonable action of the (n + 1) dimensional Poincaré group on the quantum Hilbert space in the case Y = Rn would seem to require a second order version of L . Meanwhile, a second order version of L has a Green’s function that is singular on the diagonal when dim(Y ) ≥ 1. What follows describes a strategy for dealing with a singular version of the function a that appears in (1.1). To start, fix a bounded, continuous function, c : M → R. This done, define ac (z, z ) to equal a(z, z ) when z = z and to equal c(z) when z = z . Thus, ac is continuous on the complement of the diagonal in M × M, but not continuous on the whole of M × M. Let R ≡ (r1 , . . . , r N ) ∈ R N and replace A z in (1.4) with     j j j ac (z i , z j ) A z,R ≡ rj  ∂a ∂a  + ∂ai ∂a  . (2.2) 1≤ j≤N

1≤a≤d

1≤i, j≤N

1≤a≤d

The latter operator is negative semi-definite if each r j is sufficiently positive; lower bounds are determined by the chosen point z = (z 1 , . . . , z N ). In any event, if each r j is sufficiently positive, then there is an analog, K z,R , to the measure K z that appears in (1.5) and (1.6). Note that K z,R depends implicitly on the chosen function c for the values of ac on the diagonal in M × M . Of course, one can always take c ≡ 0, but as is illustrated below, there may be better choices. A physicist might view the choice of c as the choice of a ‘normalization scale’. The plan now is to look for a suite of operators,

L N ,R : N ∈ {1, . . .} and R ≡ (r1 , . . . , r N ) ∈ R N , (2.3) where any given L N ,R provides a linear map from a subspace in C 0 (× N P) to C 0 (× N P). This suite of operators should have the following properties: • For fixed N , all {L N ,R } are defined on the same dense domain in C 0 (× N P) and the latter should include the constant function 1. • Suppose that F ∈ C 0 (× N P) is a function from the common domain of {L N ,R }. If z ∈ × N M has distinct entries, and if R is such that K z,R is well defined, then (L N ,R F)K z,R |s=1 dp ×N P

is independent of R when all r j are sufficiently large.


29

• If N > 1 and if F is independent of p N , and if each r j is sufficiently large, then L N ,R F is also independent of p N and its value at any given ( p1 , . . . , p N −1 ) is that of L N −1,R F , where F ( p1 , . . . , p N −1 , ·) and R ≡ (r1 , . . . r N −1 ). (2.4) C 0 (×

In what follows, T N ⊂ N P) denotes the common domain of the set {L N ,R }. A collection of the desired sort of operators is exhibited below in the case where P is a compact Lie group. It is not likely that a collection {L N ,R } as just described exists in the generic case, even with the second point replaced by the weakened version that demands only the existence of a reasonable limit in (2.4) as one or more of the r j tend to infinity. In any event, assume in what follows that {L N ,R } does exist as described in (2.4). Fix an integer N > 1 and suppose that F is a function on × N P. A point z ∈ × N M and F together define a function on the space of maps from M to P by assigning to a map φ the value of F at (φ(z 1 ), . . . , φ(z N )). The latter function is denoted by (z,F) . Introduce F to denote the vector space over C whose elements are finite linear combinations of the constant function and functions such as (z,F) with z having distinct entries and with F taken from the domain T N . Distinct versions of (·) that appear in any given from F need not use the same integer N . Associate to each (z,F) the number

(z,F) ≡ (L N ,R F)K z,R |s=1 dp. (2.5) lim r1 ,...,r N →∞ × P N

Use these assignments to F’s given as the basis for the definition of a linear functional on F. The value of the latter on any given ∈ F is denoted by in what follows. Keep in mind that there is an implicit dependence on the chosen function c : M → R. A measure on a space defines, by integration, a linear functional on some subset of the continuous functions. Conversely, certain sorts of linear functions are guaranteed to arise from measures (see, e.g. Chapter 14 in [R]). Even so, no claim is made here that the linear function · comes from a measure on the set of maps from M to P. The functional itself serves the purposes at hand. The case of interest in this article, and the case assumed without further notice in all that follows takes P to be a compact Lie group. Meanwhile, {∂a } is taken to be an orthonormal basis of left invariant vector fields on P for a fixed, bi-invariant metric; and the integration on P is defined with the volume form from this same metric. Here is a collection {L N ,R } to use in this Lie group case: Let T N denote the vector space of finite linear combinations of functions that send p = ( p1 , . . . , p N ) to f 1 ( p1 ) . . . f N ( p N ), where each f j is an eigenfunction of the Laplacian on P , this being the operator 1≤a≤d ∂a ∂a . For F ∈ T N , set    L N ,R F ≡ exp − (2.6) rj  ∂a ∂a  F. 1≤i≤N

1≤a≤n

Note that the exponentiated operator that appears on the right-hand side of ( 2.6) has the ‘wrong’ sign; it is the kernel operator to the backwards heat equation on × N P. As such, it can be defined only on a rather small domain in C 0 (× N P). However, it is defined for any choice of {R j } on the domain T N . The fact that the operator 1≤a≤d ∂a ∂a on a

30

C. H. Taubes

compact Lie group commutes with each version of ∂a guarantees that the conditions in (2.4) are obeyed. This same fact about commutators implies a simple relationship between the respective versions of · that arise from two different choices for the function c on M. To state the relation, suppose that the function a on M × M has been specified, and then use · c to denote the version of · that is defined by a given function c : M → R. Let c and c both denote functions on M. Given a positive integer N and a point z = (z 1 , . . . , z N ) ∈ × N M, define R(z) ∈ × N R to so that its k th component is c(z k ) − c (z k ). When F ∈ T N and when z has pairwise distinct entries, set F (z) to equal L N ,R(z) F. Granted this notation, then the c and c versions of · are related as follows: When F ∈ T N , and z has pairwise distinct entries,

(2.7) (z,F) c = (z,F (z) c . As will now be explained, the functional · has a certain finite dimensional flavor in the case at hand. To start, fix a positive integer N and suppose that V1 , . . . , V N are each a finite direct sum of eigenspaces of the Laplacian on P. Let V ≡ V 1 × . . . × V N denote the corresponding vector subspace of C ∞ (× N P). Thus, a function from V is a linear combination of functions that send any given p = ( p1 , . . . , p N ) to f 1 ( p1 ) . . . f N ( p N ), where each f k is from the corresponding Vk . To finish setting the stage, let z ∈ × N M denote a point with pairwise distinct components, and let R ≡ (r1 , . . . , r N ) ∈ × N R. The first remark is that the operator A z,R that is depicted in (2.3) maps V to itself. This is also true for the operator j Aˆ z ≡ ac (z i , z j ) ∂ai ∂a . (2.8) 1≤i≤ j≤N

1≤a≤d

Both statements hold because each k ∈ 1, ..., N version of 1≤a≤d ∂ak ∂ak commutes with Aˆ z and preserves V. Granted the preceding observation, let exp( Aˆ z )V ∈ End(V) denote the exponential of the restriction to V of the operator in (2.11). The relevance of exp( Aˆ z )V stems from the the following observation: If F ∈ V, then (z,F) = (exp( Aˆ z )V F)( p, . . . , p)dp. (2.9) P

There is one further ‘commutation’ property that plays a key role in what follows. This is stated as Lemma 2.1. Let N denote a positive integer. Then each a ∈ {1, . . . , d} version of fα ≡ k with all of the N 2 operators on × N P from the set 1≤k≤N ∂a commutes

i j . In addition, 1≤b≤d ∂b ∂b 1≤i, j≤N

(fa U )( p, . . . , p)dp = 0

(2.10)

p

for all U ∈ C ∞ (× N P). Proof of Lemma 2.1. The collection {fa }1≤a≤d generate the diagonal action of P on × N P that simultaneously multiplies each factor on the right by the same group j element. The fact that each fa commutes with each operator of the form 1≤b≤d ∂bi ∂b


31

follows because each of the latter is invariant under this diagonal action of P. Meanwhile, the identity in (2.10) follows by virtue of the fact that the full diagonal in × N P is mapped to itself by this same P action. Note that (2.9) and the preceding lemma lead to a simple formula for · in the case N = 2 or N = 3. Consider first the case N = 2, and take F = f 1 ( p1 ) f 2 ( p2 ) with f 1 and f 2 eigenfunctions of the Laplacian. Then (z,F) = 0 unless f 1 and f 2 have the same eigenvalue. In the latter case, ˆ ((z 1 ,z 2 ), f1 f2 ) = e(2a(z 1 ,z 2 )−c(z 1 )−c(z 2 )) E f 1 ( p) f 2 ( p)dp, (2.11) P

where Eˆ denotes the common eigenvalue of − 1≤a≤d ∂a ∂a . In the case that N = 3, take F = f 1 ( p1 ) f 2 ( p2 ) f 3 ( p3 ) with each f k being an eigenfunction of the Laplacian. Now −α1 Eˆ 1 −α2 Eˆ 2 −α3 Eˆ 3 ((z 1 ,z 2 ,z 3 ), f1 f2 f3 ) = e f 1 ( p) f 2 ( p) f 3 ( p)dp, (2.12) P

where − Eˆ 1 is f 1 ’s eigenvalue, α1 = c(z 1 ) + a(z 2 , z 3 ) − a(z 1 , z 2 ) − a(z 1 , z 3 ); and Eˆ 2 , Eˆ 3 , α2 and α3 are defined analogously. This section ends here with an analysis of the behavior of (z,R) as the distance between two components of z shrinks to zero. The upcoming Proposition 2.2 summarizes the story. A three part digression follows directly to set the stage. Part 1. Suppose that N ≥ 2 is given, and that a function F ∈ × N P has been specified that sends any given point ( p1 , . . . , p N ) to f 1 ( p1 ) . . . f N ( p N ), where each f k is an eigenfunction of the Laplacian on P. Suppose that z = (z 1 , z 2 , . . . , z N ) ∈ × N M is such that z 3 ,. . .,z N are pairwise distinct. Meanwhile, z 1 = z 2 and both are very near a point z 0 ∈ M that is distinct from z 3 ,. . .,z N −1 and z N . I use z ∈ × N −1 M to denote the point (z 0 , z 3 , . . . , z N ). Part 2. Let V1 and V2 denote the respective eigenspaces for the Laplacian on P that contain f 1 and f 2 . The representation theory for the diagonal action of P on ×2 P can be used to decompose the space V1 × V2 as a finite direct sum of spaces, ⊕v Wv , where the restriction to the diagonal in P × P identifies Wv with a direct sum of copies of a single eigenspace of the Laplacian on P. This notation is such that pairs Wv and Wv

with v = v map via this restriction homorphism to distinct eigenspaces of the Laplacian on P. I use φ in what follows to denote this restriction homomorphism from V1 × V2 to C ∞ (P). Thus, the image of φ on any given Wv is a particular eigenspace for the Laplacian on P. This understood, let Eˆ v denote the eigenvalue of − 1≤a≤d ∂a ∂a on φ(Wv ). Part 3. For each k ≥ 3, let Vk denote the eigenspace of the Laplacian on P that contains f k , and set V = V 1 × . . . × V N . This done, use φv to denote the homomorphism from V to φ(Wv ) × V3 × . . . × V N that is defined as follows: Let G = g1 g2 . . . g N , where each gk ∈ Vk , and let (g1 g2 )v denote the component of g1 g2 in Wv . Now set φ(G) = φ((g1 g2 ))g3 . . . g N . With the stage now set, here is the promised

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C. H. Taubes

Proposition 2.2. With the point z = (z 0 , z 3 , . . . , z N ) ∈ × N −1 M fixed, then ˆ ˆ ˆ (z,F) = e(a(z 1 ,z 2 )−c(z 0 ))( E 1 + E 2 − E v ) (z ,φv (F)) + o(1)

(2.13)

v

as z 1 and z 2 converge to z 0 . In particular, if a(·, ·) is positive when its two entries are close and diverges as their distance converges to zero, then ˆ ˆ ˆ (z,F) = e(a(z 1 ,z 2 )−c(z 0 ))( E 1 + E 2 − E 0 ) (z ,φ0 (F)) + o(1) ; (2.14)

here the notation uses Eˆ 0 to denote the smallest of the numbers in the set Eˆ v , and φ0 to denote the corresponding version of φv . By the way, the leading order term on the right-hand side of (2.14) can vanish. This is the case, for example, when zero is the only constant function in the C ∞ (P) image of φ(W0 ) × V3 × . . . × V N via the homomorphism that comes by restricting to the full diagonal in × N −1 P. Note as well that the o(1) term that appears in (2.13) is bounded by a constant times the distance between z 1 and z 2 when the function a in (1.1) and the function c are both smooth. However, in general, the o(1) term that appears in (2.14) need not be on the order of distance between z 1 and z 2 . In particular, the o(1) term in (2.14) will shrink slower than this distance when the function a has a reasonably mild divergence on approach to the diagonal. Such will be the case, for example, when the function a diverges as a small, positive multiple of the absolute value of the log of the distance between its two entries. Proof of Proposition 2.2. Any k > 2 version of either a(z 1 , z k ) or a(z 2 , z k ) can be written as a(z 0 , z k ) + o(1). Likewise, both c(z 1 ) and c(z 2 ) can be written as c(z 0 ) + o(1). This then allows the endomorphism Aˆ z from (2.8) and ( 2.9) to be written as Aˆ z = 2(a(z 1 , z 2 ) − c(z 0 )) ∂a1 ∂a2 + Aˆ + o, ˆ (2.15) 1≤a≤d

where the notation has Aˆ ≡ c(z 0 ) (∂a1 + ∂a2 )(∂a1 + ∂a2 ) + 2 a(z 0 , z k ) (∂a1 + ∂a2 )∂ak 1≤a≤d

+

3≤i, j≤N

ac (z i , z j )

k≥3 j ∂ai ∂a ;

1≤a≤d

(2.16)

1≤a≤d

and where oˆ is an endomorphism of V of size o(1). of the first two The plan now is to study Aˆ z by viewing oˆ as a small perturbation terms on the right-hand side of (2.15). In this regard, note that 1≤a≤d ∂a1 ∂a2 commutes with Aˆ and this makes the story for their sum relatively straight forward. To start this story, note that the homomorphism φ intertwines any given of ∂a1 + ∂a2 with the version 1 2 corresponding ∂a . This implies that the endomorphism 1≤a≤d ∂a ∂a acts on Wv as multiplication by the constant 21 ( Eˆ 1 + Eˆ 2 − Eˆ v ). Meanwhile, the endomorphism Aˆ preserves any given Wv × V3 × . . . × V N . To describe its action here, write Wv as a direct sum of subspaces that are each mapped isomorphically by φ onto φ(Wv ). Then, the action of Aˆ


33

on any one of these subspaces is identical to that of the z = (z 0 , z 3 , . . . , z N ) ∈ × N −1 M version of Aˆ z on φ(Wv ) × V3 × . . . × V N . Granted the preceding, a straightforward application of finite dimensional perturbation theory to exp( Aˆ Z )V finds that the ρ versions of (z,F) when z 1 is very close to z 2 have the form that is depicted in (2.13) and (2.14). 3. Quantum Field Theories Consider now the case where M = R×Y and P is a compact Lie group. Take the vector fields {∂a } and integration as in the previous section. Meanwhile, take the function a on (R×Y )×(R×Y ) to be a positive, constant multiple of a Green’s function for an operator on R×Y that has the form depicted in (2.1). To keep the story relatively short, I assume in what follows that L is a second order, elliptic operator that is self-adjoint with dense domain in L 2 (Y ). I also assume that the L 2 kernel of L is either trivial or consists of the constant functions. To be more specific about the function a in the case where Y is compact, introduce an orthonormal basis, {ηα }, of eigenfunctions of L; here Lηa = −E α2 ηα with E α ≥ 0. Write a point z ∈ R×Y as z = (t, y). Granted this notation, suppose z and z are distinct points in R×Y and set

• a((t, y), (t , y )) = σ α 2E1 α e−E α |t−t | ηα (y)ηa (y ) if all eigenvalues E α are non-zero. 1 1 −E α |t−t |

• a((t, y), (t , y )) = −σ 2Vol(Y ηα (y)ηα (y ) {α:E α >0} 2E α e ) t − t +σ if L annihilates the constants. (3.1) Here, σ is a positive constant. The only noncompact example considered below is that where Y = Rn and L is either the Laplacian on L 2 (Rn ) or differs from the latter by a negative constant. In all cases, the function c that defines the values of ac on the diagonal is taken to be independent of the R factor in R × Y . My purpose here is to explain how the Osterwalder-Schrader construction takes as input the functional · and returns a Hilbert space with a strongly continuous, unitary action of R whose generator is a self-adjoint positive semi-definite operator. To begin the Osterwalder-Schrader construction, reintroduce the vector space F, and let F 0 ⊂ F denote the set of functions that consists of the constant function and those of the form (z,F) , where, z ∈ × N M has pairwise distinct entries and F ∈ C ∞ (× N P) decomposes as a product of eigenfunctions for the Laplacian on P. Thus, F takes any given p = ( p1, . . . , p N ) to f 1 ( p1 ) . . . f N ( p N ), where each f k is an eigenfunction for 1≤a≤d ∂a ∂a . Here, N can be any positive integer. Now let F++ ⊂ F denote the subspace that is generated by the constant function and by the subset in F 0 that are labelled by (z, F), where each entry of z has positive first coordinate. This is to say that z = (z 1 , . . . , z N ) and each z k = (tk , yk ), where tk ∈ R is positive. Define F−− ⊂ F to denote the analogous subspace whose generators consist of the constant function and those (z,F) , where each entry of z has negative first component. Note that when − is in F−− and + is in F++ then − + is in F. As a consequence, multiplication of functions defines a vector space homomorphism from F−− ⊗ F++ to F. The abelian group R acts on × N (R×Y ) by simultaneously translating the R-coordinate of each entry. The image of a given point z under the action of τ ∈ R is denoted in what follows by τ · z. For example, if (t, y) ∈ R×Y , then τ · (t, y) = (t + τ, y). This R

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C. H. Taubes

action can be used to define an R action on F 0 ; this is the action whereby τ ∈ R sends any given Fz to Fτ ·z . This action extends by linearity to an action on the algebra F. In the latter guise, the action of τ is denoted by Rτ . Note that this action induces an action of the semi-group [0, ∞) ⊂ R on the subalgebra F++ ⊂ F. The notion of reflection positivity as used here refers to a certain anti-linear involution, ∗, on F that is defined from the involution on × N (R×Y ) whose effect is to change the sign of the R factor of each entry. The latter involution is also denoted by ∗. For example, in the case that (t, y) ∈ R×Y , then ∗(t, y) = (−t, y). The involution just defined on × N (R×Y ) is used to define an anti-linear involution on F 0 , that sends any given Fz to the complex conjugate of the function F∗z . This anti-linear involution on F 0 then extends as the desired anti-linear involution of the algebra F. The latter is also denoted as ∗. Note that it maps F++ to F−− . Proposition 3.1. The linear functional · has the following two properties: • It is translation invariant in that Rτ = for any given τ ∈ R and ∈ F, • It is ‘reflection positive’ in the sense that (∗) is non-negative for any ∈ F++ . This theorem is proved below. Accept it for the moment to see where it leads. Let Q(·, ·) denote the bilinear form on F++ that is defined by associating the number (∗) to any given and from F++ . Proposition 3.1 asserts that Q is a positive semi-definite Hermitian form. Let ker(Q) denote the subspace of vectors ∈ F++ with the property that Q( , ) = 0 for all ∈ F++ . The form Q thus descends to F++ / ker(Q) as a positive definite Hermitian form; this is denoted by Q also. In what follows, H denotes the Hilbert space completion of F++ / ker(Q) using Q. As noted above, the semi-group {Rτ : τ ≥ 0} maps F++ to itself. Moreover, this action is such that Q( , Rτ ) = Q(Rτ , ). (3.2) Indeed, (3.2) is a direct consequence of the fact that the value of the function depicted in (3.1) is unchanged when both t and t are translated by the same amount. Note that (3.2) implies that Rτ preserves ker(Q) and so the assignment τ → Rτ descends to define a semi-group on F++ / ker(Q). Theorem 3.2. The Hilbert space H has a strongly continuous, self-adjoint, one-parameter contraction semi-group whose time τ > 0 member maps the image of any given in F++ to the image of Rτ . The proof of Theorem 3.2 appears below. The generator of Theorem 3.2’s semi-group is minus 1 times a self-adjoint, non-negative operator on H (see, e.g [HP]). The latter operator is called the Hamiltonian. There is one more point to make here, this regarding the dependence of Theorem 3.2’s Hilbert space and semi-group on the choice for function c : Y → R that is used to define · by giving the values for ac on the diagonal in (R×Y ) × (R×Y ). For this purpose, reintroduce the notation that is used in (2.7). Granted this notation, the assignments (z,F) → (z,F (z)) extend by linearity to give an invertible, linear map from L++ to itself. This map is denoted here by J . As a consequence of (2.7 ), the corresponding c and c versions of Q are such that

Q c (, ) = Q c (J , J ).

(3.3) to the c

version. MoreThus, J descends to define an isometry from the c version of H over, as J commutes with the action on L++ of the 1-parameter semigroup {Rτ : τ ≥ 0},


35

so this isometry intertwines the c and c versions of the semi-group and its self-adjoint generator. A change from c to c might be viewed by a physicist as a change in a choice of renormalization scale for the quantum field theory. As just demonstrated, two such choices lead to equivalent Hilbert spaces and Hamiltonians. Proof of Proposition 3.1. It is sufficient to establish the reflection positivity condition solely for the linear combinations of functions from F 0 that are all defined using the same integer N . Indeed, this is a consequence of (2.4). With the preceding understood, fix N and let denote in what follows a finite set of distinct pairs of the form (z, F), where z and F are as follows: First, z is a point in × N (R×Y ) whose entries have positive first coordinate. Meanwhile, F, a function on × N P, has the form F( p1 , . . . , p N ) = f 1 ( p1 ) . . . f N ( p N ), where each f k is an eigenfunction of the Laplacian on P . Here, N = N ( ) is a positive integer that depends only on . When F and F are functions on × N P, the function on (× N P) × (× N P) that sends any given ( p, p ) to F( p)F ( p) is denoted in what follows by F × F . Since is finite, all functions on (× N P) × (× N P) of the form F¯ × F , where F and F come from lie in some N = 2N version of the subspace V ∈ C ∞ (×2N P) that appears in (2.9). As a consequence, the reflection positivity condition for is obeyed if and only if (exp( Aˆ (∗z,z ) )V · ( F¯ × F ))( p, . . . , p)dp ≥ 0. (3.4) (z,F),(z ,F )∈

P

Three cases will be considered below. The first is that where Y is compact and L has trivial kernel. The second case is that where Y is compact and L annihilates the constant functions. The third case is that where Y = Rn and L = − +m 2 with m ≥ 0. Case 1. This is the case where a is given in the top line of (3.1). The translation invariance required in the first point of Proposition 3.1 follows directly from the fact that the chosen function c is independent of R and the function a in (3.1 ) is unchanged if both t and t are translated the same amount. The arguments from Sect. 5 of [T] can be used here in this case to prove (3.4). In fact, the arguments in this case are simpler by virtue of the fact that exp( Aˆ (∗z,z ) )V is an endomorphism of a finite dimensional vector space. What follows is a brief summary of the arguments for the case at hand. To start, the operator Aˆ (∗z,z ) that appears in (3.4) can be written as tα,a (z) tα,a (z ) , (3.5) Aˆ (∗z,z ) = Aˆ z + Aˆ z + σ 1≤a≤d α

¯ while where the notation is as follows: First, Aˆ z depends only on z and acts only on F,

Aˆ z depends only on z and acts only on F . Second, the sum indexed by the Greek letter α is the same sum over an orthonormal eigenbasis for the operator L that appears in (3.1). Finally, 1 −E α ti tα,a (z) = e ηα (yi )∂ai , (3.6) 2E α 1≤i≤N

and tα,a (z ) is identical save that z is used and that ∂ai replaces ∂ai . Thus, tα,a (z) acts solely on F¯ and tα,a (z ) acts solely on F .

36

C. H. Taubes

The next observation is that for the purposes of establishing (3.4), all terms in (3.5) and (3.6) can be viewed as elements in End(V). Note in particular that the sum in (3.6) is absolutely convergent as an element in End(V). This understood, the decomposition in (3.6) is used to write exp( Aˆ (∗z,z ) )V as an absolutely convergent sum that has the schematic form

1 M × M + σ M1−s tα,a Ms × M1−s tα,a Ms ds 1≤a≤d α

+σ2

1≤a,b≤d α,β

0

0

1 s1 0

M1−s1 tα,a Ms1 −s2 tβ,b Ms2

× M1−s1 tα,a Ms1 −s2 tβ,b Ms2

ds2 ds1 + . . . .

(3.7)

Here, the unprimed terms are defined by z and operator only on F¯ , while the primed terms are defined by z and operate only on F . Also, M = exp( Aˆ z ) and M = exp( Aˆ z ), each viewed as an endomorphism of a particular finite dimensional subspace of C ∞ (× N P). Note that any given term in (3.7) is a sum of integrals of endomorphisms where each endomorphism sends F¯ × F to a function that has the form Uz F¯ × Uz F with the assignment z → Uz indicating a certain endomorphism valued function on the portion of × N (R×Y ) where all the components have positive R coordinate. As a consequence, each term in (3.7) makes a non-negative contribution to (3.4). Indeed, such is the case because 2 ( p, . . . p)dp. (Uz F¯ × Uz F )( p, . . . , p)dp = (U F) z P (z,F)∈ (z,F),(z ,F )∈ P (3.8) Case 2. The translation invariance required by Proposition 3.1 follows by virtue of the fact that the function c on R×Y comes from Y and the function that is depicted in the second line of (3.1) is left unchanged when both t and t are translated by the same amount. The argument for (3.4) in this case uses the following strategy: With fixed, and a positive real number, m, chosen, each version of the Aˆ (∗z,z ) ∈ End(V) that appears in ( 3.4) is replaced by an endomorphism, Aˆ m (∗z,z ) ∈ End(V), with two key properties. First, the argument from Case 1 proves the version of (3.4) that has Aˆ m (∗z,z ) replacing A(∗z,z ) . Second, given ε > 0, then

¯ exp( Aˆ m (∗z,z ) )V ( F × F ) ( p, . . . , p)dp P (3.9) exp( Aˆ (∗z,z ) )V ( F¯ × F ) ( p, . . . , p)dp < ε − P

when m is sufficiently small. To obtain Aˆ m ˆ m (z, z ) denote the version of the function in the first line of (∗z,z ) , let a (3.1) that appears when L is replaced by L − m 2 . With aˆ m understood, define j ˆm ˆm

Aˆ m aˆ m (∗z i , z j ) ∂ai ∂a , (3.10) (∗z,z ) = A z + A z + σ 1≤i, j≤N

1≤a≤d


37

where Aˆ m z ≡

σ + c(z i ) ∂ai ∂ai + σ 2m

1≤i≤N

1≤a≤d

aˆ m (z i , z j )

1≤i = j≤N

j

∂ai ∂a ,

(3.11)

1≤a≤d

k and where Aˆ m z is defined similarly using z and with all versions of ∂a replaced by their primed counterparts. The argument from Case 1 proves the reflection positivity condition in (3.4) for any given positive m. The proof that (3.9) holds has four steps.

Step 1. Return to the milieux of (2.7) and (2.8) and let N and V be as described in the associated discussion. Now, let ai j 1≤i, j≤N denote a set of (N )2 real numbers. Any i j ˆ operator of the form Aˆ ≡ 1≤i, j≤N ai j 1≤a≤d ∂a ∂a maps V to itself and so exp( A) ˆ V . Suppose now is well as a linear map from V to V. The latter is denoted as exp( A) defined δ > 0 with that ai j has been the following and also some ε > 0. Then there exists

given

significance: If ai j is a second set of real numbers with 1≤i, j≤N ai j − ai j < δ, ˆ V − exp( Aˆ )V ≤ ε. then exp( A) Step 2. The collection aˆ m m>0 has the following key property: Fix δ > 0 and let Oδ ⊂ ×2 (R×M) denote the set of pairs (t, y), (t , y ) such that t − t > δ and |t| + |t | < 1/δ. Then 1 m aˆ − converges uniformly to aˆ on Oδ as m → 0. (3.12) 2m m>0 Hold onto this last observation for a moment. N

Now, let z and z denote points in × N (R×Y ) that come from , and define the ˆ ˆ

≡ 4N 2 numbers a

i j so that the endomorphism A on V from Step 1 is A(∗z,z ) .

Meanwhile, let ai j denote the 4N 2 numbers that arise when the function aˆ that is used 1 to define Aˆ (∗z,z ) is replaced by some very small but positive m version of aˆ m − 2m . Let

Aˆ denote the corresponding endomorphism of V. The conclusions of Step 1 with (3.12) imply the following: Given ε > 0, then | exp( Aˆ (∗z,z ) )V − exp( Aˆ )V | < ε when m is sufficiently small. σ . This set of numbers defines the operator Aˆ m Step 3. Let aimj denote ai j + 2m (∗z,z ) that appears in (3.10). This step argues that   ˆ

 σ fa fa  . (3.13) exp( Aˆ m (∗z,z ) )V = exp( A )V exp 2m 1≤a≤d

V

ˆ σ To establish (3.13), note first that Aˆ m 1≤a≤d fa fa . Thus, (3.13) follows (∗z,z ) = A + 2m from the claim that Aˆ and 1≤a≤d fa fa commute. Meanwhile, the latter claim follows from Lemma 2.1.

38

C. H. Taubes

Step 4. Now let V ∈ V. Then exp( Aˆ m )V V ( p, . . . , p)dp = exp( Aˆ )V V ( p, . . . , p)dp P

(3.14)

P

by virtue of (3.13) and (2.10). Granted (3.14), then (3.9) follows from the conclusions of Step 2. Case 3. Here, Y = Rn . In the cases where n > 1 or where n = 1 and m > 0, the function a(·, ·) on (R × Rn ) × (R × Rn ) can be written as σ a, ˆ where aˆ is given by the Fourier integral: 1 1

aˆ (t, y), (t , y ) = e−|t−t |E(k) eik(y−y ) dk, (3.15) (2π )n Rn 2E(k) 1

where E(k) ≡ (|k|2 +m 2 ) 2 . The convention here is to take m1 ≥ 0. The n = 1 and m = 0 case has a = σ a, ˆ where aˆ (t, y), (t , y ) is the function − 4π ln (t − t )2 + (y − y )2 . In all cases, σ is a positive constant. The translation invariance is again due to the fact that the function c is independent of the R factor in R × Rn and the the relevant versions of a(·, ·) are unchanged when both t and t are simultaneously translated by the same amount. The arguments that are used in Case 1 above can be applied to prove the second point in Proposition 3.1 after one preliminary step. To describe the preliminary step, consider first the situation when n > 1 or when n = 1 and m = 0. In this case, Aˆ (∗z,z ) ∈ End(V) has a form that is very similar to the one depicted in ( 3.5): 1 1 tk,a (z)tk,a (z ) dk, (3.16) Aˆ (∗z,z ) = Aˆ z + Aˆ z + σ n (2π ) Rn 2E(k) 1≤a≤d

1

where E(k) ≡ (|k|2 + m 2 ) 2 and tk,a (z) ≡

e−E(K )ti e−ik·yi ∂ai .

(3.17)

1≤i≤N

The important point here is that the integral that appears in (3.16) is absolutely convergent and defines an operator in End(V). This understood, the argument used for Case 1 can be repeated in an essentially verbatim fashion after changing certain sums to integrals. Turn next to the case where n = 1 and m = 0. This case requires the lemma that follows. Lemma 3.3. When m > 0, let aˆ m be the corresponding n = 1 version of (3.15). Given ε > 0 and ρ > 1, then all sufficiently small but positive m versions of aˆ m enjoy the following property: If z, z ∈ R × R are points such that 1/ρ < |z − z | < ρ, then m aˆ (z, z ) − κ + 1 ln |z − z | < ε. (3.18) 2π Here, κ is a constant that depends only on ε and ρ.


39

The proof of this lemma is straightforward and left to the reader. To see where Lemma 3.3 leads, remark that with fixed in (3.4), there exists some number ρ >> 1 such that the conditions in the lemma hold for all points that appear as a component of any z = (z 1 , . . . , z N ) from any pair in . This understood, suppose that ε < 0 has been chosen and that aˆ m is given as in Lemma 3.3 with m > 0 but very small. Let a m ≡ σ aˆ m . Given a pair (z, F) and (z , F ) from , use Aˆ m (∗z,z ) to denote the element in End(V) that is defined by   (c(z i ) + σ K) ∂ai ∂ai + (c(z i ) + σ K) ∂ai ∂ai  1≤i≤N

+



1≤a≤d

a m (z i , z j )

1≤i = j≤N

+

1≤i, j≤N

1≤a≤d

∂ai ∂a + a m (z i , z j ) j

1≤a≤d

a m (∗z i , z j )

 j ∂ai ∂a 

1≤a≤d

j ∂ai ∂a .

(3.19)

1≤a≤d

ˆm With Am (∗z,z ) as above, introduce the corresponding exp( A(∗z,z ) )V . Now there are two key points. First, the arguments for the n = 1 and m > 0 case just given prove that the Aˆ m (·) version of (3.4) holds. Second, with ε > 0 given and fixed, any sufficiently small m version of Aˆ m (·) obeys (3.9 ). Indeed, Lemma 3.3 m

ˆ ˆ implies that (3.9) holds if A(∗z,z ) is replaced by A ≡ Aˆ m

) − σ κ 1≤a≤d fa fa , where (∗z,z fa ≡ 1≤i≤N (∂ai + ∂ai ). Meanwhile, Lemma 2.1 implies that such a replacement does not affect the integral that appears in (3.9). Proof of Theorem 3.2. The issue here is whether the one parameter semigroup on F++ given by the set of transformations {Rτ }τ >0 descends to H as a strongly continuous, self-adjoint, contraction semigroup. As noted previously, each such Rτ descends to H as a symmetric operator with dense domain F++ / ker(Q). As is argued momentarily, Q(Rτ , Rτ ) ≤ Q(, )

(3.20)

for all ∈ F++ and τ ≥ 0. This implies that Rτ extends to H as a bounded, self-adjoint operator. Given that Rτ Rτ = Rτ +τ on F++ , standard arguments (see, eg. [Ka] or [HP]) prove that these extensions define a strongly continuous, 1-parameter, self-adjoint, contraction semigroup. The argument for (3.20) is given momentarily. The proof depends on the following: Lemma 3.4. Let ∈ F++ and τ ∈ [0, ∞). Then there exists υ ≥ 0 such that Q(Rτ , Rτ ) ≤ υ for all τ ≥ 0. The proof of this lemma is given below. Here is how to prove (3.20): Note first that Q(Rτ , Rτ ) = Q(, R2τ ) and thus is less than the square root of Q(R2τ , R2τ )Q(, ). Now, bound Q(R2τ , R2τ ) in terms of Q(R4τ , R4τ ) using this same inequality but with τ replaced by 2τ . Continue in this vein to conclude that 1

1

1

3

Q(Rτ , Rτ ) ≤ Q(R2τ , R2τ ) 2 Q(, ) 2 ≤ Q(R4τ , R4τ ) 4 Q(, ) 4 −k

−k

≤ . . . ≤ lim Q(R2k τ , R2k τ )2 Q(, )1−2 . k→∞

(3.21)

40

C. H. Taubes

Lemma 3.4 guarantees that the limit on the right is no greater than Q(, ). Proof of Lemma 3.4. The existence of a τ -independent bound can be deduced by first writing = α bα α , where bα ∈ C and where each α is either constant or has the form (z,F) . Writing in this way exhibits the fact that it is sufficient to find a τ independent bound for any given (z, F) version of Q(Rτ (z,F) , Rτ (z,F) ). The existence of the latter bound is argued next. Consider first the cases where Y is compact and L has no kernel, where Y = Rn>1 , and where Y = R and m > 0. To start, fix some very large number r , and use R ∈ ×2N R to denote the diagonal point (r, . . . , r ). When r is sufficiently large, then Q(Rτ (z,F) , Rτ (z,F) ) =

(× N P)×(× N P)

L 2N ,R F¯ × F K (∗τ ·z,τ ·z),R dp.

(3.22)

Meanwhile, the supremum norm of L 2N ,R F¯ × F is bounded by some constant multiple of υ r where υ > 1 depends only F and z. It thus follows from (3.22) that this same υ r bounds Q(Rτ (z,F) , Rτ (z,F) ) since the function K (·) is non-negative and its integral over ×2N P is equal to 1. This understood, here is the key observation: When Y is compact and L has trivial kernel, or when Y = Rn>1 , or when Y = R and m > 0, then the same constant r can be used for all τ ≥ 0. Indeed, this can be deduced using the top line in ( 3.1) or (3.15) as the case may be. The key point is that both expressions lead to functions a(·, ·) with the following property: The assignment of a ((−t − τ ,y) , (t + τ, y ) to τ ∈ [0, ∞) for fixed (t, y) and (t , y ) defines a bounded function on [0, ∞) if both t and t are positive. Note that this is not true of the function that appears in the second line of (3.1), nor is it true for the Green’s function in the case Y = R and m = 0. To elaborate on the story in these last cases, remark that when the lower line in (3.1) is relevant, the large τ versions of r must be at least some constant times τ . In the case 1 when Y = R and the Green’s function is − 2π ln z − z , then the large τ versions of r must be at least some constant times ln(τ ). By the way, the chain of inequalities in (3.21) leads to the desired bound Q(Rτ (z,F) , Rτ (z,F) ) ≤ Q (, ) in the case that the large τ versions of r are linear in ln(τ ). This is because the k → ∞ limit of k2−k is zero. To finish the proof of Lemma 3.4, suppose now that either Y is compact and L has a kernel, or else Y = R and m = 0. To start the argument for these cases, remark that the version of (2.9) that gives the integral in (3.22) is defined using an endomorphism, Aˆ (∗τ ·z,τ ·z) , of V that has the form

ac (z i , z j )

1≤i, j≤N

+

j

∂ai ∂a +

1≤a≤d

a(∗τ · z i , τ · z j )

1≤i, j≤N

ac (z i , z j )

1≤i, j≤N

j

∂ai ∂a

1≤a≤d

j ∂ai ∂a .

(3.23)

1≤a≤d

This is to say that the integral on the right-hand side of (3.23) is the same as P

(exp( Aˆ (∗τ ·z,τ ·z) )V ( F¯ × F))( p, . . . , p)dp.

(3.24)


41

To continue, observe that τ only appears in the far right term in (3.23). Moreover, it follows from the form of a(·, ˆ ·) that the large τ behavior of this term can be written as j j ∂ai ∂a + bi j (τ ) ∂ai ∂a , (3.25) −υ(τ ) 1≤i, j≤N 1≤a≤d

1≤i, j≤N

1≤a≤d

σ 2σ where υ(τ ) = Vol(Y ) τ or 2π ln(τ ), and where bi j (τ ) is bounded as τ → ∞. Here, the case with υ (τ ) linear in τ arises when Y is compact and L annihilates the constants; and the other case arises when Y = R and m = 0. What follows is the crucial point: The leftmost term in (3.25) is

j j ∂ai ∂a + ∂ai ∂a ,

1 1 − υ(τ ) fa fa + υ(τ ) 2 2 1≤a≤d

(3.26)

1≤i, j≤N 1≤a≤d

where fa ≡ 1≤i≤N (∂ai + ∂ai ). This understood, Lemma 2.1 implies that the leftmost term in (3.26) can be dropped without affecting the integral in (3.24). This is to say that (3.24) is valid with Aˆ (∗τ ·z,τ ·z) replaced by j j Aˆ ≡ ac (z i , z j ) ∂ai ∂a + ac (z i , z j ) ∂ai ∂a

1≤i, j≤N

+

1≤i, j≤N

1≤a≤d

bi j (τ )

1≤a≤d

1≤i, j≤N

1 j ∂ai ∂a +υ(τ ) 2

1≤a≤d

(∂ai ∂a +∂ai ∂a ). (3.27) j

j

1≤i, j≤N 1≤a≤d

Since υ(τ ) > 0 forlarge τ, the operator at the far right in (3.27) is negative semi-definite. Granted that bi j (τ ) are bounded functions of τ , this last fact implies that the right-hand side of (3.24) enjoys a τ independent bound. 4. The Action of the Poincaré Group The purpose of this section is to describe a unitary action of the n + 1 dimensional Poincaré group on the Hilbert space H from Theorem 3.2 in a case where Y = Rn . In particular, this case takes the function a that appears in (1.1) to be a positive multiple of the Green’s function for the operator d2 + −m 2 , dt 2

(4.1)

where is the (negative definite) Laplacian on Rn and where m ≥ 0. In particular, a = σ a, ˆ where aˆ is given by (3.15) when n > 1 and when n = 1 and m > 0. In the case 1 n = 1 and m = 0, then aˆ = − 4π ln (t − t )2 + (y − y )2 . Meanwhile, the constant function on R × Rn is used for the definition of the function ac that appears in (2.2) and implicitly in ( 2.9). The notation used below for the Poincaré group writes the latter as the semi-direct product of the group of translations, R × Rn , with the Lorentz group S O(1, n). The ‘time’ translations are those of the 1-parameter subgroup along the R factor in the translation group R × Rn . Meanwhile, the ‘spatial translations’ are those from the Rn factor. The subgroup S O(n) ⊂ S O(1, n) is identified here as the subgroup that fixes the left-hand R factor in R × Rn .

42

C. H. Taubes

The first point to make is that · is invariant under the action on F of the semi-direct product of R × Rn and S O(n + 1) that is induced by the latter’s action on R × Rn . To elaborate, let N be a positive integer and let z ∈ × N (R × Rn ). Let b ∈ R × Rn and let U ∈ S O(n + 1) and use to denote the point (b, U ) in the semi-direct product group. Write · z to designate the point that is obtained from z by acting by simultaneously on each of its factors. Then (z,F) = (z,F) (4.2) for all pairs (z, F) with z ∈ × N (R × Rn ) and with F ∈ C ∞ (× N P) such that z’s entries are pairwise distinct and F decomposes as a product of f 1 . . . f N , where f k is an eigenfunction of the Laplacian on the k th factor × N P. Indeed, (4.2) follows by virtue of the fact that the function ac is a function only of the Euclidean distance in R × Rn between its two arguments. The action of the subgroup Rn S O(n) preserves F++ and commutes with the involution ∗ that defines the quadratic form Q. As a consequence, the action of this subgroup descends as a unitary group action on the Hilbert space H. This subgroup’s action also commutes with Theorem 3.2’s contraction semi-group because the Rn S O(n) action on F commutes with the action of R that sends τ ∈ R and (z,F) ∈ F 0 to (τ z,F) . As noted above, the generator of Theorem 3.2’s contraction semi-group is self-adjoint. As a consequence, the square root of −1 times this operator generates a 1-parameter, unitary group action on H that commutes with the action of Rn S O(n). The latter 1-parameter unitary group together with the aforementioned action of Rn S O(n) supply a unitary action on the Hilbert space of the semi-direct product (R × Rn ) S O(n). This last group appears directly as a subgroup in the Poincaré group and is identified now with the latter. Doing so specifies the desired action of much of the Poincaré group; all that is left as yet to define are the ‘Lorentz boosts’, these group elements that act on R × Rn so as to mix the time and space directions. A ‘pure’ Lorentz boost as defined here is determined by a vector v ∈ Rn with Euclidean norm less than 1. The pure Lorentz boost defined by v fixes the point (0, y) when y is orthogonal to v and it sends any given (t, 0) to γ (t, vt), where γ = (1 − v · v)−1/2 . Here, v · v is used to denote the inner product of v with itself using the Euclidean inner product on Rn . Meanwhile, it sends (0, v) to γ (v · v, v). To fill out the whole Poincaré group, it is sufficient to first find unitary operators on H that correspond to the pure Lorentz boosts, and then verify that the pure boost operators multiply one against another and against the (R × Rn ) S O(n) operators according to the multiplication law for the Poincaré group. As might be expected, the pure Lorentz boosts are defined using the part of the S O(n + 1) action on F that comes from elements that mix the R and Rn factors in R × Rn . Such an element in S O(n + 1) is deemed a ‘Euclidean boost’ in what follows. A ‘pure’ Euclidean boost as defined here is determined by a vector v ∈ Rn . This element in S O(n + 1) fixes points of the form (0, y) when y is orthogonal to v, sends (t, 0) to λ(t, vt), where λ ≡ (1 + v · v)−1/2 , and it sends (0, v) to the point λ(−v · v, v). Note that the pure boost defined by v sends any point (t, y) with t > |v||y| to a point with positive first coordinate. However, given v = 0 in Rn , there exist points (t, y) with t > 0 that are mapped by v’s boost to points with negative first coordinate. Thus, pure boosts do not preserve F++ . This last point is what makes the story interesting. When v ∈ Rn , let Oˆ v ∈ S O(n + 1) denote the pure boost that is defined by v. With r ≡ |v| fixed, there exists a subspace, Ur ⊂ F ++ , that is mapped to F++ by the action of Oˆ v . Moreover, these Rn labelled subspaces can be defined so as to have the following properties: (4.3) If r > r then Ur ⊂ Ur , and ∪r >0 Ur = F++ .


43

Indeed, take Ur to be the subspace that is generated by the constant function and functions (z,F) , where each of the N factors of z ∈ × N (R × Rn ) have the form (t, y) with t > r |y|. To continue, remark that the pure boosts have the following property with respect to the involution, ∗, on R × Rn that changes the sign of the R coordinate: ∗ Oˆ v = Oˆ −v ∗ .

(4.4)

Since Oˆ −v = Oˆ v−1 , it thus follows from (4.2) that Q( , Oˆ v ) = Q Oˆ v ,

(4.5)

whenever and are both in Ur with r > |v|. This last equation has two implications: First, the operator Oˆ v annihilates the kernel of Q. Second, if |v| < 1, then Oˆ v descends to H as a symmetric operator on the domain Ur / ker(Q) for r > 2|v|/(1 − |v|2 ). Here, this last bound on r follows from the fact that Oˆ v Oˆ v = Oˆ v with v = 2v/(1 − v · v). To make the next point, fix a vector ∈ F++ , then Oˆ v is in F++ when |v| is sufficiently small. For such v, the pairing Q( , Oˆ v ) is well defined for any given

in the Hilbert space. Moreover, the value of the resulting function v → Q( , Oˆ v ) extends from a neighborhood of 0 in Rn as a holomorphic function on a neighborhood of 0 in Cn . This follows from the form of aˆ using (2.9) and some standard, finite dimensional perturbation theory. An immediate consequence is that the assignment v → Oˆ v defines a holomorphic map from a neighborhood of 0 in Cn to the Hilbert space H. Of interest with regards to defining a Lorentz boost are the vectors Oˆ v when v is purely imaginary with Hermitian norm less than 1. Thus, where v = iw with w ∈ Rn and w · w < 1. In this regard, the operator Oˆ iw is defined on the image in H of Ur when r > |w|. It is a consequence of (4.4) and ( 4.5) that Oˆ iw is a unitary operator on this domain. Were the Ur dense in H, then Oˆ iw would extend by continuity over the whole of H as a unitary operator. Such an extension would serve as a pure Lorentz boost for the desired Poincaré group action. Since Ur does not have dense image in H, a larger domain must be found. For this purpose, fix and in F++ and remark that the analyticity near 0 in Cn of the map v → Q( , Oˆ v ) has the following consequence: Take v to be real with length 1. Then the assignment to ∈ F++ of the θ → 0 limit of θ1 − Oˆ tan(θ)v defines a symmetric operator on H with domain F++ / ker(Q). Indeed, this follows because d Q( , Oˆ tan(θ)v )|θ=0 when , ∈ F++ . • limθ→0 Q , θ1 ( − Oˆ tan(θ)v ) = dθ • limθ→0 Q( θ1 ( − Oˆ tan(θ)v ), θ1 ( − Oˆ tan(θ)v )) = when ∈ F++ .

d2 dθ 2

Q(, Oˆ tan(θ)v )|θ=0 (4.6)

Use L v in what follows to denote the operator just defined. As is argued next, L v has a self-adjoint extension. To start the argument, introduce hv to denote the hermitian form on F++ / ker(Q) that is defined by polarizing the quadratic, non-negative functional that sends to Q(L v , L v ). Thus, Q(L v , L v ) =

d2 Q(, Oˆ tan(θ)v )|θ=0 . dθ 2

(4.7)

44

C. H. Taubes

According to Theorem 1.27 in Chapter VI.5 of [Ka], the form hv is closable on the domain F++ / ker(Q). This implies that L 2v has a Friedrichs extension; this a non-negative self adjoint operator whose dense domain contains a core in F++ / ker(Q). Now, let |L v | = (L 2v )1/2 . The domain of |L v | is the domain of the closure of the form hv . In particular, F++ / ker(Q) is a core for |L v |. On F++ / ker(Q), the operator T = L v + 2 |L v | is non-negative and symmetric. The arguments just invoked to define the self-adjoint extension of L 2v can be used to endow T with a self-adjoint extension whose domain is that of |L v |. This understood, T − 2|L v | extends L v from F++ / ker(Q) as a closed and self-adjoint operator on H. Standard constructions (see, e.g. Chapter IX.1.2 in [Ka]) now provide a strongly continuous, 1-parameter group of unitary operators on H with generator i · L v . When τ ∈ R, the corresponding version of exp(iτ L v ) is denoted by Uˆ tanh(τ )v . Note that in the case that ∈ F++ / ker(Q) and τ has small absolute value, then Uˆ tanh(τ )v = Oˆ tan(iτ )v . Indeed, such is the case by virtue of the fact that Uˆ − tanh(τ )v Oˆ tan(iτ )v has zero derivative with respect to τ . The 1-parameter group τ → Uˆ tanh(τ )v is to be identified with the 1-parameter group of pure Lorentz boosts as defined by multiples of the unit vector v. To verify that the multiplication between w = w operators Uˆ w and Uˆ w is as required for Lorentz group elements, it turns out that it is sufficient to consider the issue on the domain Ur / ker(Q) for r sufficiently large. This point is explained momentarily. In the meantime, note that the operators Uˆ w and Uˆ w on any sufficiently large r version of Ur / ker(Q) are analytic continuations of corresponding versions of Oˆ (·) ; and this implies that they obey the desired multiplication law. A similar argument shows that the various versions of Uˆ (·) have the desired properties with regards to the already defined action of the (R × Rn ) S O(n) subgroup of the Poincaré group. To argue that it is enough to consider the Uˆ w Uˆ w on a large r version of Ur / ker(Q), suppose that ∈ F++ has been specified. Then Rτ is in Ur for all sufficiently large τ . Suppose that Uˆ w Uˆ w = Uˆ on Ur / ker(Q), where Uˆ denotes the operator on H that corresponds to the appropriate element along some other 1-parameter subgroup of the Lorentz group. Thus Q( , (Uˆ w Uˆ w − Uˆ )Rτ ) = 0 for all τ sufficiently large and all

in H. However, if

is any fixed vector in H , then the assignment τ → Q(

, Rτ ) defines a real analytic function on (0, ∞) with continuous extension to τ = 0 (see, e.g. Chapter IX.1.6 in [Ka].) Thus, Q( , (Uˆ w Uˆ w − Uˆ )) = 0 for all ∈ H. This then implies that Uˆ w Uˆ w = Uˆ on the dense domain F++ / ker(Q); and thus Uˆ w Uˆ w = Uˆ on the whole of H. 5. Field Theories for Quotient Spaces Suppose that P is a compact Lie group as in the previous sections. To keep things simple, I will assume that P is a simple Lie group. Note, however, that what transpires below holds in greater generality. As before, take {∂a } to be an orthonormal basis of left invariant vector fields on P with respect to a chosen bi-invariant metric. Integration on P is defined using this same metric’s volume form. Now, suppose that G ⊂ P is a subgroup and use G\P to denote the quotient of P by the action of G via multiplication on the left. My purpose here is to explain how the constructions in the previous sections can be used to obtain a quantum field theory for the space of maps from a manifold, Y , into G\P.


45

For this purpose, suppose first that M is a given manifold and reintroduce the vector space F as defined in Sect. 2 for the pair M and P. Thus, F is generated by the constant function from M to P and by functions of the form (z,F) , where z ∈ × N M has pairwise discrete entries and F is in the domain T N . Here N can be any positive integer. The domain T N consists of finite linear combinations of functions that decompose so as to send any given p = ( p1 , . . . , p N ) ∈ × N P to f 1 ( p1 ) . . . f N ( p N ), where each f k is an eigenfunction for the Laplacian on P . As is explained next, it is a consequence of the Peter-Weyl theorem that T N contains functions that are pulled up from × N G\P. To see that T N has functions from × N G\P, note first that the Peter-Weyl theorem asserts that the eigenspaces of the Laplacian on P are in 1-1 correspondence with the irreducible representations of P. To make this correspondence explicit, suppose that V is an irreducible representation of P with a hermitian metric that makes for a unitary P action. Let ρV : P → U (V ) denote the corresponding homomorphism of Lie groups. If η, ν are any two elements in V , then the function p → η† ρV ( p)ν

(5.1)

is an eigenfunction for the Laplacian on P. With V fixed, all such eigenfunctions have the same eigenvalue and these eigenfunctions span the corresponding eigenspace. This understood, remark that the representation V decomposes as a direct sum of irreducible representations of the subgroup G . Functions of the form depicted in (5.1) in the case that η is in the trivial G-representation are functions that are pulled up from G\P via the projection map. Granted the preceding, let F G ⊂ F denote the subspace that is generated by the constant function and those of the form (z,F) with z as before and with F decomposing as f 1 . . . f N , where each f k is a G-invariant eigenfunction of the Laplacian on P. Let G denote the corresponding subspace of F . The quadratic form Q restricts to F as F++ ++ ++ a Hermitian, non-negative form. Let ker G (Q) denote the kernel of this restricted form, G / ker G (Q) using Q. and let HG denote the completion of F++ G and ker G (Q), The action of the semigroup {Rτ : τ ≥ 0} on F++ preserves both F++ G G and so descends as an action on the dense domain F++ / ker (Q) in HG . The argument given for Theorem 3.2 works as well here to prove that this semigroup action extends to an action on HG of a strongly continuous, self-adjoint 1-parameter contraction semi-group. This semi-group is generated by a non-negative, self-adjoint operator. In addition, the argument given in the previous section can be applied here in the case where Y = Rn to prove that HG has an action of the (n + 1)-dimensional Poincaré group whose time translation subgroup is generated by the square root of −1 times the generated afore-mentioned contraction semi-group. The structure just described can be viewed as a quantum field theory for the space of maps from Y to G\P. 6. When the Domain is a Riemann Surface My purpose in this section is to exhibit certain properties of · in the case when M is a compact Riemann surface or one with some number of punctures. To motivate the ensuing discussion, note that physicists have generally agreed on a list of properties that would allow, were they satisfied, the collection of dim(M) = 2 versions of · to be deemed the normalized correlation functions for a conformal field theory. Such a list of properties was first summarized by Segal [Se1, Se2]. The discussion that follows

46

C. H. Taubes

refers to Gawedzki’s presentation of the list in his second lecture from [Ga]. Some of the properties on the list refer only to a single Riemann surface, and the forthcoming Propositions 6.1–6.4 assert that the latter sort are satisfied by · . To set the stage, I treat a surface with n punctures as a compact surface, M, together with a set, ϑ, of n distinct points in M, these are the missing points in the original unpunctured surface. With it understood that M is compact, assume that M has a given Riemannian metric and take the function a that appears in (1.1) to be a positive multiple of a certain Green’s function for the Laplacian on M. To be precise here, I use a(z, ˆ z )

to denote the value of this Green’s function at points z = z in M − ϑ. With z ∈ M − ϑ fixed, then aˆ z (·) ≡ a(·, ˆ z ) is a distributional solution on M to the equation 1 1 −aˆ z = δz − δw + (2 − n) ; (6.1) 2 area(M) w∈ϑ

here is the negative definite Laplacian, δu is the Dirac delta function with mass 1 at the point u ∈ M, and the area of M is computed using the metric’s area measure. In particular, aˆ z is the unique solution to (6.1) whose integral over the whole of M is zero when computed using the metric’s area measure. By way of an example, suppose that M is the round 2-sphere with area 4π . Write M = C ∪ {∞} and take ϑ = {0, ∞}, then z − z 1

a(z, ˆ z)=− . (6.2) ln 2π |z|1/2 |z |1/2 ˆ z ) with σ a positive constant. With aˆ understood, set a(z, z ) ≡ σ a(z, 1 An important point in what follows is that a(·, ˆ z ) diverges as − 2π ln(dist(·, z )) near

the point z In fact, 1 ˆ z ) + ln dist(z, z ) lim a(z, (6.3) 2π z→z exists and defines a function of the point z that varies smoothly over M − ϑ. Indeed, this follows from the fact that any metric on a surface is locally conformally flat. In what follows, c denotes the function from M − ϑ to R that is depicted in (6.3). Use this same function c to define the function ac that appears in (2.2). In the propositions that follow, N is a positive integer, z is a point in × N (M − ϑ) with pairwise distinct entries and F is a function on × N P that sends any given p = ( p1 , . . . , p N ) to f 1 ( p1 ) . . . f N ( p N ) where each f k is a real valued eigenfunction of the Laplacian on P. Note that Eˆ k is used to denote the absolute value of the eigenvalue of fk . To make contact with the axioms in Gawedzki’s article [Ga], let us agree that what Gawedzki calls a ‘primary field’ is any real valued eigenfunction of the Laplacian on P. Thus, the collection { f k } constitutes a set of primary fields. Note that there is no nontrivial analog here of Gawedzki’s partition function Z. This is because our functional · gives normalized correlation functions by virtue of the fact that it assigns the value 1 to the element 1 ∈ F. The first proposition here describes how · is affected by a diffeomorphism. It verifies that · satisfies the axioms that are given by Eqs. (2.2) and (2.6) in [Ga]. Proposition 6.1. Let · denote the version of (2.5) that is defined using the pull-back of the original metric on M via a given diffeomorphism. Let ϑ denote the inverse image of ϑ under this diffeomorphism. When z ∈ × N (M − ϑ), let z ∈ × N (M − ϑ ) denote the inverse image of z via the induced diffeomorphism of × N M. Then (z ,F) = (z,F) .


47

The proof of Proposition 6.1 and the subsequent propositions are given at the end of this section. The next proposition describes how · is affected by a conformal change to the given metric on M. For this purpose, suppose that u is a smooth function on M and that the conformal change is such that the norm on T M given by the new metric is eu/2 times the norm as defined by the original metric. Let · denote the original version of (2.5) and let · u denote the version that is defined by the new metric. Proposition 6.2. Let u be a smooth function on M. Then   ˆ (z,F) u =  e− E k σ u(z k )/4π  (z,F) .

(6.4)

1≤k≤N

Note that this proposition verifies that · satisfies the axiom that is given by Eq. (2.4) in [Ga]. By the way, the requirement that is stated in Eq. (2.32) of [Ga] follows from Eqs. (2.2) and (2.4) of [Ga]; thus it is also satisfied by · . The next two propositions describe the change in · with a first order change in the metric that is neither a conformal change nor one that is tangent to the metric’s diffeomorphism group To make this notion precise, let m denote a given symmetric orbit. section of T ∗ M T ∗ M, and consider the 1-parameter family of metrics parametrized by a small real number, τ , and defined so that the square of the τ version norm is obtained from the original by adding τ m(·, ·). Such a 1-parameter family of metrics provides the corresponding 1-parameter families of versions of (2.5), and a τ version is denoted in what follows as · τ . I know no general formula for the τ -dependence of these linear forms. However, with z and F fixed, then the function τ → (z,F) τ is an analytic function of τ near τ = 0 when the support of m is disjoint from the set of points that define the components of z. This follows from (2.8) and (2.9) using perturbation theory to analyze the τ dependence of the Green’s function for the Laplacian on M. In particular, the derivatives of (z,F) τ with respect to τ can be readily computed at τ = 0 for such m. The axioms for a conformal field theory make demands on the behavior of these derivatives as the support of m approaches some component of z. To compare the singular behavior for the first derivative with that required for a conformal field theory, it proves d useful to write the derivative dτ (z,F) τ |τ =0 as if its value was that of the L 2 inner product between the symmetric tensor m and a certain section over M − ϑ ∪ {z k }1≤k≤N of T ∗ M T ∗ M that extends to all of M as a distribution. The latter section is symmetric and traceless, and so determined by its type (1, 0)2 portion; this as defined by the complex structure on M that comes from the τ = 0 metric. The type (1, 0)2 part of this 1 1 distribution at w ∈ M is denoted in what follows by 4π t(w)(z,F) − 4π t(w) (z,F) . 1 The factors of 4π are traditional in the conformal field theory literature.

Proposition 6.3. The assignment (w, z) → t(w), (z,F) − t(w) (z,F) defines a holomorphic section over the complement of the diagonals in(M − ϑ) × (× N (M − ϑ)) of the pull-back via projection to the first factor of M of 2 T 1,0 M. Moreover, this section is equivariant with respect to the action of the group of orientation preserving diffeomorphisms of M. The following is also true: Let k ∈ {1, . . . , N } and let x denote a holomorphic coordinate for a neighborhood of z k in M such that x = 0 is mapped to 2 2 z k and such that the given

back as |d x| to order |x| . Then the pull-back metric pulls at x of the section w → t(w), (z,F) − t(w) (z,F) has the form

48

C. H. Taubes

1 ˆ 1 1 ∂ σ Ek 2 + 4π x x ∂z k

(z,F) (d x)2 + . . . ,

(6.5)

where the three dots indicate terms that are bounded as x → 0. This proposition verifies that · is compatible with the conditions that are required by Eqs. (2.26) and (2.28)–(2.31) in [Ga]. The axioms for a conformal field theory also make demands on the

τ second derivatives and higher order derivatives at τ = 0 of the function τ → (z,F) . These can also be verified in the present case. The proposition that follows summarizes the story for the second derivatives. To conform with convention, the second derivative at τ = 0 of the τ function τ → (z,F) is written as if it were obtained by integrating m ⊗ m against a tensor valued distribution on M × M. The latter is determined by its (1, 0)2 ⊗ (1, 0)2 and (1, 0)2 ⊗ (0, 1)2 parts. The (1, 0)2 ⊗ (1, 0)2 part at a point (w, w ) ∈ M × M is denoted by the rather cumbersome 2

1 t(w)t(w )(z,F) − t(w) t(w )(z,F) − t(w ) t(w)(z,F) 4π

+ t(w) t(w ) (z,F) . (6.6) Meanwhile, the (1, 0)2 ⊗ (0, 1)2 part at a point (w, w ) is denoted by the analog of (6.6) that is obtained by replacing each occurrence of t(w ) with ¯t(w ). (The notation here best conforms to that used by Gawedzki.) The axioms of conformal field theory specifically require certain singularities in these distributions along the diagonal in M × M, these implied by Eq. (2.27) in [Ga]. The next proposition asserts that the desired singularities do exist. Proposition 6.4. Suppose that w ∈ M is distinct from all components of z. Let x denote a holomorphic coordinate on a neighborhood of w such that x = 0 is mapped to w and such that the given metric pulls back as |d x|2 + O |x|2 . Then the pull-back at x = 0 of the section depicted in (6.6) has the form

2 1 ∂ t (·) (z,F) − t(·) (z,F) |w (d x)2 + . . . , + (6.7) 2 x x ∂w where the three dots indicate terms that are bounded as x → 0. Meanwhile, the pull-back of the joint t(w) and ¯t(w ) version of (6.6) has no singularity at x = 0. The following are true and not hard to prove: The section that is denoted by (6.6) is jointly holomorphic with respect to both w and w . Meanwhile, the joint t(w) and ¯t(w ) version of (6.6) describes a section that is holomorphic with respect to w and anti-holomorphic with respect to w . The remainder of this section is occupied with the proofs of the preceding four propositions. Proof of Proposition 6.1. The assertion follows because the pull-back of aˆ is the solution to the pull-back metric’s version of ( 6.1) as defined using the set ϑ . Proof of Proposition 6.2. The proof begins by describing how the Green’s function, a, ˆ changes when the metric on M is changed by a conformal transformation. For this purpose, let aˆ denote the Green’s function for the original metric and let aˆ u denote the


49

Green’s function for the new metric. These two functions on M × M are related as follows: aˆ u z, z = aˆ z, z + q u (z) + q u (z ), (6.8) where q u (z) =

1 2−n 2 area(M)

area(M) u(z ) aˆ z, z 1 − dz + k u e u (M) area M

(6.9)

with k u a suitable constant. Here, the integration u measure dz is that defined by the u original metric. Meanwhile, area (M) ≡ M e dz denotes the area of M as computed using the conformally transformed metric.

For each k ∈ {1, . . . , N }, let Vk denote the eigenspace for the Laplacian on P that contains the function f k ; then set V ≡ V 1 × . . . × V N . With V understood, agree now that Aˆ z and Aˆ uz denote the respective old and new versions of the endomorphism of V that appears in (2.8) and (2.9). The claimed relation between · and · u asserts that exp Aˆ z F ( p, . . . , p)dp V P   ˆ − E k σ u(z k )/4π   exp Aˆ uz = e F ( p, . . . , p)dp. V

P

1≤k≤N

(6.10) To establish (6.10), note that (6.10) and the formula for c given by (6.3) imply that

Aˆ uz = Aˆ z + 2σ

 

1≤a≤d

+

1≤i≤N



q u (z i )∂ai  

1≤i≤N

1 σ u(z i ) 4π

 ∂ak 

1≤k≤N

∂ai ∂ai .

(6.11)

1≤a≤d

Now, keep in mind that each version of ∂ai ∂ai commutes with any given ∂ak . As a consequence, the rightmost term on the right-hand side of the equality in (6.7) can be replaced by − 1≤k≤N Eˆ k σ u(z k )/4π . This replacement accounts for the factor that multiplies the integral on the right-hand side of (6.10). This understood, (6.8) follows with a proof that the left-hand side of (6.10) is unchanged when Aˆ z is replaced by Aˆ z ≡ Aˆ z + 2σ

 

1≤a≤d



q u (z i )∂ai  

1≤i≤N

1≤k≤N

To prove that such is the case, reintroduce the collection   

fa ≡

1≤k≤N

∂ak

   1≤a≤d

 ∂ak  .

(6.12)

50

C. H. Taubes

of operators that appear in Lemma 2.1, but viewed as endomorphisms of V. To keep the formula that follows relatively uncluttered, introduceas shorthand ba to denote each u i a ∈ {1, . . . , d} version of 2 1≤a≤d 1≤i≤N q (z i )∂a . Then 1 exp( Aˆ z )V = exp( Aˆ z )V + 2σ exp (1 − τ ) Aˆ z ba fa exp(τ Aˆ z )V dτ. V

0

1≤a≤d

(6.13) Now, each fa commutes with the corresponding ba , and with both Aˆ z and Aˆ z . This understood, then (6.13) implies that exp Aˆ z F ( p, . . . , p)dp V P = exp Aˆ z F ( p, . . . , p)dp + fa Ua dp, (6.14) P

where

1

Ua ≡ 2σ 0

V

1≤a≤d

P

exp (1 − τ ) Aˆ z ba exp(τ Aˆ z )V F ( p, . . . , p)dτ. V

(6.15)

The proof ends by using (2.10) to prove the the rightmost sum in (6.14) is zero.

Proof of Proposition 6.3. To derive a formula for t(·)(z,F) − t(·) (z,F) , digress for a minute to study the τ dependence of the Green’s function on M. For this purpose, note that the small τ metric’s version of the Laplacian on M has the form 1 τ = − τ m¯ − ∗ m 0 (·, ) + O(τ 2 ). (6.16) 2 The notation here uses to denote the τ = 0 Laplacian, the τ = 0 covariant derivative and ∗ its adjoint. Thus, = − ∗ . Meanwhile, m¯ and m 0 are the trace of m and the trace free part of m, these as defined with respect to the τ = 0 metric. Assume in what follows that m = m 0 ; this because the trace of m defines a deformation of the τ = 0 metric in its conformal equivalence class. By virtue of (6.16), the small τ metric’s Green’s function can be written as " # τ

m|w , (d 2 a)| ˆ z )+τ ˆ (z,w) ⊗ d 1 aˆ |(w,z ) dw + O(τ 2 ), (6.17) aˆ (z, z ) = a(z, M ∗ T M. where the notation uses ·, · to denote the τ = 0 metric’s inner product on T ∗ M Meanwhile, d 1 denotes the exterior derivative along the first factor in M × M while d 2 denotes the exterior derivative along the second. It follows as a consequence of (6.17) that

t(·)(z,F) − t(·) (z,F) %(1,0)2 $ d 2 a| = 4π σ ˆ (zi ,w) ⊗ d 1 a| ˆ (w,z k ) 1≤i,k≤N

× P

here

2 [·](1,0)



1

 exp(1 − τ ) Aˆ z

0

denotes the type

V

(1, 0)2

 ∂ai ∂ak exp(τ Aˆ z )V F  ( p, . . . , p)dτ dp,

1≤a≤d

part of the given section of

T ∗M

(6.18) T ∗ M.


51

With (6.18) in hand, the first point to make is that with z fixed, then (6.18) defines a holomorphic section over M − ϑ ∪ {z k }1≤k≤N of the square of the holomorphic cotangent bundle. To see why such is the case, note that the exterior derivative on M can ¯ where ∂ is the projection of the exterior derivative onto the (1, 0) be written as ∂ + ∂, summand in T ∗ MC and ∂¯ is the projection to the (0, 1) summand. Furthermore, if h is any function on M , then the 2-form that is obtained by multiplying the area form by ¯ h is equal to 4∂∂h. This understood, it then follows that $ %(1.0)2 1 ¯∂ d 2 a| ˆ (zi ,w) ⊗ d a| ˆ (w,z k ) =

1 2 a| ˆ (zi ,w) ⊗∂ 1 a| ˆ (w,z k ) +∂ 2 a| ˆ (zi, w) ⊗1 a| ˆ (w,z k ) . 4

(6.19)

Here, ∂ 1 aˆ and ∂ 2 aˆ denote the projections of d 1 aˆ and d 2 aˆ onto the (1, 0) parts of the cotangent bundle of the relevant factor of M. Granted (6.1), this last equation implies that

∂¯ t(·)(z,F) − t(·) (z,F) |w % $ 1 ∂ 2 a| = π σ (1 − n) ˆ (z k ,w) + ∂ 1 a| ˆ (w,z k ) 2 1≤k≤N   1  exp(1−τ ) Aˆ z × fa ∂ak exp(τ Aˆ z )V F ( p, . . . , p)dτ dp, (6.20) P 0

V 1≤a≤d

where fa ≡ 1≤i≤N ∂ai . Since fa commutes with each ∂ai and also with Aˆ z , itfollows that the expression on the right-hand side of (6.20 ) can be written as 1≤a≤d P fa Ua ( p, . . . , p)dp, where each Ua is a function on × N P with values in T 1,0 M. This understood, the vanishing of the right-hand side of (6.20) follows from Lemma 2.1. To establish (6.5), view a(z ˆ 1 , w) with z 1 fixed and w near z 1 as a function, aˆ 1 , of x. 1 Likewise, view a(w, ˆ z 1 ) as a function, aˆ 2 , of x. Of course both differ from − 2π ln (|x|) by a function that is smooth near x = 0. Note, however that the smooth terms in aˆ 1 and aˆ 2 need not be the same. In any event, the exterior derivative of aˆ 1 is what appears as d 2 aˆ in (6.18), while that of aˆ 1 appears in (6.18) as d 1 a. ˆ These derivatives have the form 1 1 d aˆ ν=1,2 = − ¯ (6.21) − αν d x − − α¯ ν d x, 4π x 4π x¯ where α1 and α2 are smooth in a neighborhood of x = 0. It follows directly from (6.21) that the most singular portion of the x → 0 limit of the expression in (6.18) is given by

1 1 ˆ σ E 1 (z,F) 2 (d x)2 . 4π x

(6.22)

To identify the lower order but divergent term that appears in (6.18) in the limit as w → z 1 , note that the terms α1 and α2 that appear in (6.21) sum at x = 0 to give the differential of c/σ at the point z 1 . Granted this identification, the remaining divergences in (6.18) have the form of 1 υ(d x)2 , (6.23) x

52

C. H. Taubes

where υ is defined to be ∂ 1 a| ˆ (z 1 ,z k ) + ∂ 2 a| ˆ (z k ,z 1 ) Eˆ 1 ∂c − σ 2≤k≤N



1

× P

0

 exp(1 − τ ) Aˆ z

V

 ∂a1 ∂ak exp(τ Aˆ z )V F  ( p, . . . , p)dτ dp.

1≤a≤d

(6.24) Here, ∂c denotes the projection of the exterior derivative of c at z 1 onto the (1, 0) portion of T ∗ M, and ∂ 1 aˆ and ∂ 2 aˆ are, as before, the analogous projections for the forms d 1 aˆ and d 2 a. ˆ It is left as an exercise for the reader to verify that the expression in ( 6.24) is identical to the (1, 0) portion of the differential at z 1 of (z,F) when the latter is considered as a function of the first component of z with the remaining components fixed. Proof of Proposition 6.4. The appearance of the required singular behavior near the diagonal comes from the second of a τ at τ = 0 that appear via the chain derivatives

τ rule when computing those of (z,F) . In particular, they come from the τ -derivative & '(1,0)2 at τ = 0 of the expression d 2 aˆ τ |(z 1 ,w) ⊗ d 1 aˆ τ |(w,z k ) ; this is the a τ analog of the term that appears before the integral in (6.18). This understood, (6.17) can be used to compute the latter derivative. Algebraic manipulation of the resulting expression then gives (6.7). The assertion about the t and ¯t version of (6.6) is proved with this same τ derivative calculation after an appeal to (6.1). The detailed manipulations are reasonably straightforward in both cases and left to the reader. 7. Remarks on Conformal Field Theories The remaining properties for a conformal field theory make demands that require a Hilbert space assignment to a surface with boundary. It is not clear whether these other properties are satisfied. For example, a conformal field theory must assign a vector in Theorem 3.2’s Hilbert space to a surface with a connected boundary and a marked boundary point. It is not clear that the right sort of assignment exists in the case at hand. A conformal field theory also requires that Theorem 3.2’s contraction semi-group consists of trace class operators at positive times. It is not known whether this last condition is satisfied. Note that these are not the only questionable properties. The first subsection that follows elaborates on this manifold with boundary issue. The subsequent subsection initiates a study of the spectrum of the generator of Theorem 3.2’s contraction semi-group. On the positive side of the ledger, there is a pair of unitary S L(2; R) actions on Theorem 3.2’s Hilbert space that fits appropriately into the conformal field theory story. These actions are briefly described in the final subsection. a) Manifolds with boundary. Suppose that M0 is a compact Riemann surface with connected boundary whose metric is flat on some neighborhood of the boundary. I shall also assume that the boundary circle is totally geodesic and has length 2π . Suppose that a fiducial point has been specified in the boundary circle. This data is supposed to yield a vector in Theorem 3.2’s Hilbert space via a prescription that is described momentarily.


53

As remarked at the outset, it is not clear at present whether this prescription can be implemented here. The description of the desired vector assignment follows in four parts. Part 1. The given manifold with boundary can be viewed in a canonical way as the complement of disk in a compact manifold. To see how this comes about, note that there exists a positive number, ε, and coordinates, (t, θ ), on a neighborhood of the boundary with t ∈ (−ε, 0] and θ ∈ R/(2π Z) with the following properties: First, the boundary is the t = 0 slice. Second, the metric appears in these coordinates as dt 2 + dθ 2 . Finally, the point θ = 0 corresponds to the fiducial boundary point. Note that these coordinates are canonically associated with the given data. Now identify this neighborhood of the boundary with the portion of C where the holomorphic coordinate, u, has norm greater than or equal to 1 and less than eε by the map that sends (t, θ ) to u = e−(t+iθ) . Use this identification to glue the |u| ≤ 1 disk in the Riemann sphere C to M0 . Let M denote the resulting compact manifold and w ∈ M the point that corresponds to the origin in the attached disk. I use D ⊂ M in what follows to denote the attached disk. Take the metric on D to be flat and such that the boundary of M is the circle of radius 1. Extend this metric over the rest of M so as to be conformal on M0 to the given metric. Part 2. Suppose next that N is a positive integer and z ∈ × N (R × S 1 ) is a point with pairwise distinct entries and such that each entry has positive first coordinate. The gluing of the |u| ≤ 1 portion of C to M0 identifies z with a point in × N (D − w), and thus with a point in × N (M − w). Meanwhile, let F ∈ C ∞ (× N P) denote a function of the usual sort, one that assigns a given point p = ( p1 , . . . , p N (w) ) to f 1 ( p1 ) . . . f N ( p N ), where each f k is an eigenfunction for the Laplacian on P. The pair z and F defines a function, (z,F) in M’s version of the linear space F. Of course, with z viewed in × N (R×S 1 ), the pair (z, F) also defines an element in Theorem 3.2’s space F++ . Part 3. Let aˆ M denote the Green’s function on M for the data given by the metric and the one marked point, w. Thus, aˆ obeys the following version of (6.1): 1 1 M aˆ z = δz − δw + . (7.1) 2 area(M) Suppose that both z and z are distinct points in D − w. This understood, both points can be viewed simultaneously as points in the |u| < 1 part of C and so aˆ M (z, z ) can be written as u − u 1 1 M

2 2 |u| u aˆ (z, z ) = − ln 1/2 + s(u, u ). (7.2) − + 2π |u| |u |1/2 4area(M) Here, u and u are the respective images in C of z and z and s is the real part of a symmetric, holomorphic function on the radius 1 polydisk in C × C. As can be seen from (7.2), the function c : M → R that defines M’s function ac in (2.9) pulls back to C − 0 from D − w so as to send the holomorphic coordinate u to 1 1 |u|2 + s(u, u). ln |u| − 2π 2area(M)

(7.3)

54

C. H. Taubes

Part 4. Let · Mdenote the version of the linear form in (2.5). Meanwhile, let Eˆ k denote the value of − 1≤a≤d ∂a ∂a on the function f k . The assignment of

M ˆ ec(z k ) E k (z,F)

(7.4)

1≤k≤N

to the pair (z, F) extends as a linear functional on Theorem 3.2 ’s vector space F++ . This functional is denoted by J in what follows. Pretend for the moment that J annihilates the kernel of Theorem 3.2’s bilinear form Q. Then J define a linear functional on a dense domain in the Hilbert space H. If this functional on H is bounded, then it defines a vector in H. The latter is the required conformal field theory vector. The issue of whether J annihilates the kernel of Q requires understanding the affect of the s(u, u ) term in (7.2) on the endomorphism exp( Aˆ z )V that appears in (2.9). b) The spectrum of the Hamiltonian. This section initiates a study of the Hamiltonian from Theorem 3.2 for the case where Y = S 1 , the function a has the form a = σ a, ˆ where aˆ is Green’s function for the Laplacian on R × S 1 , and the function c is given by (6.3). The conclusion here is that the spectrum of the Hamiltonian is not as simple as pure point with no accumulations. To begin, identify R × S 1 with C − 0 as in the previous subsection by first writing S 1 as R/(2π Z) and then mapping a given (t, θ ) ∈ R × R/(2π Z) to the point u = e−(t+iθ) in C. This identifies the Green’s function, aˆ with the function on ×2 (C − 0) that is depicted in (6.2). The function c in this case is identically zero. This version of a and c ≡ 0 are used in what follows to define · for R × S 1 and thus the Hilbert space in the corresponding version of Theorem 3.2. This Hilbert space is denoted below as H and the Hamiltonian by H. Thus, −H generates Theorem 3.2’s contraction semi-group. Supposing that τ ≥ 0, the action on H of exp(−τ H) is such that Q (z,F) , exp(−τ H)(z ,F ) =

exp( Aˆ (∗z,qz ) )V ( F¯ × F ) ( p, . . . , p)dp, P

(7.5) where q ≡ e−τ and where q ·z ≡ qz 1 , . . . , qz N . Here, z and z are points in × N (C−0) whose components have norm less than 1 and are pairwise distinct. Meanwhile, F and F are, as usual, functions on × N P that decompose as a product of eigenfunctions of the Laplacian from each factor. To say more about the q-dependence of the right-hand side of (7.5), let { f k }1≤n≤k denote the eigenfunction on P that define F and, for each k, let Vk denote the eigenspace

that f k . As usual, let V F denote V1 × . . . × Vk . Use F to define the analogous contains f k , Vk and V F . Thus, V = V F × V F . This done, note that σ Aˆ (∗z,qz ) = Aˆ z + Aˆ z − 2π

1≤i, j≤N

j ln (qz j z¯ i )1/2 − (qz j z¯ i )−1/2 ∂ai ∂a , (7.6) 1≤a≤d

where the notation is such that Aˆ z and the unprimed versions of ∂ai involve only the factor of V F in V; meanwhile A z and all ∂ai involve only the factor of V F in V.


55

To continue, note that the sums with q that appear in (7.6) can be written as j j ln(1 − qz j z¯ i ) ∂ai ∂a + ln(1 − q¯ z¯ j z i ) ∂ai ∂a

1≤i, j≤N



1≤a≤d

1≤i, j≤N

1≤i≤N



1≤a≤d

 j 1  1 j  − ln |z i | ∂ai ∂a  − ∂ai ln z j ∂a  2 2 1≤a≤d 1≤i≤N 1≤ j≤N 1≤a≤d 1≤i≤N 1≤ j≤N   j 1  − ln(q q) ¯ ∂ai ∂a  . (7.7) 2 1≤a≤d



1≤ j≤N

The introduction of q¯ is for reference later; q = q¯ for now. Note that the natural logs that appear in the two leftmost terms in (7.7) are defined by the power series expansion that writes ln(1 − b) = − n=1,2,... n −1 bn for points b ∈ C with norm less than 1. This understood, the part of (7.3) that is not analytic in q is a term that has the form   j σ  ln(q q) ¯ (7.8) ∂ai ∂a  . 4π 1≤a≤d

1≤i≤N

1≤ j≤N

ˆ This endomorphism of V is denoted in what follows as O.

ˆ ˆ ˆ As it turns out, O commutes with A z , A z and the two leftmost terms in (7.7). However, Oˆ does not commute with all of (7.7). This last conclusion has certain ramifications that will now be explored. To start, imagine for the moment that H has pure point spectrum with finite multiplicities and with no accumulation points. Were this the case, then (7.5) with q real could be written as a convergent series with each term a power of q, thus as n E q −E . (7.9) E

Moreover, the sum in (7.9) would be indexed by the set of eigenvalues of H; and each version of n E would be a q -independent constant. Just such a sum would arise were the operator Oˆ to commute with all of the terms in (7.7). To see this, first expand each of the natural logarithms that appear in the two leftmost terms of (7.7) as respective power series in q and q. ¯ This expansion allows the operator in (7.6) to be written as ˆ Aˆ z + Aˆ z + b(q, q) ¯ + O,

(7.10)

where b is the sum of a convergent power series in q and one in q. ¯ Use standard perturbation theory to expand the exponential of the latter as an infinite power series in b ˆ With Oˆ lacking, the resulting whose zeroth order term is exp( Aˆ z )V exp( Aˆ z )V exp( O).

expansion has the form 0≤n,n mn,n q −n q¯ −n , where the sum is over non-negative integer pairs (n, n ) and where the coefficients are independent of q and q. ¯ The presence of a commuting version of Oˆ replaces this last expansion with

un,n ,λ q −(n+λ) q¯ −(n +λ) , (7.11) 0≤n,n λ

56

C. H. Taubes

where λ ranges over the set of eigenvalues of Oˆ on V. The real q version of (7.11) has the form of the sum in (7.9). Now consider what happens when Oˆ does not commute with b . As is explained next, this leads to a version of (7.11) where certain coefficients u(·) are functions of τ that have convergent expansions at large τ in powers of 1/τ . To see how this comes about, again write the exponential of the endomorphism in ( 7.10) as a convergent power series in b. The first two terms in this expansion are ˆ exp( Aˆ z + Aˆ z + O) 1 exp (1 − s) Aˆ z + Aˆ z + Oˆ b exp s Aˆ z + Aˆ z + Oˆ ds. +

(7.12)

0

Of interest here is the rightmost term. In particular the latter can intertwine distinct eiˆ Suppose that such is the case and let µ1 and µ2 denote the genspaces of Aˆ z + Aˆ z + O. corresponding eigenvalues of Aˆ z + Aˆ z + Oˆ on the respective initial and final eigenspaces. The integral in (7.12) has the form of a power series in q and q¯ times eµ2 − eµ1 . µ2 − µ1

(7.13)

Granted this, suppose that the initial and final eigenspaces have distinct Oˆ eigenvalues. In this regard, remember that Oˆ commutes with Aˆ z and Aˆ z . Write the initial Oˆ eigenvalue as τ λ1 and the final Oˆ eigenvalue as τ λ2 with both λ1 and λ2 independent of τ . With τ large, (7.13) can be written as 1 α2 q λ2 − α1 q λ1 (1 + . . .), (7.14) τ λ2 − λ1 where the missing terms are O(τ −2 ). Here, α1 and α2 are constants that are determined by the eigenvalues of Aˆ z + Aˆ z on the respective initial and final eigenspaces. The power of 1/τ in (7.14) indicates that the spectrum of H is not pure point without accumulations. By the way, the existence of accumulations in the spectrum of H is suggested by the following observation: Consider the Y = S 1 case and a has the form depicted in the top line of (3.1) where the indexing set for the sum is the integers, each ηα (y) is exp(iαy), each non-zero version of E α is |α|, and E 0 = m with m positive but very small. The arguments just given prove that the function τ → Q((z,F) exp(−τ H)(z ,F ) ) for these m > 0 cases has a large τ expansion as depicted in (7.9), where each n E is constant. Moreover, the values of E that appear have the form of a sum of the elements from some finite set of elements chosen from the set {E α }α∈Z . In particular, finite sets with any given number of E 0 ’s can appear with a fixed number of nonzero α versions of E α . As a consequence, the set of such E sees eigenvalues accumulate as m → 0. To tie up a loose end, note that the introduction of q¯ in (7.7 ) makes sense in the context of the circle action on H that is induced by the group of rotations of the S 1 factor of R×S 1 . To be precise here, the action of ϕ ∈ R/(2π Z) on any given (z,F) ∈ F++ sends the latter to the vector (z ,F) , where z ∈ × N (C − 0) is obtained from z by multiplying the latter’s components by e−iϕ . This action preserves F++ and is isometric with respect to the bilinear form Q. In particular, it maps the kernel of Q to itself and so it descends as a circle action on H. As Q is preserved by the circle action on F++ , the


57

resulting action on H is unitary. As the action is strongly continuous, Stone’s theorem [St] finds it generated by an operator of the form −iP, where P is a self-adjoint operator on H. Note that P commutes with H because the circle action on F++ commutes with the action of the semi-group {Rτ }τ ≥0 . Supposing that τ > 0, then Q((z,F) , exp(−τ H − iϕP)(z ,F ) ) is given by the right-hand side of (7.5), where q is now e−(τ +iϕ) . Note that (7.6) holds now as does (7.7) but with q as just described. Thus q need not equal q. ¯ In principle, the expression in (7.11) can be used to study the spectrum of H ± P. c) A pair of group actions on H. What follows is meant as a brief description of a pair of actions on H of the universal covering group of S L(2; R). One of these actions is a bonafide S L(2; R) action; the other is not. The story starts with the group S L(2; C) and its standard action on the Riemann sphere, C ∪ ∞, as the group of conformal diffeomorphisms of the round metric. Let z ∈ R × S 1 denote a point with positive first component. As explained previously, z can be viewed as a point in C∗ with norm less than 1. This understood, there exists an open neighborhood, U ⊂ PSL(2; C), of the identity element such that each element in U maps z to a point in C∗ with norm less than 1. The neighborhood U depends only on |z|. When q ∈ U , use qz to denote the point where q sends z. Now let z ∈ (z 1 , . . . , z N ) ∈ × N (R × S 1 ) be a point with pairwise distinct components, each with norm less than 1 when R × S 1 is viewed as C∗ . Set Uz ⊂ S L(2; C) to denote the intersection of the versions of U that are defined by the components of z. When q ∈ Uz , use q · z to denote the point (qz 1 , . . . , qz N ). Meanwhile, let F ∈ C ∞ (× N P) denote the usual sort of function, one that decomposes so as to send p = ( p1 , . . . , p N ) to f 1 ( p1 ) . . . f N ( p N ), where each f k is an eigenfunction of the Laplacian on P . Let Eˆ k denote −1 times f k ’s eigenvalue. When q ∈ Uz , set ˆ e− E k σ u(z k )/4π (qz,F) . (7.15) T [q] · (z,F) ≡ 1≤k≤N

Here, u(·) is determined by q as follows: Write a b q= c d

(7.16)

and u(w) = ln(|aw − b| · |cw − d|) − ln |w|. Now let z ∈ × N (R × S 1 ) be a second point with pairwise distinct components, each with norm less than 1; and let F ∈ C ∞ (× N P) denote a function of the same sort as F. If q ∈ Uz ∩ Uz , then it follows from Proposition 6.2 that & ' Q(T q ∗ (z ,F ) , T [q] (z,F) ) = Q((z ,F ) , (z,F) ), (7.17) where q ∗ is q∗ ≡

d¯ b¯

c¯ a¯

(7.18)

when q is given by (7.16). Granted (7.18), agree to identify S L(2; R) ⊂ S L(2; C) with the subset of matrices as depicted in (7.16), where b = c, ¯ d = a. ¯ Elements of this sort have q ∗ = q. According to (7.17), the elements near the identity in S L(2; R) act unitarily on a domain in H.

58

C. H. Taubes

Moreover, the intersection of these domains is dense in H. This understood, an argument much like the one in Sect. 4 finds a unitary action of S L(2; R) on H with the operator iP as one of its generators. Meanwhile, there is an embedded R × S 2 in S L(2; C), where q ∗ = q −1 ; this is the subset of matrices with b = −c¯ with both a and d being real. Any such element near the identity in S L(2; C) acts in a hermitian fashion on a domain in H. Arguing as in Sect. 4, the square root of −1 times the tangent space at 1 to this R × S 1 generates a unitary action of the universal cover of S L(2; R) on H. One of the generators of this action is iH. These two actions are those mentioned at the start of this section. 8. Free Field Theories The purpose of this final section is to describe a linear functional on F that comes directly from a Gaussian measure on the space of maps from a given manifold M to the Lie algebra of P. This Gaussian measure can then be used to define a ‘free quantum field’ theory whose Hilbert space is a quotient of certain versions of the space F++ that appears in Theorem 3.2. The ensuing discussion has six parts. Part 1. Fix a manifold M and the group P. As before, use p to denote the Lie algebra of P. A Gaussian measure on Maps(M; p) is defined by a positive definite, integral kernel, q, on M × M with possibly singular behavior on the diagonal. Integrals with respect to the corresponding Gaussian measure are defined initially by their values on functions from Maps(M; p) to C of the following sort: Let h denote a continuous, complex valued, compactly supported map top. Now on Maps(M; p) let h denote the function that sends a given map φ to exp R×Y 1≤a≤d h a (z)φa (z)dz . Here, {h a }1≤a≤d are the components of h with respect to the basis of p that corresponds to the chosen basis, {∂a }1≤a≤d , of left invariant vector fields on P. The integral of h using the Gaussian measure is defined to be   exp  q(z 1 , z 2 )a1 ,a2 h a1 (z 1 )h a2 (z 2 )dz  . (8.1) M×M 1≤a a ≤d 1, 2

The integrals so defined are extended linearly to the vector space of finite linear combinations of functions such as h . These integrals are denoted below by (·). An example of interest for quantum field theory aficionados is the case M = R × Y and q equal to σ a, ˆ where aˆ is the Green’s function that is used in Sect. 3 to define · , and σ is a positive constant. Part 2. Let z = (z 1 , . . . , z N ) ∈ × N M denote a point with pairwise distinct entries. Meanwhile, let (a1 , . . . , a N ) ∈ × N {1, . . . , d}. As it turns out, functions on Maps(M; p) of the form φ → φa1 (z 1 ) . . . φa N (z N ) (8.2) are integrable. In fact, let f denote the function on × N p that sends a given N -tuple τ ≡ (τ1 , . . . , τ N ) to τ1a1 . . . τ N a N . Then φa1 (z 1 ) , . . . φa N (z N )     j q z i , z j a ,a ∂ai 1 ∂a2  f  . (8.3) = exp  1 2 1≤i≤ j≤N 1≤a1 ,a2 ≤d τ¯ =0


59

If q is Holder continuous across the diagonal, then the equality in (8.3) makes good sense even when components of z are not pairwise distinct (see, e.g. Theorem 3.2 in [T].) In this case, the formula on the right-hand side of (8.3) defines the Gaussian measure’s integral of any function on Maps(M; p) that sends φ to f (φ(z 1 ), . . . , φ(z N )) with f a smooth, bounded function on × N p. In particular, (8.3) makes good sense when f is the pull-back via a smooth map from × N p to × N P. This last observation leads to the ‘Gaussian measure’ on Maps(M; P) that is described in [T]. Part 3. Of particular interest here is the case where q is singular on the diagonal in M × M. A strategy employed in this case replaces q with qγ , where qγ (z, z ) = q(z, z ) when z = z and qγ (z, z) ≡ γ (z) with γ a continuous function on M. As the operator

hγ ,z ≡

1≤i≤ j≤N 1≤a1 ,a2 ≤d

j

qγ (z i , z j )a1 a2 ∂ai 1 ∂a2

(8.4)

is unbounded from below when z has a pair of components that are close in M, the qγ version of the exponential that appears in ( 8.3) can not be defined for such z as the time 1 element in a contraction semi-group on L 2 (M; × N p). This n issue is often avoided by 1 defining exp(hγ ,z ) f by the series H D60 n=0,1... n! hγ ,z f . Of course, the power series definition makes sense only for a very prescribed set of functions f since the series in question does not generally converge. For example, the definition makes sensewhen f is a polynomial in the various components n 1 of τ. For such f , the series n=0,1... n! hγ ,z f has but a finite set of non-zero terms. For a second example, note that the series in question converges when f ( τ) = , with each k exp k τ = (k , . . . , k ) a fixed vector in p. The ia ia i 11 id 1≤i≤N 1≤a≤d series in this case sums to   exp  qγ (z i , z j )a1 a2 kia1 k ja2  . (8.5) 1≤i≤ j≤N 1≤a1 ,a2 ≤d

A definition of the Gaussian integral of f (φ(z 1 ), . . . , φ (z N )) using qγ is what is called a ‘normal ordering’ prescription. Part 4. There are yet other functions f for which sense can be made of the right-hand side of (8.3), these relevant to Maps(M; P). To say more, suppose that f is a smooth, τ |2 ) f ( τ ) has a bounded function on × N p. When ε > 0, the function τ → exp(−ε | Fourier transform, a function on × N p that is denoted by tε f . Now, fix a point z ∈ × N M with pairwise distinct entries. Also, fix r > 0 and consider





qγ (z i , z j )a1 a2 kia1 kia2  (tε exp  N p:k 1, and by (3.15) with m positive when Y = R. Here, σ0 is a positive constant. Meanwhile, take γ to be a function that is pulled up from Y . In all these cases, the Gaussian expectation on F has the following properties: • It is translation invariant in that (Rτ ) = () for any given τ ∈ R and ∈ F, • It is ‘reflection positive’ in the sense that ((∗) ) is non-negative for any ∈ F++ . Use the bilinear form on F++ that is defined so as to send a given Q G to denote pair , to (∗) . The action on F++ of the 1-parameter semi-group {Rτ }τ ≥0 descends as a strongly continuous, self-adjoint, contraction semigroup to the Hilbert space that is obtained by completing F++ / ker(Q G ) using the norm that is defined by QG . This proposition follows from what is known about Gaussian measures. For example, it can be deduced using the results in Chapter 6 of [G-J]. Note that Proposition 8.2 says nothing about the cases where aˆ is given by either the bottom line in (3.1) or by the Green’s function 2 2 1 − ln t − t + y − y

4π


61

when Y = R. To say more about this absence, remark that the proof of Proposition 3.1 for these cases uses (2.10) at a key juncture. The steps that use (2.10) for the proof of Proposition 3.1 require here a different condition; this is the vanishing of each a ∈ {1, . . . , d} version of ∂ f 1 ( p exp(τ1 )) . . . f N ( p exp(τ N ))dp. (8.9) ∂τi,a P 1≤ j≤N

However, the expression in (8.9) is not in general equal to zero. As a parenthetical remark, one can try to circumvent this problem with the use of an alternate definition of (·) that builds in the required analog of (8.9). In particular, one can imagine a definition that starts by replacing (8.6) with   exp  qγ (z i , z j )a1 a2 kia1 k ja2  N p:k 0 take the case where 1 −E |t−t | in ( y −y ) e n (8.11) a (t, y) , t , y = e 2E n n∈Z

1/2 with E n = 0 for all n. For example, one can consider E n = n 2 + m 2 , or E n equal to |n| when n = 0 and E 0 = m. In any event, assume that {E n }n∈Z has no accumulation points. Take the function c to be identically zero. Suppose that N is a positive integer and that { f 1 , . . . , f N } is a set of eigenfunctions for the Laplacian on P. As usual F denotes the function on × N P that sends ( p1 , . . . , p N ) to f 1 ( p1 ) . . . f N ( p N ). Let z ∈ × N (R × S 1 ) denote a point with distinct components, each with positive R factor. The resulting function (z,F) on Maps(M; P) is in F++ and so defines an element in Theorem 3.2’s Hilbert space. Let −H again denote the generator of Theorem 3.2’s contraction semigroup. As noted for an example at the end of Sect. 7b, there exists a large τ expansion of the form Q((z,F) , exp(−τ H)(z,F) ) = n E e−Eτ , (8.12) E

where the values of E are the sums of the elements in some finite set chosen from {E α }α∈Z . Meanwhile, let HG denote the Hamiltonian for the case described by Proposition 8.2 where q = a and γ = 0. An argument very similar to that used in Sect. 7b finds that the function τ → Q G (z,F) , exp(−τ HG )(z,F) on [0, ∞) has a large τ , asymptotic

62

C. H. Taubes

expansion that also takes the form that is depicted in (8.12). Moreover, the values of E that appear for the Gaussian case come from the same set as those that appear in the case from Theorem 3.2. Note, however, that corresponding versions of n E can differ. Part 7. This last part of the section contains the P = SU (2) version of the Proof of Lemma 8.1. Fix an irreducible representation, V , for the group P. Such a representation determines an eigenspace for the Laplacian on SU (2); functions in this eigenspace have the form given in (5.1 ). The ψ p pull-back of (5.1) is †

τ → η p ρV (exp (τ )) ν,

(8.13)

here η p is shorthand for ρV ( p)η. Lemma 8.1 follows from the claim that the ε → 0 limit of positive ε versions of the function   † k → exp i ka τa − ε |τ |2  η p ρV (exp (τ )) νdτ (8.14) p

1≤a≤d

is a compactly supported distribution on p. Since the function in (8.13) is smooth and bounded on p, the ε → 0 limit of the function in (8.14) defines a distribution on p. This understood, the issue is whether the distribution has compact support. The arguments that follow establish that such is the case when P = SU (2). To start, suppose that k = 0, and let oˆ 3 ∈ su(2) denote a unit vector that is tangent to k. Now introduce oˆ + and oˆ − to denote vectors in the complexified Lie algebra with the following properties: They are orthogonal to oˆ 3 , they have norm squared equal to 21 , they ' & † , and oˆ 3 , oˆ ± = ±2i oˆ ± . Use r ∈ R and w ∈ C to parameterize are such that oˆ + = oˆ − su(2) via the map that sends a given (z, w) to r oˆ 3 + w oˆ + − w¯ oˆ − . Consider now when V = C2 is the fundamental representation. The representation sends oˆ ± to matrices with square 0. As a consequence, ρC2 (exp(τ )) = exp(ir oˆ 3 ) 1 +

1 0

ds we−ir s oˆ + − we ¯ ir s oˆ − + . . . ,

(8.15)

where the (2n + 1)st term is |w|2n we−ir (s1 −s2 +s3 −...+s2n+1 ) oˆ + oˆ − . . . oˆ + 2n+1 S∈

and the 2n th term is

− we ¯ ir (s1 −s2 +s3 −...+s2n+1 ) oˆ − oˆ + . . . oˆ − ds1 . . . ds2n+1

(8.16)

|w|2n e−ir (s1 −s2 +s3 −...+s2n+1 ) oˆ + oˆ − . . . oˆ − 2n S∈ − eir (s1 −s2 +s3 −...+s2n+1 ) oˆ − oˆ + . . . oˆ + ds1 . . . ds2n .

(8.17)

Here, k is the k-dimensional simplex where 0 ≤ sk ≤ . . . ≤ s1 ≤ 1.


63

The term depicted in (8.16) contribute zero to (8.14). Meanwhile, the term in (8.17) contributes π 3/2 n! 1 † 2 exp − + s − . . . + s ds1 . . . ds2n η p πˆ + ν + 1 − s ) (κ 1 2 2n εn+3/2 S∈ 4ε 2n π 3/2 n! 1 † 2 + n+3/2 exp − (κ − 1 + s1 − s2 + . . . − s2n ) ds1 . . . ds2n η p πˆ − ν. 2n ε 4ε s∈ (8.18) Here, κ is defined so that k = κ oˆ 3 , and πˆ ± are the respective projections onto the ±i eigenspaces of oˆ 3 . To continue, suppose that κ ≥ 2. In this case, (κ ± (1 − s1 + s2 − . . . + s2n )) has size at least 1. Under this condition, any given version of (8.18) has limit zero as n → ∞. Even so, it is important to consider the convergence of their sums. For this purpose, remark that the integrals in (8.18) are no greater than a κ and n independent multiple of (κ − 1)−(2n+3) n! (n + 3/2)n e−n / (2n)!.

(8.19)

Here, factor 1/(2n)! is the volume of 2n . Meanwhile, Stirling’s approximation finds the expression in (8.19) on the order of 2−n (κ − 1)−2n−3 for large n. What with any given integer n version of (8.18) limiting to zero as ε → 0, this last observation implies that (8.14) limits uniformly to zero as ε → 0 in both the C0 and L 1 topology on the complement of the radius 2 ball in su(2). The case where dim(V ) > 2 can be handled using the following View C2 as device: 2 the defining representation of SU (2). Fix m ≥ 1 and let W ≡ m C . View W as a representation of SU (2) via the simultaneous action of any given element on all of the k summands. This representation is not irreducible, but any given irreducible representation is contained as a summand in some large m version of W . Granted this, it follows that the ε → 0 limit of (8.14) has compact support as a distribution in the variable k if such is the case for the analogous limit where ρV is replaced by ρW in the case where W contains V as a summand. To see that such is the case, it is enough to consider the case; where, η and ψ are decomposable, thus η = η1 ⊗ . . . ⊗ ηm and ψ = ψ1 ⊗ . . . ⊗ ψm . For such η and ψ, −ε |τ |2 p† 2 p† exp(−ε |τ | )η ρW (exp(τ ))ψ = ×1≤i≤m exp ηi ρC2 (exp (τ )) ψi . m (8.20) This implies that the Fourier transform of the function on the left-hand side of (8.20) is an m-fold convolution of Fourier transforms of the sort just computed for the representation C2 . As the latter have ε → 0 limit equal to zero in both the C 0 and L 1 topologies in the complement of the radius 2 ball in su(2), it follows that the Fourier transform of the function on the left-hand side of (8.20) limits to zero as ε → 0 in both the C 0 and L 1 topologies in the complement of the radius 2(m + 1) ball in su(2). References [Ga] [GJ]

Gawedzki, K.: Lectures on conformal field theory and strings. In: Quantum Fields and Strings: A Course for Mathematicians, Providance, RI: Amer. Math. Soc., 1999 Glimm, J., Jaffe, A.: Quantum Physics. New York: Springer-Verlag, 1981

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[H] [HP] [Ka] [OS] [R] [Se1] [Se2] [St] [SV] [T]

C. H. Taubes

Hörmander, L.: Pseudo-differential operators and hypoelliptic equations. Proc. Symp. Pure Math. 10, Providance, RI: Amer. Math. Soc., 1966, pp. 138–183 Hille, E., Phillips, R.S.: Functional Analysis and Semigroups. AMS Colloq. Publ. 31, Providance, RI: Amer. Math. Soc., 1957 Kato, T.: Perturbation Theory of Linear Operators. New york: Springer-Verlag, 1984 Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions I. Commun. Math. Phys. 42, 83–112 (1973) Royden, H.L.: Real analysis. Third edition. New york: Macmillan Publishing Company, 1988 Segal, G.: Two-dimensional conformal field theories and modular functon. In: IXth International Congress on Mathematical Physics (Swansea 1988), Bristol: Adam Hilger Pub., 1989, pp. 22–37 Segal, G.: The definition of conformal field theory. In: Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., 308, Cambridge: Cambridge Univ. Press, 2004, pp. 421–577 Stone, M.H.: On one parameter unitary groups in Hilbert space. Ann. Math. 33, 643–648 (1932) Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. New York: Springer-Verlag, 1979 Taubes, C.H.: Constructions of measures and quantum field theories on mapping spaces. J. Diff. Geom. 70, 23–57 (2005)

Communicated by A. Connes


Communications in


Quantum States on Harmonic Lattices Norbert Schuch, J. Ignacio Cirac, Michael M. Wolf Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany. E-mail: [email protected] Received: 5 October 2005 / Accepted: 23 March 2006 Published online: 15 July 2006 – © Springer-Verlag 2006

Abstract: We investigate bosonic Gaussian quantum states on an infinite cubic lattice in arbitrary spatial dimensions. We derive general properties of such states as ground states of quadratic Hamiltonians for both critical and non-critical cases. Tight analytic relations between the decay of the interaction and the correlation functions are proven and the dependence of the correlation length on band gap and effective mass is derived. We show that properties of critical ground states depend on the gap of the point-symmetrized rather than on that of the original Hamiltonian. For critical systems with polynomially decaying interactions logarithmic deviations from polynomially decaying correlation functions are found. 1. Introduction The importance of bosonic Gaussian states arises from two facts. First, they provide a very good description for accessible states of a large variety of physical systems. In fact, every ground and thermal state of a quadratic bosonic Hamiltonian is Gaussian and remains so under quadratic time evolutions. In this way quadratic approximations naturally lead to Gaussian states. Hence, they are ubiquitous in quantum optics as well as in the description of vibrational modes in solid states, ion traps or nanomechanical oscillators. The second point for the relevance of Gaussian states is that they admit a powerful phase space description which enables us to solve quantum many-body problems which are otherwise (e.g., for spin systems) hardly tractable. In particular, the phase space dimension, and with it the complexity of many tasks, scales linearly rather than exponentially in the number of involved subsystems. For this reason quadratic Hamiltonians and the corresponding Gaussian states also play a paradigmatic role as they may serve as an exactly solvable toy model from which insight into other quantum systems may be gained. Exploiting the symplectic tools of the phase space description, exact solutions have been found for various problems in quantum information theory as well as in quantum

66

N. Schuch, J. I. Cirac, M. M. Wolf

statistical mechanics. In fact, many recent works form a bridge between these two fields as they address entanglement questions for asymptotically large lattices of quadratically coupled harmonic oscillators: the entropic area law [1–3] has been investigated as well as entanglement statics [4–6], dynamics [7–9] and frustration [10, 11]. In the present paper we analytically derive general properties of ground states of translationally invariant quadratic Hamiltonians on a cubic lattice. We start by giving an outlook and a non-technical summary of the main results. The results on the asymptotic scaling of ground state correlations are summarized in Table 1. We note that related investigations of correlation functions were recently carried out in [12, 13] for finite dimensional spin systems and in [1, 14] for generic harmonic lattices with non-critical finite range interactions. Quadratic Hamiltonians. In Sect. 2, we start by introducing some basic results on quadratic Hamiltonians together with the used notation. Translationally invariant systems. In Sect. 3, we show first that every pure translational invariant Gaussian state is point symmetric. This implies that the spectral gap of the symmetrized rather than the original Hamiltonian determines the characteristic properties of the ground state. We provide a general formula for the latter and express its covariance matrix in terms of a product of the inverse of the Fourier transformed spectral function and the Hamiltonian matrix. Non-critical systems. Section 4 shows that if the Hamiltonian is gapped, then the correlations decay according to the interaction: a (super) polynomial decay of the interaction leads to the same (super) polynomial decay for the correlations, and (following Ref. [1]) finite range interactions lead to exponentially decaying correlations. Correlation length and gap. Section 5 gives an explicit formula for the correlation length for gapped 1D-Hamiltonians with finite range interactions. The correlation length ξ is expressed in terms of the dominating zero of the complex spectral function, which close Table 1. Summary of the bounds derived in the paper on the asymptotic scaling of ground state correlations, depending on the scaling of the interaction (left column). Here n is the distance between two points (harmonic oscillators) on a cubic lattice of dimension d. O denotes upper bounds, O ∗ tight upper bounds, and the exact asyptotics. The table shows the results for generic interactions—special cases are discussed in the text interaction local

non-critical O ∗ e−n/ξ d = 1: ξ ∼ √ 1

m ∗

O n −∞ 1 O α n α > 2d + 1 c nα d=1

critical v =

O n −∞ O

1

1 n2 log n

d = 1 : O∗ d >1: O

n d+1

n ν−d α>ν∈N

1 α≥2: α n

  12 ,c>0 n α =3: √  log n , c < 0 n2 1 α >3: 2 n

Quantum States on Harmonic Lattices

67

to a critical point is in turn determined by the spectral gap and the effective mass m ∗ at the band gap via ξ ∼ (m ∗ )−1/2 . When the change in the Hamiltonian is given by a global scaling of the interactions this proves the folk theorem ξ ∼ 1/. Critical systems. Section 6 shows that for generic d-dimensional critical systems the correlations decay as 1/n d+1 , where n is the distance between two points on the lattice. Whereas for sufficiently fast decreasing interactions in d = 1 the asymptotic bound is exactly polynomial, it contains an additional logarithmic correction for d ≥ 2. Similarly for d = 1 a logarithmic deviation is found if the interaction decays exactly like −1/n 3 . 2. Quadratic Hamiltonians and their Ground States Consider a system of N bosonic modes which are characterized by N pairs of canonical operators (Q 1 , P1 , . . . , Q N , PN ) =: R. The canonical commutation relations (CCR) are governed by the symplectic matrix σ via N

0 1 , Rk , Rl = iσkl , σ = −1 0 n=1

and the system may be equivalently √ described in terms of bosonic creation and annihilation operators al = (Q l + i Pl )/ 2. Quadratic Hamiltonians are of the form H=

1 Hkl Rk Rl , 2 kl

where the Hamiltonian matrix H is real and positive semidefinite due to the Hermiticity and lower semi-boundedness of the Hamiltonian H. Without loss of generality we neglect linear and constant terms since they can easily be incorporated by a displacement of the canonical operators and a change of the energy offset. Before we discuss the general case we mention some important special instances of quadratic Hamiltonians: a well studied 1D example of this class is the case of nearest neighbor interactions in the position operators of harmonic oscillators on a chain with periodic boundary conditions 1 2 Q i + Pi2 − κ Q i Q i+1 , κ ∈ [−1, 1]. 2 N

Hκ =

(1)

i=1

This kind of spring-like interaction was studied in the context of information transfer [7], entanglement statics [4–6] and entanglement dynamics [9]. Moreover, it can be considered as the discretization of a massive bosonic continuum theory given by the Klein-Gordon Hamiltonian 1 L/2 ˙ 2 + φ(x) 2 + m 2 φ(x)2 d x, HKG = φ(x) 2 −L/2 2 where the coupling κ is related to the mass m by κ −1 = 1+ 21 mNL [5]. Other finite range quadratic Hamiltonians appear as limiting cases of finite range spin Hamiltonians via the Holstein–Primakoff approximation [15]. In this way the x y-spin model with transverse magnetic field can for instance be mapped onto a quadratic bosonic Hamiltonian in the limit of strong polarization where a (σx + iσ y )/2. Longer range interactions appear naturally for instance in 1D systems of trapped ions. These can either be implemented as

68


Coulomb crystals in Paul traps or in arrays of ion microtraps. When expanding around the equilibrium positions, the interaction between two ions at position i and j = i is— cQ Q in harmonic approximation—of the form |i−i j|3j , where c > 0 (c < 0) if Q i , Q j are position operators in radial (axial) direction [16]. Let us now return to the general case and briefly recall the normal mode decomposition [17]: every Hamiltonian matrix can be brought to a diagonal normal form by a congruence transformation with a symplectic matrix S ∈ Sp(2N , R) = {S|Sσ S T = σ }:1

I J

εi 0 0 0 ⊕ , εi > 0, SH S = 0 εi 0 1 T

i=1

(2)

j=1

where the symplectic eigenvalues εi are the square roots of the duplicate nonzero eigenvalues of σ H σ T H . The diagonalizing symplectic transformation S has a unitary representation U S on Hilbert space which transforms the Hamiltonian according to U S HU S† =

1 2

I

Q i2 + Pi2 εi +

1 2

i=1

J

P j2 =

j=1

I

† ai ai + 21 εi + i=1

1 2

J

P j2 .

(3)

j=1

Hence, by Eq. (3) the ground state energy E 0 and the energy gap can easily be expressed in terms of the symplectic eigenvalues of the Hamiltonian matrix: E0 =

1 2

I

εi , =

i=1

mini εi , J = 0 . 0, J >0

(4)

The case of a vanishing energy gap = 0 is called critical and the respective ground states are often qualitatively different from those of non-critical Hamiltonians. For the Hamiltonian Hκ , Eq. (1), this happens in the strong coupling limit |κ| = 1 − 2 → 1, and in the case of 1D Coulomb crystals a vanishing energy gap in the radial modes can be considered as the origin of a structural phase transition where the linear alignment of the ions becomes unstable and changes to a zig-zag configuration [18–20]. Needless to say, these phase transitions appear as well in higher dimensions and for various different configurations [21]. Ground and thermal states of quadratic Hamiltonians are Gaussian states, i.e, states having a Gaussian Wigner distribution in phase space. In the mathematical physics literature they are known as bosonic quasi-free states [22, 23]. These states are completely characterized by their first moments dk = tr ρ Rk (which are w.l.o.g. set to zero in our case) and their covariance matrix (CM) γkl = tr ρ Rk − dk , Rl − dl + , (5) where {·, ·}+ is the anticommutator. The CM satisfies γ ≥ iσ , which expresses Heisenberg’s uncertainty relation and is equivalent to the positivity of the corresponding density operator ρ ≥ 0. In order to find the ground state of a quadratic Hamiltonian, observe that

(4) (5) 1 εi = E 0 = inf tr[ρH] = 41 inf tr[γ H ]. (6) 2 i

ρ

γ

1 Note that we disregard systems where the Hamiltonian contains irrelevant normal modes.


69

By virtue of Eqs. (2,3) the infimum is attained for the ground state covariance matrix  

I J

s 0 1 0  S, ⊕ γ = lim S T  (7) 0 1 0 s −1 s→∞ i=1

j=1

which reduces to γ = S T S in the non-critical case. Note that the ground state is unique as long as H does not contain irrelevant normal modes [which we have neglected from the very beginning in Eq. (2)]. In many cases it is convenient to change the order of the canonical operators such that R = (Q 1 , . . . , Q N , P1 , . . . , PN ). Then the covariance matrix as well as the Hamiltonian matrix can be written in block form HQ HQ P H= . H QT P H P In this representation a quadratic Hamiltonian is particle number preserving iff H Q = H P and H Q P = −H QT P , that is, the Hamiltonian contains only terms of the kind ai† a j + a †j ai . In quantum optics terms of the form ai† a †j , which are not number preserving, are neglected within the framework of the rotating wave approximation. The resulting Hamiltonians have particular simple ground states: Theorem 1a. The ground state of any particle number preserving Hamiltonian is the vacuum with γ = 1, 1 and the corresponding ground state energy is given by E 0 = 41 tr H . Proof. Number preserving Hamiltonians are most easily expressed in terms of creation and annihilation operators. For this reason we change to the respective complex representation via the transformation 1 11 −i1 0 X 1 . H → H T = ¯ , = √ 1 X 0 2 11 i1 In this basis H is transformed to normal form via a block diagonal unitary transformation U ⊕ U¯ which in turn corresponds to an element of the orthogonal subgroup of the symplectic group Sp(2N , R) ∩ SO(2N ) U(N ) [24]. Hence, the diagonalizing S in Eqs. (2,7) is orthogonal and since J = 0 due to particle number conservation, we have γ = S T S = 1. 1 E 0 follows then immediately from Eq. (6). Another important class of quadratic Hamiltonians for which the ground state CM takes on a particular simple form corresponds to the case H Q P = 0 where there is no coupling between the momentum and position operators: Theorem 1b. For a quadratic Hamiltonian with Hamiltonian matrix H = H Q ⊕ H P the ground state energy and the ground state CM are given by −1/2 1/2 1/2 −1/2 E 0 = 21 tr H Q H P , γ = X ⊕ X −1 , X = H Q H Q H P H Q H Q . (8) = H P H Q ⊕ H Q H P , the symplectic of H are given Proof. Since σ H σ T H eigenvalues √ √ by the eigenvalues of H Q H P and thus E 0 = 21 tr H Q H P . Moreover, by the uniqueness of the ground state and the fact that E 0 = 41 tr[γ H ] with γ from Eq. (8) we know that γ is the ground state CM (as it is an admissible pure state CM by construction).

70


Finally we give a general formula for the ground state CM in cases where the blocks in the Hamiltonian matrix can be diagonalized simultaneously. This is of particular importance as it applies to all translational invariant Hamiltonians discussed in the following sections. Theorem 1c. Consider a quadratic Hamiltonian for which the blocks H Q , H P , H Q P of the Hamiltonian matrix can be diagonalized simultaneously and in addition H Q P = H QT P . Then with Eˆ = H Q H P − H Q2 P we have (9) ˆ −1 σ H σ T . ˆ E 0 = 21 tr[E], = λmin Eˆ , γ = (Eˆ ⊕ E) (10) ˆ and = λmin Eˆ . Proof. Since σ H σ T H = Eˆ 2 ⊕ Eˆ 2 we have indeed E 0 = 21 tr[E] Positivity γ ≥ 0 is implied by H ≥ 0 such that we can safely talk about the symplectic eigenvalues of γ . The latter are, however, all equal to one due to (γ σ )2 = −1 1 so that γ is an admissible pure state CM. Moreover it belongs to the ground state since 1 4 tr[H γ ] = E 0 . 3. Translationally Invariant Systems Let us now turn towards translationally invariant systems. We consider cubic lattices in d dimensions with periodic boundary conditions. For simplicity we assume that the size of the lattice is N d . The system is again characterized by a Hamiltonian matrix Hkl , where the indices k, l, which correspond to two points (harmonic oscillators) on the lattice, are now d-component vectors in ZdN . Translational invariance is then reflected by the fact that any matrix element Akl , A ∈ {H Q , H P , H Q P } depends only on the relative position k − l of the two points on the lattice, and we will therefore often write Ak−l = Akl . Note that due to the periodic boundary conditions k − l is understood modulo N in each component. Matrices of this type are called circulant, and they are all simultaneously diagonalized via the Fourier transform 2πi 1 Fαβ = √ e N αβ , α, β ∈ Z N , such that N  

2πi   An e− N m n  , Aˆ := F ⊗d AF †⊗d = diag 

n∈ZdN

m

ZdN .

where m n is the usual scalar product in It follows immediately that all circulant matrices mutually commute. In the following, we will show that we can without loss of generality restrict ourselves to point-symmetric Hamiltonians, i.e., those for which H Q P = H QT P (which means that H contains only pairs Q k Pl + Q l Pk ). For dimension d = 1 this is often called reflection symmetry. Theorem 2. Any translationally invariant pure state CM is point symmetric. Proof. For the proof, we use that any pure state covariance matrix can be written as Q Q P X XY , = = T Q P Y X X −1 + Y X Y P


71

where X ≥ 0 and Y is real and symmetric [25]. From translational invariance, it follows that all blocks and thus X and Y have to be circulant and therefore commute. Hence, T , i.e., is point symmetric. Q P = X Y = Y X = Q P Let P : ZdN → ZdN be the reflection on the lattice and define the symmetrization operation S(A) = 21 (A + P AP) such that by the above theorem S(γ ) = γ for every translational invariant pure state CM. Then due to the cyclicity of the trace we have for any translational invariant Hamiltonian inf tr H γ = inf tr S(H )γ . γ

γ

Hence, the point-symmetrized Hamiltonian S(H ), which differs from H by the offdiagonal block S(H Q P ) = 21 (H Q P + H QT P ) has both the same ground state energy and the same ground state as H . Together with Theorem 1c this leads us to the following: Theorem 3. Consider any translationally invariant quadratic Hamiltonian. With Eˆ = 1/2 the ground state CM and the corresponding ground H Q H P − 41 (H Q P + H QT P )2 state energy are given by −1 ˆ E 0 = 21 tr[E], γ = Eˆ ⊕ Eˆ σ S(H )σ T . (11) It is important to note that the energy gaps of H and S(H ) will in general be different. In particular H might be gapless while S(H ) is gapped. However, as we will see ˆ of the in the following sections, the properties of γ depend on the gap = λmin (E) symmetrized Hamiltonian rather than on that of the original H . For this reason we will in the following for simplicity assume H Q P = H QT P . By Thm. 3 all results can then also be applied to the general case without point symmetry if one only keeps in mind that is the gap corresponding to S(H ). Note that the eigenvalues of Eˆ are the symplectic eigenvalues of S(H ), i.e., E = ⊗d ˆ †⊗d is the excitation spectrum of the Hamiltonian. This is the reason for the F EF notation where E resides in Fourier space and Eˆ in real space, which differs from the normal usage of the hat throughout the paper. Correlation functions. According to Eqs. (9,10,11) we have to compute the entries of functions of matrices in order to learn about the entries of the covariance matrix. This is most conveniently done by a double Fourier transformation, where one uses that ˆ and we find f (M) = f ( M), 2πi 1 − 2πi nr ˆ r s e N sm . [ f (M)]nm = d e N [ f ( M)] (12) N r,s As we consider translationally invariant systems, M is circulant and thus Mˆ is diagonal. We define the function

ˆ M(φ) = Mn e−inφ (13) n∈ZdN

ˆ such that M(2πr/N ) = Mˆ r,r . As f (M) is solely determined by its first row, we can write 1 2πi nr/N ˆ [ f (M)]n = d e f ( M(2πr/N )). (14) N d r ∈Z N

72


In the following we will use the index n ∈ Zd for the relative position of two points on the lattice. Their distance will be measured either by the l1 , l2 or l∞ norm. Since we are considering finite dimensional lattices these are all equivalent for our purpose and we will simply write n. In the thermodynamic limit N → ∞, the sum in Eq. (14) converges to the integral

1 inφ ˆ ˆ [ f (M)]n = dφ f ( M(φ)) e with M(φ) = Mn e−inφ , (15) (2π )d T d d n∈Z

where T d is the d-dimensional torus, i.e., [0, 2π ]d with periodic boundary conditions. The convergence holds as soon as |Mn | < ∞ [which holds e.g. for Mn = O(n−α ) ˆ with some α > d] and f is continuous on an open interval which contains the range of M. k d ˆ ˆ From the definition (15) of M, it follows that M ∈ C (T ) (the n times continuously differentiable functions on T d ) whenever the entries Mn decay at least as fast as n−α for some α > k + d, since then the sum of the derivatives converges uniformly. Particularly, if the entries of M decay faster than any polynomial, then Mˆ ∈ C ∞ (T d ). In the following the most important function of the type f ◦ Mˆ will be the spectral function ! (16) e−inφ [H Q H P ]n − [H Q2 P ]n . E(φ) = n∈Zd

Asymptotic notation. As the main issue of this paper is the asymptotic scaling of correlations, we use the Landau symbols o, O, and , as well as the symbol O ∗ for tight bounds: f (x) – f (x) = o(g(x)) means lim g(x) = 0, i.e., f vanishes strictly faster than g for x→∞ x → ∞; " " " f (x) " – f (x) = O(g(x)), if lim sup " g(x) " is finite, i.e., f vanishes at least as fast as g; x→∞

– f (x) = (g(x)), if f (x) = O(g(x)) and g(x) = O( f (x)) (i.e., exact asymptotics); – f (x) = O ∗ (g(x)), if f (x) = O(g(x)) but f (x) = o(g(x)), i.e., g is a tight bound on f .2 If f is taken from a set (e.g., those functions consistent with the assumptions of a theorem) we will write f = O ∗ (g) if g is a tight bound for at least one f (i.e., the best possible universal bound under the given assumptions). If talking about Hamiltonians, the scaling is meant to hold for all blocks, e.g., if the interaction vanishes as O(n−α ) for n → ∞, this holds for all the blocks H Q , H P , and H Q P = H PT Q . The same holds for covariance matrices in the non-critical case. By the shorthand notation f (n) = o(n−∞ ), we mean that f (n) = o(n−α ) ∀α > 0. Note finally that the Landau symbols are also used in (Taylor) expansions around a point x0 where the considered limit is x → x0 rather than x → ∞. 4. Non-Critical Systems In this section, we analyze the ground state correlations of non-critical systems, i.e., those which exhibit an energy gap > 0 between the ground and the first excited state. 2 In order to see the difference to , take an f (x) = g(x) for even x, f (x) = 0 for odd x, x ∈ N. Although f does not bound g, thus f (x) = O(g(x)), the bound g is certainly tight. A situation like this is met, e.g., in Theorem 5, where the correlations oscillate within an exponentially decaying envelope.


73

Simply speaking, we will show that the decay of correlations reflects the decay of the interaction. While local (super-polynomially decaying) interactions imply exponentially (super-polynomially) decaying correlations, a polynomial decay of interactions will lead to the same polynomial law for the correlations. According to Theorem 3, we will consider a translationally invariant system with a point-symmetric Hamiltonian (H Q P = H QT P ). Following (10,11), we have to determine the entries of (Eˆ −1 ⊕ Eˆ −1 )σ H σ T , with Eˆ = (H Q H P + H Q2 P )1/2 . In Lemma 1 we will first show that it is possible to consider the two contributions independently, and as the asymptotics of σ H σ T is known, we only have to care about the entries of Eˆ −1 , i.e., we have to determine the asymptotic behavior of the integral 1 (Eˆ −1 )n = dφ E −1 (φ)einφ where E = ( Hˆ Q Hˆ P + Hˆ Q2 P )1/2 . (2π )d T d Lemma 1. Given two asymptotic circulant matrices A, B in d dimensions with polynomially decaying entries, An = O(n−α ), Bn = O(n−β ), α, β > d. Then (AB)n = O ∗ (n−µ ), µ := min{α, β}. Proof. With Q η (n) := min{1, n−η }, we know that |An | = O(Q α ) and |Bn | = O(Q β ), and " "   " "

" " A0, j B j,n "" ≤ |A j ||Bn− j | = O  Q α ( j)Q β (n − j) . (17) |(AB)n | = "" " j " j j We consider only one half space j ≤ n− j, where we bound Q β (n− j) ≤ Q β (n/2). As Q α ( j) is summable, the contribution of this half-plane is O Q β (n/2) . The other half-plane gives the same result with α and β interchanged, which proves the bound, while tightness follows by taking all An , Bn positive. We now determine the asymptotics of (Eˆ −1 )n for different types of Hamiltonians. Lemma 2. For non-critical systems with rapidly decaying interactions, i.e., as o(n−∞ ), the entries of Eˆ −1 decay rapidly as well. That is, > 0 ⇒ (Eˆ −1 )n = o(n−∞ ). Proof. As the interactions decay as o(n−∞ ), Hˆ • ∈ C ∞ (T d ) (• = Q, P, P Q), and thus E 2 = Hˆ Q Hˆ P + Hˆ Q2 P ∈ C ∞ (T d ). Since the system is gapped, i.e., E ≥ > 0, it follows that also g := E −1 ∈ C ∞ (T d ). For the proof, we need to bound 1 dφ g(φ)einφ (Eˆ −1 )n = (2π )d T d by n−κ for all κ ∈ N. First, let us have a look at the one-dimensional case. By integration by parts, we get $π # π 1 1 1 −1 inφ ˆ g(φ)e − dφ g (φ)einφ , (E )n = 2π in 2πin −π φ=−π

74


where the first part vanishes due to the periodicity of g. As g ∈ C ∞ (T 1 ), the integration by parts can be iterated arbitrarily often and all the brackets vanish, such that after κ iterations, π 1 −1 ˆ (E )n = dφ g (κ) (φ)einφ . 2π(in)κ −π % As g (κ) (φ) is continuous, the integral can be bounded by |g (κ) (φ)|dφ =: Cκ < ∞, such that finally |(Eˆ −1 )n | ≤

Cκ nκ

∀κ ∈ N ,

which completes the proof of the one-dimensional case. The extension to higher dimensions is straightforward. For a given n = (n 1 , . . . , n d ), integrate by parts with respect to the φi for which |n i | = n∞ ; we assume i = 1 without loss of generality. As g(·, φ2 , . . . , φd ) ∈ C ∞ (S 1 ), the same arguments as in the 1D case show " " κ " ∂ " 1 " " dφ = Cκ . g(φ) |(Eˆ −1 )n | ≤ κ " " d κ d ∂φ (2π ) |n | nκ T

1

∞

1

For systems with local interactions, a stronger version of Lemma 2 can be obtained: Lemma 3. For a system with finite range interaction, the entries of Eˆ −1 decay exponentially. This has been proven in [1] for Hamiltonians of the type H = V ⊕ 1, 1 exploiting a result on functions of banded matrices [26]. Following Eqs. (9,11) the generalization to arbitrary translational invariant Hamiltonians is straightforward by replacing V with H Q H P − H Q2 P . In fact, it has been shown recently that the result even extends to non translational invariant Hamiltonians of the form in Theorem 1 b [14]. Finally, we consider systems with polynomially decaying interaction. Lemma 4. For a 1D lattice with H = V ⊕ 11 > 0 and an exactly polynomially decaying interaction & i= j : a Vi j = , 2 ≤ ν ∈ N, i = j : |i−bj|ν Eˆ −1 decays polynomially with the same exponent, (Eˆ −1 )n = (V 1/2 )n = (|n|−ν ). Hamiltonians of this type appear, e.g., for the vibrational degrees of freedom of ions in a linear trap, where ν = 3. (9) Proof. We need to estimate (Eˆ −1 )n = (V −1/2 )n =

Vˆ (φ) = a + 2b

∞

cos(nφ) n=1

nν

1 2π

% 2π 0

Vˆ −1/2 (φ)einφ dφ. Note that

= a + 2b Re Liν (eiφ ) > 0,

(18)

n ν where Liν (z) = n≥1 z /n is the polylogarithm. The polynomial decay of coeffiν−2 1 cients implies Vˆ ∈ C (S ), and as the system is non-critical, Vˆ −1/2 ∈ C ν−2 (S 1 ).


75

As Liν has an analytic continuation to C\[1; ∞), Vˆ ∈ C ∞ ((0; 2π )) and thus Vˆ −1/2 ∈ C ∞ ((0; 2π )). We can therefore integrate by parts ν − 1 times, and as all brackets vanish due to periodicity, we obtain $ 2π # ν−1 d 1 ˆ −1/2 (φ) einφ dφ, (Eˆ −1 )n = (19) V 2π(in)ν−1 0 dφ ν−1 and dν−1 ˆ −1/2 Vˆ (ν−1) (φ) 3(ν − 2)Vˆ (ν−2) (φ)Vˆ (1) (φ) (φ) = − + + g(φ). V ν−1 dφ 2 Vˆ (φ)3/2 4Vˆ (φ)5/2

(20)

Note that the second term only appears if ν ≥ 3, and g only if ν ≥ 4. As g(φ) ∈ C 1 (S 1 ), its Fourier coefficients vanish as O(n −1 ), as can be shown by integrating by parts. The second term can be integrated by parts as well, the bracket vanishes due to continuity, and we remain with ( ' 1 2π 3(ν − 2)Vˆ (ν−1) (φ)Vˆ (1) (φ) + h(φ) einφ dφ, in 0 4Vˆ (φ)5/2 with h ∈ C (S 1 ). [For ν = 3, a factor 2 appears as (Vˆ (1) ) = Vˆ (ν−1) .] As we will show later, Vˆ (ν−1) is absolutely integrable, hence the integral exists, and thus the Fourier coefficients of the second term in Eq. (20) vanish as O(n −1 ) as well. Finally, it remains to bound 2π ˆ (ν−1) (φ) inφ V e dφ. (21) ˆ 2 V (φ)3/2 0 As Liν (x) = Liν−1 (x)/x, it follows from Eq. (18) that V (ν−1) (φ) = 2b Re i ν−1 Li1 (eiφ ) = 2b Re −i ν−1 log(1 − eiφ ) , where the last step is from the definition of Li1 . We now distinguish two cases. First, assume that ν is even. Then, V (ν−1) (φ) ∝ Im log(1 − eiφ ) = arg(1 − eiφ ) =

(φ − π ) 2

on (0; 2π ), hence the integrand in Eq. (21) is bounded and has a bounded derivative, and by integration by parts, the integral Eq. (21) is O(n −1 ). In case ν is odd we have " " " " V (ν−1) (φ) ∝ Re log(1 − eiφ ) = log "1 − eiφ " = log(2 sin(φ/2)) on (0; 2π ). With h(φ) := Vˆ −3/2 (φ)/2, the integrand in Eq. (21) can be written as Vˆ (ν−1) (φ)h(φ) ∝ log(2 sin(φ/2)) h(0) + log(2 sin(φ/2)) [h(φ) − h(0)]. The first term gives a contribution proportional to 2π 1 log(2 sin(φ/2)) cos(nφ)dφ = − 2n 0

(22)

76


as it is the back-transform of − 21 n≥1 cos(nφ)/n. For the second term, note that h ∈ C 1 (S 1 ) for ν ≥ 3 and thus h(φ)−h(0) = h (0)φ +o(φ) by Taylor’s theorem. Therefore, the log singularity vanishes, and we can once more integrate by parts. The derivative is 1 cot(φ/2) [h(φ) − h(0)] + log(2 sin(φ/2)) h (φ). 2 In the left part, the 1/φ singularity of cot(φ/2) is cancelled out by h(φ) − h(0) = O(φ), and the second part is integrable as h ∈ C (S 1 ), so that the contribution of the integral (21) is O(n −1 ) as well. In order to show that n −ν is also a lower bound on (Vˆ −1/2 )n , one has to analyze the asymptotics more carefully. Using the Riemann-Lebesgue lemma—which says that the Fourier coefficients of absolutely integrable functions are o(1)—one finds that all terms in (19) vanish as o(1/n ν ), except for the integral (21). Now for even ν, (21) can be integrated by parts, and while the brackets give a (n −ν ) term, the remaining integral is o(n −ν ), which proves that (Vˆ −1/2 )n = (n −ν ). For odd ν, on the other hand, the first part of (22) gives exactly a polynomial decay, while the contributions from the second part vanishes as o(n −ν ), which proves (Vˆ −1/2 )n = (n −ν ) for odd ν as well. Generalizations of Lemma 4. The preceding lemma can be extended to non-integer exponents α ∈ N: if Vn ∝ n −α , n = 0, then (Eˆ −1 )n = O(n −α ). For the proof, define α = ν + ε, ν ∈ N, 0 < ε < 1. Then Vˆ ∈ C ν−1 (S 1 ), ˆ V ∈ C ∞ ((0; 2π )), and one can integrate by parts ν times, where all brackets vanish. What remains is to bound the Fourier integral of the ν th derivative of Vˆ −1/2 by n −ε . An upper bound can be established by noting that |Vˆ (ν) (φ)| ≤ |Liε (eiφ )| = O(φ ε−1 ) and |Vˆ (ν+1) (φ)| = O(φ ε−2 ). It follows that all contributions in the Fourier integral except the singularity from Vˆ (ν) lead to o(1/n) contributions as can be shown by another integration by parts. In order to bound the Fourier integral of the O(φ ε−1 ) term, split the Fourier integral at n1 . The integral over [0; n1 ] can be directly bounded by n −ε , while for [ n1 ; 1], an equivalent bound can be established after integration by parts, using Vˆ (ν+1) = O(φ ε−2 ). This method is discussed in more detail in the proof of Theorem 10, following Eq. (44). The proof that n −ε is also a lower bound to (Eˆ −1 )n is more involved. From a series expansion of Vˆ and its derivatives, it can be seen that it suffices to bound the sine and cosine Fourier coefficients of φ ε−1 from below. As in the proof of Theorem 9, this is accomplished by splitting the integral into single oscillations of the sine or cosine and bounding each part by the derivative of φ ε−1 . For polynomially bounded interactions Vn = O(n −α ), α > 1, not very much can be said without further knowledge. With ν < α, ν ∈ N the largest integer strictly smaller than α, we know that Vˆ ∈ C ν−1 (S 1 ). Thus, one can integrate by parts ν − 1 times, the brackets vanish, and the remaining Fourier is o(1) using the Riemann-Lebes integral gue lemma. It follows that (Eˆ −1 )n = o n −(ν−1) . In contrast to the case of an exactly polynomial decay, this can be extended to higher spatial dimensions d > 1 by replacing ν − 1 with ν − d, which yields (Eˆ −1 )n = o n −(ν−d) . We now use the preceding lemmas about the entries of Eˆ −1 (Lemma 2–4) to derive corresponding results on the correlations of ground states of non-critical systems. Theorem 4. For systems with > 0, the following holds: (i) If the Hamiltonian H has finite range, the ground state correlations decay exponentially.


77

(ii) If H decays as o(n−∞ ), the ground state correlations decay as o(n−∞ ) as well. (iii) For a 1D system with H = V ⊕ 1, 1 where V decays with a power law |n|−ν , ν ≥ 2, the ground state correlations decay as (|n|−ν ). Proof. In all cases, we have to find the scaling of the ground state γ which is the product γ = (Eˆ −1 ⊕ Eˆ −1 )σ H σ T , Eq. (10). Part (i) follows directly from Lemma 3, as multiplying with a finite-range σ H σ T doesn’t change the exponential decay, while (ii) follows from Lemma 2, the o(n−∞ ) decay of σ H σ T , and Lemma 1. To show (iii), note that for H = V ⊕ 1, 1 the ground state is γ = V −1/2 ⊕ V 1/2 , and from Lemma 4, O(n −ν ) follows. For Vˆ −1/2 , Lemma 4 also includes that the bound is exact, while for Vˆ 1/2 , it can be shown by transferring the proof of the lemma one-to-one. Note that a simple converse of Theorem 4 always holds: for each translationally invariant pure state CM γ , there exists a Hamiltonian H with the same asymptotic behavior as γ such that γ is the ground state of H . This can be trivially seen by choosing H = σγσT. 5. Correlation Length and Gap In this section, we consider one-dimensional chains with local gapped Hamiltonians. We compute the correlation length for these systems and use this result to derive a relation between correlation length and gap. Theorem 5. Consider a non-critical 1D chain with a local Hamiltonian. Define the 1/2 L complex extension of the spectral function E(φ) = c cos(nφ) in Eq. (16) n n=0 as g(z) :=

L

n=0

cn

z n + z −n , 2

(9) such that g(eiφ ) = E 2 (φ) = Hˆ Q (φ) Hˆ P (φ) − Hˆ Q2 P (φ) and let z˜ be zero of g with the largest magnitude smaller than one. Then, the correlation length

ξ =−

1 log |˜z |

determines the asymptotic scaling of the correlations which is given by √ – O ∗ (e−n/ξ / n), if z˜ is a zero of order one, – O ∗ (e−n/ξ ), if z˜ is a zero of even order, – o(e−n/(ξ +ε) ) for all ε > 0, if z˜ is a zero of odd order larger than one. For the nearest neighbor interaction Hamiltonian Hκ from Eq. (1)√one has for instance √ 2 E(φ) = 1 − κ cos(φ), There√ so that g has simple zeros at z 0 = (1 ± 1 − κ )/κ. √ fore z˜ = (1 − 1 − κ 2 )/κ, and the correlations decay as (e−n/ξ / n), where ξ = −1/ log |˜z |.

78


Proof. For local Hamiltonians, the correlationsdecay as the matrix elements of Eˆ −1 [Eq. (10)]. By Fourier transforming (9), E(φ) = g(eiφ) , with g(eiφ ) = Hˆ Q (φ) Hˆ P (φ)− L cn cos(nφ) an even trigonometric polynomial (we assume c L = 0 Hˆ Q2 P (φ) = n=0 without loss of generality), and min(g(eiφ )) = 2 . We have to compute (Eˆ −1 )n =

1 2π

2π 0

1 inφ z n−1 1 e dφ = dz, √ E(φ) 2πi S 1 g(z)

(23)

where S 1 is the unit circle. The function g(z) has a pole of order L at zero and 2L zeros altogether. Since min(g(φ)) = 2 > 0, g has no zeros on the unit circle. As g(z) = g(1/z), the zeros come in pairs, and L of them are inside the unit circle. Also, the conjugate of √ a zero is a zero as well. From each zero with odd multiplicity emerges a branch cut of g(z). We arrange all the branch cuts inside the unit circle such that they go straight to the middle where they annihilate with another cut. In case L is odd, √ the last cut is annihilated by the singularity of g(z) at 0. If two zeros lie on a line, one cut curves slightly. A sample arrangement is shown in Fig. 1. Following Cauchy’s theorem, the integral can be decomposed into integrals along √ the different branch cuts and around the residues of 1/ g, and one has to estimate the contributions from the different types of zeros of g. The simplest case is given by zeros z 0 with even multiplicity 2m. In that case, define h(z) := g(z)/(z − z 0 )2m which has no zero around z 0 . The contribution from z 0 to the correlations is then given by the residue at z 0 and is

√ Fig. 1. Sample arrangement of branch cuts and poles of g inside the unit circle. From each odd order zero of g, a branch cut emerges. All cuts go to 0 where they cancel with another cut. In case their number is odd, there is an additional branch point at 0 cancelling the last cut. In case two zeros are on a line to the origin, the √ cuts are chosen curved. The integral of g around the unit circle is equal to the integral around the cuts, plus integrals around the residues which originate from the even order zeros of g


dm−1 1 (m − 1)! dz m−1

79

" z n−1 "" n−(m−1) ∝ z0 √ h(z) "z=z 0

for n − (m − 1) > 0, i.e., it scales as |z 0 |n . Note that for z 0 ∈ R, the imaginary parts originating from z 0 and its conjugate z¯ 0 exactly cancel out, but the scaling is still given by |z 0 |n = en log |z 0 | , i.e., ξ = −1/ log |z 0 | is the corresponding correlation length. If z 0 is a simple zero of g(z), we have to integrate around the branch cut. Assume first that the cut goes to zero in a straight line, and consider a contour with distance ε to the slit. Both the contribution from the ε region around zero and the ε semicircle at z 0 vanish as ε → 0, and the total integral is therefore given by twice the integral along the cut, z0 z n−1 1 dz, √ √ πi 0 z − z 0 h(z) where again h(z) = g(z)/(z − z 0 ). Intuitively, for growing n the part of the integral close to z 0 becomes more and more dominating, i.e., the integral is well approximated by the modified integral where h(z) has been replaced by h(z 0 ). After rotating it onto the real axis, this integral—up to a phase—reads |z 0 | n−1 r 1 |z 0 |n−1/2 (n) , (24) dr = √ √ √ |z 0 | − r π |h(z 0 )| 0 π |h(z 0 )| (n + 21 ) which for large n is 1 |z 0 |n √ √ +O n π |z 0 h(z 0 )|

|z 0 |n n 3/2

.

(25)

In order to justify the approximation h(z) h(z 0 ), consider the difference of the two respective integrals. It is bounded by " " " " z 0 " 1 "" "" |z|n−1 "" 1 " − dz . √ √ √ " |z − z 0 | " h(z) h(z 0 ) " " 0 ) *+ , (∗)

On [z 0 /2, z 0 ], h(z) is analytic and has no zeros, thus, |h(z)−1/2 −h(z 0 )−1/2 | < C|z−z 0 |, where C is the maximum of the derivative of h(z)−1/2 on [z 0 /2, z 0 ]. On [0, z 0 /2], the same bound is obtained by choosing C the supremum of |h(z)−1/2 − h(z 0 )−1/2 |/|z 0 /2| on [0, z 0 /2]. Together, (∗) ≤ C|z − z 0 |, and the above integral is bounded by √ |z 0 | |z 0 |n π|z 0 |n+1/2 (n) n−1 , =O C r |z 0 | − r dr = C n 3/2 2(n + 23 ) 0 i.e., it vanishes by 1/n faster than the asymptotics derived in Eq. (25), which justifies fixing h(z) at h(z 0 ). √ From Eq. (25), it follows that the scaling is e−n/ξ / n, where the correlation length is again ξ = −1/ log |z 0 |. The same scaling behavior can be shown to hold for appropriately chosen curved branch cuts from z 0 to 0 by relating the curved to a straight integral. The situation gets more complicated if zeros of odd order > 1 appear. In order to get an estimate which holds in all scenarios, we apply Cauchy’s theorem to contract the unit

80


circle in the integration (23) to a circle of radius r > |z 0 |, where z 0 is the largest zero inside the unit circle.√Then, the integrand can be bounded by Cr r n−1 (where Cr < ∞ is the supremum of 1/ g on the circle), and this gives a bound 2πCr r n−1 for the integral. This holds for all r > |z 0 |, i.e., the correlations decay faster than en log r for all r > |z 0 |. This does not imply that the correlations decay as en log |z 0 | , but it is still reasonable to define −1/ log |z 0 | as the correlation length. Theorem 6. Consider a 1D chain together with a family of Hamiltonians H () with gap > 0, where H () is continuous for → 0 in the sense that all entries of H converge. Then, the ground state correlations scale exponentially, and for sufficiently small the correlation length is

Here, m ∗ =

"

d2 E (φ) " " dφ 2 φ=φ

−1

1 ξ√ . m ∗ is the effective mass at the band gap.

√ For the discretized Klein-Gordon field (1), for example, we have = 1 − |κ|, √ m ∗ =√2 1 − |κ|/|κ|, and for √ small (corresponding to |κ| close to 1), one obtains ξ |κ|/2(1 − |κ|) 1/ 2. Hence, the ξ ∝ 1/ law holds if the coupling is increased relative to the on-site energy (in which case m ∗ ∝ ). More generally, if we expand the spectral function [Eq. √ (16)] around the band gap 3 we are generically led to the dispersion relation E(k) 2 + v 2 k 2 (k ≡ φ). By the definition of the effective mass and Theorem 6 this leads exactly to the folk theorem v (26) ξ . Proof. According to Theorem 5, what remains to be done is to determine the position of the largest zero z˜ of g in the unit circle. Due to the restriction on H (), the coefficients of the polynomial g(z)z L and thus also the zeros of g continuously depend on , i.e., for sufficiently small , the zero closest to the unit circle is the one closest to the gap. In order to determine the position of this zero, we will expand g around the gap. We only discuss the generic case where the gap appears only for one angle φ0 , g(φ0 ) = . In the case of multiple occurrences of the gap in the spectrum, one will pick the gap which gives the zero closest to the unit circle, i.e., the largest correlation length. Furthermore, we assume φ0 = 0 without loss of generality. Otherwise, one considers g(ze−iφ0 ) instead of g(z), which on the unit circle coincides with the (rotated) spectrum. The knowledge on g =: u + iv (with u, v : C → R) which will be used in the proof is u(1) = 2 , u φ (1) = 0, u φφ (1) = 2 m ∗ > 0,

v(1) = 0, vφ (1) = 0, vφφ (1) = 0,

(27)

where the subscripts denote the partial derivative with respect to the respective subscript (in Euclidean coordinates z ≡ x + i y, in polar coordinates z ≡ r eiφ ). Note that 3 This makes the natural assumption that the minimum under the square root is quadratic. In fact, if it is of higher order, then m ∗ = ∞ and thus ξ = 0, which is consistent with the findings of the following section. An example of such a behavior is given by so called ‘quadratic interactions’ [2] for which H = V ⊕1, 1 where V is the square of a banded matrix.


81

z = 1 is the point where the gap appears, and that g(eiφ ) = E(φ)2 is real. Therefore, the derivatives of the imaginary part v along the circle vanish, while the derivatives of the real part u are found to be u(1) = E(0)2 = 2 , u φ (1) = 2E(0)E (0) = 0, and u φφ (0) = 2E (0)2 + 2E(0)E (0) = 2/m ∗ , where m ∗ = 1/E (φ) is the effective mass at the band gap. We need to exploit the relation between Euclidean and polar coordinates, gx (1) = gr (1) ; g y (1) = gφ (1), gx x (1) = grr (1) ; g yy (1) = gφφ (1) + gr (1), and the Cauchy–Riemann equations u x = v y , u y = −vx , and gx x + g yy = 0, which together with the information (27) lead to u(1) = 2 ; v(1) = 0 ; u x (1) = u y (1) = vx (1) = v y (1) = 0 ; 2 u x x (1) = − 2 m ∗ ; u yy (1) = m ∗ ; vx x (1) = 0 ; v yy (1) = 0.

Note that it is not possible to derive information about the mixed second derivates using only the information (27). However, as long as vx y does not vanish at 1, v will only stay zero in direction of x or y, but not diagonally. Since 2 > 0 and 2/m ∗ > 0, the closest zero is—to second order—approximately located along the x axis. By intersecting with √ the parabola 2 − m∗ (x − 1)2 , one finds that the zero is located at x0 ≈ 1 − m ∗ . For √ ∗ small √ , the correlations thus decay with correlation length ξ ≈ −1/ log(1− m ) ≈ ∗ 1/ m . 6. Critical Systems In the following, we discuss critical systems, i.e., systems without an energy gap, = 0.4 In that case, the Hamiltonian will get singular and some entries of the ground state covariance matrix will diverge, which leads to difficulties and ambiguities in the description of the asymptotic behavior of correlations. We will therefore restrict to Hamiltonians of the type H = V ⊕ 1, 1 for which the ground state CM is γ = V −1/2 ⊕ V 1/2 . While the Q part diverges, the entries of the P-block stay finite. Following Thm. 1(b) the extension to interactions of the form H = H Q ⊕ H P is straightforward. In order to compute the correlations we have to determine the asymptotics of V 1/2 , i.e., 1 1/2 (V )n = Vˆ (φ)einφ dφ. (2π )d T d 4 Note that there are different meanings of the notion criticality referring either to a vanishing energy gap or to an algebraic decay of correlations. In this section we discuss in which cases these two properties are equivalent.

82


We will restrict to the cases in which the excitation spectrum E = Vˆ has only a finite number of zeros, i.e., finitely many points of criticality. In addition, we will also consider the special case in which the Hamiltonian exhibits a tensor product structure. We proceed as follows. First, we consider one-dimensional critical chains and show that the correlations decay typically as O(n −2 ) and characterize those special cases where the correlations decay more rapidly. The practically important case of exactly cubic decaying interactions will be investigated in greater detail. Depending on the sign of the interaction this case will lead to a logarithmic deviation from the n −2 behavior. Then, we turn to higher dimensional systems and show that generically the correlations decay as n −(d+1) log n, where d is the spatial dimension of the lattice. 6.1. One dimension. First, we prove a lemma which shows that although taking the square root of a smooth function destroys its differentiability, the derivatives will stay bounded. Lemma 5. Let f ∈ C m ([−1; 1]), f (x) ≥ 0 with the only zero at x = 0, and let 2ν ≤ m be the order of the minimum at x = 0, i.e., f (k) (0) = 0 ∀k < 2ν, f (2ν) (0) > 0. √ Define g(x) := f (x). Then, the following holds: – For odd ν, g ∈ C ν−1 ([−1; 1]), and g ∈ C m−ν ([−1; 0]), g ∈ C m−ν ([0; 1]), i.e., the first m − ν derivatives (for x = 0) are bounded. – For even ν, g ∈ C m−ν ([−1; 1]). k (k) m−k ) for Proof. Using the Taylor expansion f (x) = m k=2ν ck x + ρ(x), ρ (x) = o(x ν ν k ≤ m, we express g as g(x) = (sgn x) x r (x) with . m . ρ(x) ck x k−2ν + 2ν , r (x) = / x k=2ν

√

where we used that (sgn x)ν x ν = x 2ν . Let us now consider the derivatives of r (x). While the sum leads to a O(1) contribution, the k th derivative of the remainder behaves as o(1)/x 2ν−m+k . Together, this leads to r (k) (x) = O(1), r (k) (x) = o(1)/x 2ν−m+k ,

2ν − m + k ≤ 0, 2ν − m + k ≥ 1.

Now consider the k th derivative of g(x) for x = 0, $ k # l

k d ν (k−l) (k) ν g (x) = (sgn x) x r (x) . l dx l l=0 ) *+ , sl

Assume first k ≤ ν. Then, sl ∝ O(1)x ν−l for 2ν − m + k − l ≤ 0, and sl ∝ o(1)x m−ν−k for 2ν − m + k − l ≥ 1, and as m ≥ 2ν, it follows that g (k) = O(x) for k < ν, which cancels the discontinuity originating from sgn x. For k = ν, on the contrary, sk = O(1), and sgn x introduces a discontinuity on g (k) , yet, it remains bounded and piecewise differentiable on [−1; 0] and [0; 1]. The first non-bounded sl is found as soon as m − ν − k = −1, and g ∈ C m−ν ([0; 1]) directly follows. This also implies that for m − ν − k ≥ 0, g(x)/(sgn x)ν ∈ C m−ν ([−1; 1]), i.e., the discontinuity is only due to (sgn x)ν . Since, however, this is only discontinuous for odd ν, it follows that g ∈ C m−ν ([−1; 1]) if ν even.


83

Theorem 7. Consider a one-dimensional critical chain with Hamiltonian H = V ⊕ 1, 1 where Vn = O(n −α ), α > 4 and where Vˆ has a finite number of critical points which are all quadratic minima of Vˆ . Then, (γ P )n = O ∗ (n −2 ). For Vn ∝ n −α , α > 3 it even follows that (γ P ) = (n −2 ). Note that for Vn ∝ n −α , the extrema of Vˆ are always quadratic. Proof. We want to estimate (V

1/2

1 )n = g(φ)einφ dφ, 2π S 1

(28)

where g = Vˆ 1/2 . Under both assumptions, Vˆ ∈ C 2 (S 1 ), and all critical points are minima of order 2. It follows from Lemma 5 that g is continuous with bounded derivative. Therefore, we can integrate by parts, the bracket vanishes, and we obtain 2π 1 (V 1/2 )n = − g (φ)einφ dφ. 2πin 0 0 Now, split S 1 at the zeros of g into closed intervals I j , j I j = S 1 , and rewrite the above integral as a sum of integrals over I j . As g ∈ C (I j ) (and differentiable on the inner of I j ), one can once more integrate by parts which yields

1 inφ inφ g (V 1/2 )n = − (φ)e − g (φ)e dφ . (29) Ij 2π(in)2 Ij j

Neither of the terms will vanish, but since g ∈ C (I j ), the bracket is bounded. In case Vn ∈ O(n −α ), α > 4, we have Vˆ ∈ C 3 (S 1 ), therefore g is bounded (Lemma 5), and the integrals vanish as o(1). Unless the contributions of the brackets for the different I j cancel out, the n −2 bound is tight, (V 1/2 )n = O ∗ (n −2 ). The tightness of the bound is also illustrated by the example which follows the proof. For the case of an exactly polynomial decay, we additionally have to show that g is absolutely integrable for 3 < α ≤ 4. Then, the exactness of the bound holds because the bracket in Eq. (29) does not oscillate (the critical point is either at φ = 0 or at φ = π ), and because the integral is o(1) for g ∈ L1 (S 1 ). In case the critical point is at φ = π , the latter holds since Vˆ ∈ C ∞ ((0; 2π )) implies that g is bounded at π , and Vˆ ∈ C 2 (S 1 ) that g ∈ C 2 ((−π, π )), which together proves that g is bounded on S 1 . In case the critical point is at φ = 0, the situation is more involved (and for α = 3, a logarithmic correction appears, cf. Theorem 9). Since Vˆ (3) (φ) = −Im Liα−3 (eiφ ) = O(φ α−4 ), we have Vˆ (φ) = Vˆ (0) + O(φ α−3 ), Vˆ (φ) = Vˆ (0)φ + O(φ α−2 ), Vˆ (φ) = 1 Vˆ (0)φ 2 + O(φ α−1 ). 2

With this information, g (φ) =

2 Vˆ (φ)Vˆ (φ) − Vˆ (φ)2 = O(φ α−4 ), 4V (φ)3/2

which indeed proves that g ∈ L1 (S 1 ), and thus (V 1/2 )n = (n −2 ).

84


As an example, consider again the discretized Klein-Gordon field of Eq. (1) which is critical for κ = ±1, corresponding to Vˆ (φ) = 1 ∓ cos φ. The Fourier integral is solvable √ 2 2 (sgn κ)n and yields (γ P )n = − π 4n 2 −1 = (n −2 ). Generalizations of Theorem 7. Using Lemma 5, several generalizations for the 1D critical case can be found. In the following, we mention some of them. In all cases H = V ⊕1 1 is critical. Critical points of even order. If Vn = o(n −∞ ) and the critical points are minima of order 2ν, ν even, the correlations decay as (γ P )n = o(n −∞ ). This is the case, e.g., if V = X 2 with X itself rapidly decaying. Critical points of higher order. If Vˆ has critical points of order at least 2ν, ν odd, and Vn = O(n −α ), α > 2ν + 2, then (γ P )n = O(n −(ν+1) ). Minima of different orders. If Vˆ has minima of different orders 2νi , in general the minimum with the lowest odd νi ≡ ν1 will determine the asymptotics, (γ P )n = O(n −(ν1 +1) ). As Vˆ ∈ C (2 max{νi }) (S 1 ) is required anyway, the piecewise differentiability of Vˆ 1/2 is guaranteed. Weaker requirements on V . It is possible to ease the requirements imposed on V in Theorem 7 to Vn = O(n −α ), α > 3 or Vˆ ∈ C 2 (S 1 ), respectively. The price one has to pay is that one gets an additional log correction as in the multidimensional critical case, Theorem 10. The method to bound g is the same which is used there to derive (39). The above proof does not cover the case of the relevant 1/n 3 interaction, which for instance appears for the motional degrees of freedom of trapped ions. In the following, we separately discuss this case. It will turn out that the scaling will depend on the sign of the coupling: while a positive sign (corresponding to the radial degrees of freedom) again gives a ( n12 ) scaling as before, for the negative sign (corresponding to the axial √ n degree of freedom) one gets log . n2 Theorem 8. Consider a critical 1D chain with a 1/n 3 coupling with positive sign, i.e., H = V ⊕ 1, 1 Vn = c/n 3 , V0 = 3cζ (3)/2, c > 0, with ζ the Riemann zeta function. Then, the ground state correlations scale as (γ P )n = ( n12 ). Proof. We take w.l.o.g. c = 1/2. For this sign of the coupling, the critical point is at π , Vˆ (π ) = 0. From the proof of Lemma 4, we know that Vˆ ∈ C 1 (S 1 ), Vˆ ∈ C ∞ ((0; 2π )), and that Vˆ (φ) = log(2 sin(φ/2)) on (0; 2π ). With g := Vˆ 1/2 , it follows from Lemma 5 that g ∈ C (S 1 ), g ∈ C 1 ([−π ; π ]), and g ∈ C ∞ ((0; π ]), g ∈ C ∞ ([−π ; 0)). This means that all derivatives g (k) , k ≥ 1 can exhibit jumps at the critical point π but they all remain bounded. In contrast, around φ = 0, g is continuous but g has a log divergence. Thus, the Fourier integral 1 1/2 g(φ)einφ dφ (V )n = 2π S 1 can be split at 0 and π , and then integrated by parts twice. The brackets of the first integration cancel out due to continuity of g, and one remains with π π 1 g (φ) cos(nφ) + g (φ) cos(nφ)dφ , (V 1/2 )n = 0 π(in)2 0


85

where we used the symmetry of g. One finds [g (φ) cos(nφ)]π0 = − log2 2 (−1)n , and since g is integrable, the integral is o(1) due to the Riemann-Lebesgue lemma. Together, this proves (γ P )n = ( n12 ). Theorem 9. Consider a critical 1D chain with a 1/n 3 coupling with negative sign, i.e., H = V ⊕1, 1 Vn = −c/n 3 , V0 = 2cζ (3), c > 0, with√ζ the Riemann zeta function. Then, n the ground state correlations scale as (γ P )n = log . n2 Proof. Again, take w.l.o.g. c = 1/2. For the negative sign of the interaction, the critical point is at φ = 0. Since at this point Vˆ diverges, Lemma 5 cannot be applied, and the situation gets more involved. As in the previous proof, we use that Vˆ ∈ C 1 (S 1 ), Vˆ ∈ C ∞ ((0; 2π )), and thus 1/2 ∈ C (S 1 ), Vˆ 1/2 ∈ C ∞ ((0; 2π )). Further, Vˆ (φ) = − log(2 sin(φ/2)) on (0; 2π ), Vˆ cf. the proof of Lemma 4, and with sin x = x(1 + O(x 2 )) we have Vˆ (φ) = − log(φ) + O(φ 2 ) for φ → 0 (and similarly for φ → 2π ), and therefore Vˆ (φ) = φ(1 − log φ) + O(φ 3 ), Vˆ (φ) = 41 φ 2 (3 − 2 log φ) + O(φ 4 ). As Vˆ 1/2 ∈ C (S 1 ), we can integrate by parts one time, π 1 1 (V 1/2 )n = g (φ) sin(nφ)dφ, Vˆ 1/2 (φ)einφ dφ= 2π S 1 πn 0

(30)

(31)

where we exploited the symmetry of Vˆ , and with g := Vˆ 1/2 . Then, from (30), 1 − log φ −2 + log φ φ φ2 , g (φ)= , g (φ)= √ + O +O √ √ φ(3 − 2 log φ)3/2 3 − 2 log φ | log φ| | log φ| and after another round of approximation, √ | log φ| 1 1 1 1 . , g (φ) = − 3/2 √ g (φ) = +O √ +O √ 2 φ | log φ| φ| log φ|3/2 | log φ| 2 √ This shows that the remainder g (φ) − | log φ|/2 is continuous with an absolutely integrable derivative, and by integration by parts it follows that it only leads to a contribution O(1/n) in the integral (31). Thus, √ it remains to investigate the asymptotics of the sine Fourier coefficients of h(φ) = | log φ|. For convenience, we split the integral (31) at 1, and [1; π ] only contributes with O(1/n), as h is continuous with absolutely integrable derivative on [1; π ]. On [0; 1], we have to compute the asymptotics of 1 I= − log φ sin(nφ)dφ. (32) 0

Therefore, split the integral at 1/n. The left integral can be bounded directly, and the right after integration by parts [cf. the treatment of Eq. (44)]. One gets √ √ 1/n 1 log n 1 1 log n . I≤ − log φ dφ + dφ = O + √ n n 1/n 2φ − log φ n 0 In order to prove that this is also a lower bound for the asymptotics, it suffices to show this for the integral (32) as all other contributions vanish more quickly. To this end, split

86


the integral (32) into single oscillations of the sine, Jk = [ 2πn k , 2π(k+1) ], k ≥ 0. As n √ − log φ has negative slope on (0; 1), each of the Jk gives a positive contribution to I, and thus we can truncate the integral at 21 , I≥

2π(k+1) 1 ≤2 n

− log φ sin(nφ) dφ.

(33)

Jk

√ On [0; 21 ], − log φ has a positive curvature, and thus, each of the integrals can be esti√ mated by linearly approximating − log φ at the middle of each Jk but with the slope at 2π(k+1) , which gives n Jk

− log φ sin(nφ) dφ ≥

π n2

1 2π(k+1) n

1 − log

2π(k+1) n

.

Now, we plug this into the sum (33) and bound the sum by the integral from integrand in monotonically decreasing), which indeed gives a lower bound √ √ log 2) on I and thus proves the ( log n/n 2 ) scaling.

2π 1 n to 2

1 n(

log

(the

n 2π

−

6.2. Higher dimensions. For more than one dimension, the situation is more involved. First of all, it is clear by taking many uncoupled copies of the one-dimensional chain that there exist cases where the correlations will decay as n −2 . However, these are very special examples corresponding to Hamiltonians with a tensor product structure Hi1 i2 , j1 j2 = Hi1 , j1 Hi2 , j2 . In contrast, we show that for generic systems the correlations in the critical case decay as O(n −(d+1) log n), where d is the dimension of the lattice. The requirement is again that the energy spectrum E(φ) has only a finite number of zeroes, i.e., finitely many critical points. Note that the case of a Hamiltonian with a tensor product structure can also be solved, as in that case Vˆ becomes a product of terms depending on one φi each and thus the integral factorizes. Interestingly, although the correlations along the axes decay as n −2 , −2 n−2d and the correlations in a fixed diagonal direction will decay as n −2 1 · · · nd ∝ −(d+1) thus even faster than in the following theorem. The O n log n decay of the theorem holds isotropically, i.e., independent of the direction of n. Theorem 10. Consider a d-dimensional bosonic lattice with a critical Hamiltonian H = V ⊕ 1. 1 Then the P-correlations of the ground state decay as O n−(d+1) log n if the following holds: Vˆ ∈ C d+1 [e.g., the correlations decay as O(n−(2d+1+ε) ), ε > 0],Vˆ has only a finite number of zeros which are quadratic minima, i.e., the 2 ˆ (φ) Hessian ∂∂φVi ∂φ is positive definite at all zeros. j ij


87

Proof. We have to evaluate the asymptotic behavior of the integral 1 d (Vˆ 1/2 )n = d φ Vˆ (φ) cos[nφ]. (2π )d T d Let us first briefly sketch the proof. We start by showing that it suffices to analyze each critical point separately. To this end, we show that is is possible to smoothly cut out some environment of each critical point which reproduces the asymptotic behavior. Then, we rotate the coordinate system such that we always look at the correlations in a fixed direction, and integrate by parts—which surprisingly can be carried out as often as Vˆ is differentiable, as all the brackets vanish. Therefore, the information about the asymptotics is contained in the remaining integral, and after a properly chosen number of partial integrations, we will attempt to estimate this term. Let now ζi , i = 1, . . . , I be the zeros of Vˆ . Clearly, these will be the only points which contribute to the asymptotics as everywhere else Vˆ is C d+1 . In order to separate the contributions coming from the different ζi , we will make use of so-called neutralizers [27]. For our purposes, these are functions Nξ0 ,r ∈ C ∞ (Rd → [0; 1]) which satifsy 1 : ξ − ξ0 ≤ r/2 Nξ0 ,r (ξ ) = 0 : ξ − ξ0 ≥ r and are rotationally symmetric (cf. [27] for an explicit construction). For each ζi , there exists an ri such that the balls Bri (ζi ) do not intersect. We now define the functions f i (φ) :=

Vˆ (φ) Nζi ,ri (φ), ρ(φ) :=

Vˆ (φ) −

I

f i (φ).

i=1

Clearly, ρ is C d+1 , and so is each f i except at ζi . Furthermore, each f i is still the square root of a C d+1 function. By definition, (Vˆ −1/2 )n =

I 1 1 d d φ f (φ) cos[nφ] + dd φρ(φ) cos[nφ], (34) i d d d (2π )d (2π ) T i=1 T

i.e., it suffices to look at the asymptotics of each f i separately. The contribution of ρ is O(n−(d+1) ) as can be shown by successive integrations by parts just as for the non-critial lattice (cf. the proof of Lemma 2). Let us now analyze the integrals Ii = dd φ f i (φ) cos[nφ]. Bri (ζi )

The integration range can be restriced to Bri (ζi ) as f i vanishes outside the ball. By a rotation, this can be mapped to an integral where n = (n, 0, . . . , 0), whereas f i is rotated to another function f˜i with the same properties, Ii = dd φ f˜i (φ) cos[nφ1 ]. Bri (ζi )

88


Since the integrand is continuous and thus bounded, it is absolutely integrable, and from Fubini’s theorem, one finds ζi,1 +ri ˜ cos[nφ1 ], Ii = dd−1 φ˜ dφ1 f˜i (φ1 , φ) )

Bri (ζ˜i )

ζi,1 −ri

*+

˜ Ji (φ)

,

where we separated out the integration over the first component. The vector φ˜ denotes the components 2 . . . d of φ. The extension of the integration range to a cylinder is possible as f˜i vanishes outside Bri (ζi ). Let us now require φ˜ = ζ˜i . This does not change the integral since the excluded set is of measure zero, but it ensures that f˜i is in C d+1 . This allows us to integrate the inner ˜ by parts up to d + 1 times, and each of the brackets integral Ji (φ) # $ζi,1 +ri 1 (k) ˜ ˜ cos(nφ1 − kπ/2) f i (φ1 , φ) nk φ1 =ζi,1 −ri d ˜ = ∂ d f˜i (φ1 , φ)/∂φ ˜ appearing in the k th integration step vanishes. Here, f˜i(d) (φ1 , φ) 1 is th the d partial derivative with respect to the first argument. After integrating by parts d times, we obtain

1 Ii = nd

ζi,1 + ri

d

Bri (ζ˜i )

d−1

φ˜

˜ cos[nφ1 − dπ/2]. dφ1 f˜i(d) (φ1 , φ)

(35)

ζi,1 − ri

Now we proceed as follows: first, we show that the order of integration can be inter(d) changed, and second, we show that for the function obtained after integrating f˜i over ˜ the Fourier coefficients vanish as log(n)/n. φ, (k) The central issue for what follows is to find suitable bounds on | f˜i |. Therefore, 2 d+1 define f˜i =: h i ∈ C . By virtue of Taylor’s theorem, and as h i (ζi ) = 0 is a minimum, h i (φ) = 21 (φ − ζi ) · (D2 h i (ζi ))(φ − ζi ) + o(φ − ζi 2 ) with D2 the second derivative. As the first term is bounded by 21 D2 h i (ζi )∞ φ − ζi 2 and the second vanished faster than φ − ζi 2 , we can find εi > 0 and C1 > 0 such that |h i (φ)| ≤ C1 φ − ζi 2

∀φ − ζi < εi .

(36)

By looking at the Tayor series of h i ≡ ∂h i /∂φ1 up to the first order we also find that there are εi > 0 and C2 > 0 such that |h i (φ)| ≤ C2 φ − ζi

∀φ − ζi < εi .

(37)

In addition to these upper bounds, we will also need a lower bound on |h i |. Again, by the Taylor expansion of h i around ζi , we find |h i (φ)| ≥ λmin D2 h i (ζi ) − o(φ − ζi 2 ),


89

and as all the zeros are quadratic minima, i.e., λmin D2 h i (ζi ) > 0, there exist εi > 0, C3 > 0 such that |h i (φ)| ≥ C3 φ − ζi 2

∀φ − ζi < εi .

(38)

Clearly, εi can be chosen equal in Eqs. (36–38). Note that the bounds can be chosen to be invariant under rotation of h i and thus of f˜i . This holds in particular for the εi as the remainders of Taylor series vanish uniformly. Thus, the bound we will obtain for the correlation function indeed only depends on n and not on the direction of n. (k) Now, we use the conditions (36–38) to derive bounds on | f˜i |. Therefore, note that √ from f˜i ≡ h i it follows that ( j1 )

f˜i(k) =

j1 +···+ jk =k jν =0,1,2,...

c j1 ... jk h i

(2k−1)/2

hi

( jk )

· · · hi

.

One can easily check that for each term in the numerator, the number K 0 of zeroth derivatives and the number K 1 of first derivatives of h i satisfy 2K 0 + K 1 ≥ k. By bounding all higher derivatives of h i from above by constants, we find that the modulus of each summand in the numerator, and thus the modulus of the numerator itself, can be bounded above by C φ − ζi k in the ball Bεi (ζi ) with some C > 0. On the other hand, it follows directly from (38) that the modulus of the denominator is bounded below by C φ − ζi 2k−1 , C > 0, such that in total | f˜i(k) (φ)| ≤ C

1 ; 1 ≤ k ≤ d + 1. φ − ζi k−1

(39)

Note that this holds not only inside Bεi (ζi ) but in the whole domain of f i , as outside Bεi (ζi ), f i is C d+1 and thus all the derivatives are bounded. Equation (39) is the key result for the remaining part of the proof. First, it can be used to bound the integrand in (35) by an integrable singularity (this is most easily seen in spherical coordinates, where 1/r d−1 is integrable in a d-dimensional space). Hence, the order of integration in (35) can be interchanged, and it remains to investigate the asymptotics of the integral ζi,1 + ri

1 Ii = nd gi (φ1 ) ≡

dφ1 gi (φ1 ) cos[nφ1 − dπ/2], with

(40)

ζi,1 − ri

(d) ˜ dd−1 φ˜ f˜i (φ1 , φ).

(41)

Bri (ζ˜i )

From (39), we now derive bounds on gi (φ1 ) and its first derivative. Again, we may safely fix φ1 = ζi,1 as this has measure zero. Then, using (39) we find that

ri

|gi (φ1 )| ≤ 0

C d−2 dr, (d−1)/2 Sd−1r 2 2 (φ1 − ζi,1 ) + r )

90


where we have transformed into spherical coordinates [Sd−1 is the surface of the (d −1)dimensional unit sphere] and assumed the l2 -norm. Since (φ1 − ζ1 )2 + r 2 ≥ r 2 , the integrand can be bounded once again, and we find

ri

C Sd−1 dr ((φ − ζi,1 )2 + r 2 )1/2 1 0 $ # = C − log |φ1 − ζi,1 | + log ri + ri2 + (φ1 − ζi,1 )2

|gi (φ1 )| ≤

≤ −C log |φ1 − ζi,1 |,

(42)

where in the last step we used that in (40) |φ1 − ζi,1 | < ri and that ri can be chosen sufficiently small. Next, we derive a bound on gi (φ1 ). As we fix φ1 = ζ1 , the integrand in (41) is C 1 and we can take the differentiation into the integral, gi (φ1 )

=

˜ dd−1 φ˜ f˜i(d+1) (φ1 , φ).

Bri (ζ˜i )

Again, we bound the integrand by virtue of Eq. (39) and obtain |gi (φ1 )|

ri

C Sd−1 dr 2 2 0 ((φ1 − ζi,1 ) + r ) ri arctan |φ1 −ζ C i,1 | ≤ . =C |φ1 − ζi,1 | |φ1 − ζi,1 | ≤

(43)

Finally, these two bounds will allow us to estimate (40) and thus the asymptotics of the correlations in the lattice. We consider one half of the integral (40), ζi,1 + ri

dφ1 gi (φ1 ) cos[nφ1 − dπ/2],

(44)

ζi,1

as both halves contribute equally to the asymptotics. We then split the integral at ζi,1 + ri /n. The left part gives " " " ζi,1 +r " /n " i " (42) " " dφ1 gi (φ1 ) cos[nφ1 − dπ/2]" ≤ " " " " ζi,1 "

ζi,1 +r i /n

dφ1 (− log |φ1 − ζi,1 |) ζi,1

ri − ri log ri + ri log n . = n

(45)


91

The right part of the split integral (44) can be estimated by integration by parts, " " " ζ " i,1 +ri " " " " dφ1 gi (φ1 ) cos[nφ1 − dπ/2]" " " " " ζi,1 +ri /n " "# " ζ i,1 +ri $ζi,1 +ri " " 1 1 " " ≤ " gi (φ1 ) dφ1 |gi (φ1 )| cos[nφ1 − (d + 1)π/2] "+ " " n n ζi,1 +ri /n ζi,1 +ri /n

(42,43)

≤

C

| log ri | log n + C . n n

(46)

Thus, both halves [Eqs. (45),(46)] give a log n/n bound for the integral (44), and thus the integral Ii is asymptotically bounded by log n/nd+1 following Eq. (40). As the number of such integrals in (34) is finite, this proves that the correlations of the ground state decay at least as log n/nd+1 . Acknowledgements. We would like to thank Jens Eisert, Otfried Gühne, David Pérez García, Diego Porras, Tommaso Roscilde, Frank Verstraete, and Karl Gerd Vollbrecht for helpful discussions and comments. This work has been supported by the EU IST projects QUPRODIS and COVAQIAL.

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Communications in


A Geometric Approach to the Classification of the Equilibrium Shapes of Self-Gravitating Fluids Alvaro Pelayo1 , Daniel Peralta-Salas2 1 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043,

USA. E-mail: [email protected]

2 Departamento de Física Teórica II, Universidad Complutense, 28040 Madrid, Spain.

E-mail: [email protected] Received: 10 October 2005 / Accepted: 3 March 2006 Published online: 20 July 2006 – © Springer-Verlag 2006

Abstract: The classification of the equilibrium shapes that a self-gravitating fluid can take in a Riemannian manifold is a classical problem in Mathematical Physics. In this paper it is proved that the equilibrium shapes are isoparametric submanifolds. Some geometric properties of them are also obtained, e.g. classification and existence for some Riemannian spaces and relationship with the isoperimetric problem and the group of isometries of the manifold. Our approach to the problem is geometrical and allows to study the equilibrium shapes on general Riemannian spaces. 1. Introduction Let (M, g) be an analytic, complete and connected (without boundary) Riemannian n-manifold and an open connected subset of M (bounded or not) occupied by a mass of fluid. We say that a fluid is self-gravitating if the only significative forces are its interior pressure and its own gravitation. A fluid in these conditions represents a simplified stellar model of fluid-composed star. Depending on whether the gravitational field is modelled by Poisson or Einstein equations we say that the fluid is Newtonian or relativistic. An important problem in Fluid Mechanics consists in studying the shape that a self-gravitating fluid will take when it reaches the equilibrium state. By the term shape of a fluid it is meant the topological and geometrical properties of the boundary ∂. The mathematical description of this kind of fluids only involves three physical quantities, the gravitational potential, which is a function f 1 : M → R (constant on ∂), and the density and pressure, which are two functions f 2 , f 3 : → R. The set of partial differential equations fulfilled by the functions ( f 1 , f 2 , f 3 ) is of free-boundary type because the domain is an unknown of the problem. In the relativistic case the metric tensor g is also an unknown and it must satisfy the coupling condition Rab = f 1−1 f 1;ab + 4π( f 2 − f 3 )gab , Rab standing for the Ricci tensor and; standing for the covariant derivative.

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A. Pelayo, D. Peralta-Salas

The standard approaches to the problem of classifying the equilibrium shapes of ∂ generally employ analytical techniques. In the Newtonian case maximum principles for elliptic equations are used in order to prove the existence of symmetries of the solutions ( f 1 , f 2 , f 3 ). Lichtenstein [1] and later on Lindblom [2] proved the existence of spherical symmetry (i.e. ∂ is a round sphere) when M is the Euclidean R3 , is bounded and the functions ( f 1 , f 2 , f 3 ) satisfy some physical constraints. In the relativistic case arguments involving the positive mass theorem are used for obtaining the conformal flatness of the metric tensor. As a consequence of this technique Beig&Simon [3] and Lindblom&Masood-ul-Alam [4] proved again spherical symmetry when is bounded and the solutions verify certain physical hypotheses. Despite these important results many questions remain open: What about Newtonian fluids on Riemannian manifolds? What about unbounded domains ? Do the same results hold if we drop the physical assumptions? In this paper we approach the problem in a different, more geometrical way. Indeed, since the gravitational potential f 1 is constant on ∂ then the study of the geometric properties of the level sets of f 1 is an effective procedure to classify the equilibrium shapes that the fluid can take, thus connecting our problem with the philosophy of the geometric theory of PDEs. The literature on this field is essentially focused on the study of general properties of the level sets of the solutions to differential equations, e.g. critical levels [5, 6], convexity or starshapedness of the levels [7, 8], order of vanishing and measure for level sets [9, 10], symmetries of the levels due to overdetermined boundary conditions [11–15], . . .. This work provides another contribution to the abundant literature on geometric theory of PDEs, and as far as we know, the techniques that we develop for classifying the shapes of ∂ (e.g. the analytic representation property) are new and independent of the other approaches to similar problems. It is interesting to observe that in Serrin’s paper [11, 12] it is proved that solutions to certain boundary problems exist only when the domain is a sphere (for related results see the literature on overdetermined boundary problems, e.g. [15] and references therein). However Serrin’s method, somehow related to the classical approaches to Newtonian self-gravitating fluids by Lichtenstein and Lindblom (i.e. moving plane technique and maximum principles), bear no similarity with our methods (in fact the physical motivation of Serrin’s problem is not related to static fluids). Let me summarize the organization of this paper in three blocks. – In Sects. 2 and 3 we establish the notation of the paper, prove some elementary lemmas and formulate the problem. The equations that we study include the Newtonian and the relativistic (without taking into account Eq. (1)) as particular cases. In Sect. 4 we prove that the function f 1 is analytically representable across ∂ and as a consequence of this remarkable property f 1 is proved to satisfy the equilibrium condition. This result is particularly relevant from the physical viewpoint since it allows us to classify the equilibrium shapes of a self-gravitating fluid on any Riemannian manifold. Our techniques are different from the previous ones which appeared in the literature on self-gravitating fluids, and could be of interest to PDE theorists working on (overdetermined) free-boundary problems. – In the second block we study the geometrical consequences of the equilibrium condition. In Sect. 5 we prove that the (regular) level sets of the function f 1 are isoparametric submanifolds, thus connecting the shapes of static fluids with these classical objects of differential geometry, and in Sect. 6 we obtain classification theorems for certain spaces (M, g). In Sect. 7 necessary and sufficient conditions for the existence of certain equilibrium shapes are obtained and the 3-manifolds admitting

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fluid-composed stars are classified. In Sect. 8 the relationship between the equilibrium shapes, the isoperimetric problem and the Killing vector fields of M is studied. Apart from the physical relevance (most of our statements have not been obtained through the classical approaches) our results could be of interest to differential geometers. – In Sects. 9 and 10 some open problems are discussed, the most important being the extension of our techniques to general relativistic fluids. 2. Notation and Preliminary Results Let f be a smooth, real-valued function on M. For each c ∈ f (M) ⊂ R we have a pre-image f −1 (c). Let Vci be the i th connected component of f −1 (c). The partition induced on M by f is defined as Vci . βM ( f ) = i,c

Analogously if we have an open set U ⊂ M the partition induced on U by f |U is called βU ( f ). Each Vci is called a leaf of the partition. In general the dimension of the leaves of the partition is not constant because there can exist singular fibres (∇ f = 0) and therefore β M ( f ) is a singular foliation. We say that a function f represents the partition of U if βU ( f ) = . If f is real analytic (C ω ) then is said to admit an analytic representation. Analogously we call f analytically representable on U ⊂ M if βU ( f ) admits an analytic representation. We will say that a family { f i : M → R} agrees fibrewise if β M ( f i ) = β M ( f j ) for all i, j. The following extension lemma will be useful in forthcoming sections. Lemma 1. Let f , g be real analytic functions on M. Let U ⊂ M be an open set. Then βU ( f ) = βU (g) =⇒ β M ( f ) = β M (g). Proof. Since βU ( f ) = βU (g) we have that rank(d f, dg) ≤ 1 in U . Since U is open and f , g are analytic functions this condition extends to the whole M, i.e. rank(d f, dg) ≤ 1 in M. This implies [16] that f and g are functionally dependent and hence there exists an analytic function Q : R2 → R such that Q( f, g) = 0 in M, thus showing that the partitions induced by f and g agree. Let S be a codimension one orientable submanifold in M. The metric induced by g on S is given by βab = gab − n a n b [17], where n a is the unit normal vector field to S on M. The extrinsic curvature or second fundamental form of S is defined as Hab = βac βbd n d;c = 1/2L n (βab ) [17], L n standing for the Lie derivative with respect to the unit normal vector field. If S is given as the zero-set of the function f , i.e. S = { p ∈ M : f ( p) = 0} and d f |S = 0, then the mean curvature H of S (the trace of the second fundamental form) is given by the following expression: ∇f , (2) H = div |∇ f | div and ∇ standing for the divergence and gradient operators on the Riemannian manifold.

a is the Riemann curvature tensor induced by R a on S then the Gauss theorem If Rbcd bcd implies [17] that R = R − 2Rab n a n b + (Haa )2 − Hab H ab . (3)

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From this expression we deduce the relationship between the intrinsic sectional curvature of S (K ), the sectional curvature of M restricted to S (K ) and the Gauss curvature of S ( K¯ ), namely K = K + K¯ [17]. Let us finish this section with the following lemma [18]. Lemma 2. Let f be a smooth function on an open set U ⊆ M saturated by level sets of f . If |d f | ≥ m > 0 on U then f is a (locally trivial) fibre bundle on U . Proof. The normal vector field X =

∇f (∇ f )2

is smooth on U and a symmetry of f because

L X ( f ) = 1 [19]. X is complete because (M, g) is complete and |X | ≤ a global 1-parameter group of diffeomorphisms [20].

1 m , thus defining

3. Statement of the Problem The problem (P) in which we are interested is a system of PDEs defined on M. Its form and additional regularity assumptions are inspired by the equations modelling static self-gravitating fluids, both Newtonian and relativistic [21]. The boundary of the fluid region, ∂, is assumed to be a codimension one analytic submanifold (connected or ¯ constant on ∂ and f 3 is not). The functions f 2 and f 3 are required to be analytic in , not allowed to be constant in the whole . With these assumptions in mind we state now the equations of problem (P) ( is the Laplace–Beltrami operator on the manifold). In this paper we consider only classical solutions. f 1 = F( f 1 , f 2 , f 3 ) in , H ( f 1 )∇ f 3 + G( f 2 , f 3 )∇ f 1 = 0 in , f 1 = c, c ∈ R, ∇ f 1 = 0 and f 1 ∈ Ct2 on ∂, ¯ f 1 = 0 in M − ,

(4) (5) (6) (7)

where F, G and H are (not identically zero) analytic functions. We also impose that G is not a constant. The symbol Ct2 in (6) means that f 1 is C 1 on the boundary and its tangential second derivatives f 1,i j t j are continuous for any (local) vector field t = t i ∂i tangent to the boundary. This assumption is analogous to Synge’s junction condition for the metric tensor in General Relativity [22]. The normal components of the second derivatives will not be continuous in general. Note that the constant c in (6) is not a priori prescribed and that the domain is an unknown of (P). For the sake of simplicity we have assumed that ∇ f 1 = 0 on ∂, but all the results of this paper hold only requiring that ∇ f 1 is not identically zero on the boundary. It is interesting to observe that one only needs to assume that ∂ is smooth enough, its analyticity following from general properties of elliptic free-boundary equations [23, 24]. Remark 1. The solutions to problem (P) are the functions ( f 1 , f 2 , f 3 ) and the domains satisfying all the required conditions. For certain values of F, G and H or certain manifolds (M, g) problem (P) could have no solutions at all. Since we are not interested in the existence problem we will suppose that solutions to (P) exist and characterize the structure of the level sets of these solutions. This is in strong contrast to the classical approaches where the problem of existence and uniqueness is first considered and then the geometrical restrictions arise.


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Let us prove three important (elementary) lemmas which will be used in the following sections. The first one is a consequence of [25–27], and is stated without proof. Lemma 3. The solutions f 1 to (P) are analytic on M − ∂. Lemma 4. The solutions f 1 , f 2 and f 3 of (P) agree fibrewise in . Proof. From (5) it follows that d G(df2f,3 f3 ) = −d Hd(ff11 ) = 0 and hence G , f2 d f 2 ∧ d f 3 = 0 (the subscript denotes, as usual, partial differentiation). Therefore ∇ f 2 and ∇ f 3 are linearly dependent. Also from (5) d f 3 = − G(Hf(2f,1f)3 ) d f 1 and taking the exterior derivative in this equation we get (G , f2 d f 2 +G , f3 d f 3 )∧d f 1 = 0. The linear dependence of ∇ f 2 and ∇ f 3 implies the linear dependence of all ∇ f 1 , ∇ f 2 and ∇ f 3 . This fact and the analyticity of the functions imply that they agree fibrewise in the whole . For any point p ∈ ∂ consider a small enough open neighborhood U ⊂ M. Define the open sets Uin = U ∩ = ∅ and Uout = U ∩ (M − ) = ∅. Lemma 5. f 2 and f 3 are functions of f 1 in Uin . Proof. Suppose that ∇ f 1 does not vanish in U (it is always possible by continuity and the fact that (∇ f 1 )|∂ = 0). If we cover U with a local coordinate system (x1 , . . . , xn ) we can assume, without loss of generality, that f 1,x1 = 0. The implicit function theorem guarantees the following steps in Uin : x1 = X 1 ( f 1 , x2 , . . . , xn ) =⇒ f 2 = f 2 (X 1 ( f 1 , x2 , . . . , xn ), x2 , . . . , xn ) ≡ F2 ( f 1 , x2 , . . . , xn ) =⇒ f 3 = f 3 (X 1 ( f 1 , x2 , . . . , xn ), x2 , . . . , xn ) ≡ F3 ( f 1 , x2 , . . . , xn ). It is easy to check that Fi,x2 = . . . = Fi,xn = 0, i = 2, 3. One only has to take into account the implicit function theorem and Lemma 4. Hence in Uin we get that f 2 = F2 ( f 1 ) and f 3 = F3 ( f 1 ), where F2 and F3 are analytic functions of their argument. Since the pressure, the density and the gravitational potential induce the same partition in Uin we can reduce our study to β M ( f 1 ), which has the advantage of being defined on the whole M. The following sections are devoted to understand the topological and geometrical properties of this partition, which will only depend on the base manifold (M, g). 4. Analytic Representation and Proof of the Classification Theorem When working with partitions of U ⊆ M it is reasonable to try to represent them by functions as good as possible, e.g. analytic functions. The general programme would be to substitute the “pathological” function g : U → R, which represents the partition, by an analytic function f satisfying βU (g) = βU ( f ). In general a partition cannot be analytically represented. Even in the case in which the function g is analytic in the whole M except for the fibre g −1 (0) (as happens with the potential f 1 solution to (P)), an analytic representation does not need to exist. The following examples illustrate the difficulties which may arise.

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Example 1. Let f : Rn → R be defined as f (x) = |x|2 − 1 (|.| standing for the Euclidean norm) and let h : R → R be given by h(t) = t exp(−1/t 2 ) if t = 0 and h(0) = 0. Then the function g = f + h ◦ f is C ∞ on Rn , analytic on Rn − g −1 (0) and agrees fibrewise with f . So f is an analytic representation of g. Example 2. Let g : Rn → R be a smooth function defined in coordinates (x1 , . . . , xn ) as x1 (1 + x2 exp(−1/x12 )) if x1 > 0 g(x1 , . . . , xn ) = x1 if x1 ≤ 0 It is not difficult to prove that g does not admit analytic representation in any neighborhood of the fibre {x1 = 0}. Similar examples can be constructed when the “pathological” fibre is compact. In spite of these results, the gravitational potential f 1 , which is generally not analytic on the boundary, has a remarkable property: its partition can be represented by analytic functions in a neighborhood of ∂. Before proving this result let us comment on a physical reason supporting this property. Since ∂ is an analytic submanifold and (∇ f 1 )|∂ = 0 then there exists an analytic function I : M → R, arbitrarily close to f 1 (in the C ∞ strong topology), such that I −1 (0) = ∂ [28]. This proves that a small perturbation of the gravitational potential makes its partition analytically representable. Since two arbitrarily close partitions are indistinguishable from the physical viewpoint, this suggests that β M ( f 1 ) admits itself analytic representation. The rigorous proof of this result is a modification of a technique developed by Lindblom in [29]. Theorem 1 (Analytic representation property). Let f 1 be any solution to (P), then β M ( f 1 ) = β M (I ) where I : M → R is a function analytic on the whole M. Proof. Let U be a small enough neighborhood of p ∈ ∂. On account of Lemma 5 we can write the equations defining (P) in terms of just f 1 : ˆ f 1 ) in Uin , f 1 = F( 2 f 1 = c, c ∈ R, ∇ f 1 = 0 and f 1 ∈ Ct on ∂ ∩ U, f 1 = 0 in Uout .

(8) (9) (10)

Consider a vector field ξ = ξ i ∂i which is a symmetry of f 1 in Uin , i.e. ξ( f 1 ) = 0. Note ¯ that ξ can always be chosen analytic. The assumption that f 2 and f 3 are analytic in ˆ implies that F in Eq. (8) is an analytic function of f 1 . It follows that the interior solution f 1 , satisfying Eq. (8) and the boundary condition f 1 = c, is analytically continuable across the boundary [30], although its continuation does not generally coincide with the exterior solution. Consequently ξ i and ξ,i j can be extended to ∂ as analytic functions and in fact ξ( f 1 )|∂ = 0. Now let us extend the vector field ξ beyond the free-boundary in such a way that the extension is analytic in the whole U . The components of the new vector field ξˆ in U are defined by the following boundary problem: ξˆ k +

2Dµ Dk f 1 µ k Rik D i f 1 k ξˆ = 0 , (D ξˆ ) + f 1,k f 1,k

(11)


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k = 1, . . . , n, provided with the boundary conditions on ∂ given by the values (ξˆ i )|∂ = (ξ i )|∂ and (ξˆ,i j )|∂ = (ξ,i j )|∂ of the interior symmetry. Note that the symbol D stands for the covariant derivative and Rik is the Ricci curvature. In order that (11) be well defined assume, without loss of generality, that in U all the components of ∇ f 1 are non zero. Since all the terms of (11) are analytic in U in and U out , the boundary conditions and ∂ are also analytic and the equation is linear and elliptic there exists a (unique) extension which defines an analytic vector field in U [25, 26] (Cauchy-Kowalewsky theorem). This analytic vector field has the property of being a symmetry of f 1 in U . Indeed, the first step consists in the following computation 2Dµ Dk f 1 µ k D µ Dµ Dk f 1 k ξˆ . (12) (ξˆ k f 1,k ) = Dµ D µ (ξˆ k f 1,k ) = f 1,k ξˆ k + (D ξˆ ) + f 1,k f 1,k Note now that Dµ D µ Dk f 1 = Dk ( f 1 ) + Rik D i f 1 and since f 1 is solution of (P) we get from Eq. (11) and (12) that (ξˆ k f 1,k ) = Fˆ ( f 1 )ξˆ k f 1,k in Uin , (ξˆ k f 1,k ) = 0 in Uout .

(13) (14)

The boundary conditions are the following. First note that f 1,k ξˆ k is C 1 on ∂. Indeed, recall that on ∂ f 1 is C 1 and its tangential second derivatives are continuous, therefore f 1,k ξˆ k and ( f 1,k ξˆ k )i = f 1,ik ξˆ k + f 1,k ξˆ,ik are also continuous on the boundary. The same applies to f 1,k ξ k . Since f 1,k ξ k = 0 in Uin then ( f 1,k ξ k )|∂ = 0 and ∂i ( f 1,k ξ k )|∂ = 0, and hence the boundary conditions are ( f 1,k ξˆ k )|∂ = 0 and ∂i ( f 1,k ξˆ k )|∂ = 0. Note now that Eq. (13) is analytic in U¯ in because the interior solutions f 1 , f 2 , f 3 can be analytically continued across the boundary. Holmgren’s theorem for linear analytic elliptic equations [31] implies that the (unique) C 1 solutions to (13) and (14) provided with the boundary conditions are ξˆ k f 1,k = 0 in Uin and ξˆ k f 1,k = 0 in Uout , thus showing that ξˆ k f 1,k = 0 in the whole U . Summarizing we have a (local) Lie algebra of n − 1 independent analytic symmetries of f 1 in U . From this Lie algebra we can reconstruct the partition βU ( f 1 ) via Frobenius theorem, the analyticity of ξˆ implying that the partition is analytic [32]. This ensures the existence of an analytic function Iˆ : U → R such that βU ( f 1 ) = βU ( Iˆ). By Lemma 5 it follows that Iˆ = F( f 1 ), thus showing that Iˆ extends to a saturated neighborhood of ∂, N (∂), as analytic function Iˆ : N (∂) → R. This result and the analyticity of f 1 in M − ∂ prove that β M ( f 1 ) is analytically representable across any leaf, and therefore there exists an analytic extension I : M → R of Iˆ such that β M ( f 1 ) = β M (I ), thus proving the claim. Concerning the physical meaning of the analytic representation property (ARP) we must say the following. From the proof of Theorem 1 it follows that the interior symmetries propagate across the free-boundary and remain symmetries of the exterior solution. This implies that a physical matching on a free-boundary (at least in static situations) does not only guarantee the continuity of the gravitational field but also the dependence between the external and internal properties of the fluid.

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Definition 1. An analytic function I : M → R is of equilibrium on U ⊆ M if I , (∇ I )2 and I agree fibrewise on U . A partition induced by an equilibrium function is called an equilibrium partition. Let us now prove the classification theorem. Theorem 2 (Classification theorem). If f 1 is a solution to the problem (P) then β M ( f 1 ) is an equilibrium partition. Proof. If p ∈ ∂ and U is a small neighborhood of p then f 2 and f 3 can be expressed ˆ f 1 ) in Uin . In Uout as functions of f 1 in Uin (see Lemma 5), thus implying that f 1 = F( the potential satisfies f 1 = 0. Theorem 1 ensures the existence of an analytic function I agreeing fibrewise with f 1 . The same technique as in Lemma 5 can be applied in order to show that f 1 = R(I ) in U . From Eq. (7) it follows R(I ) = 0 in Uout , which is equivalent to R

(I )(∇ I )2 + R (I )I = 0, I (∇ I )2

= (I ) in Uout . Since I is analytic so are I and (∇ I )2 and hence ˆ be the analytic continuation of to U (which indeed exists is analytic in U . Let

and therefore I (∇ I )2

(15)

ˆ ) in U and in particular in because of the analyticity of I and (∇II )2 ). Then (∇II )2 = (I Uin . On the other hand ˆ ) (16) R

(I )(∇ I )2 + R (I )I = F(I in Uin and together with (15) implies that I and (∇ I )2 depend only on I . The argument applies to the whole of U by analyticity and in fact the property of agreeing fibrewise extends (Lemma 1) to the whole M (although generally I and (∇ I )2 can be written as functions of I only locally). Since I and f 1 agree fibrewise we have that β M ( f 1 ) is an equilibrium partition. Theorem 2 provides a complete characterization of the level sets of the solutions to problem (P). In the following sections we will explore the geometrical and topological meaning of the equilibrium condition. This theorem applies to Newtonian self–gravitating fluids thus characterizing their equilibrium shapes on any Riemannian manifold. It also works for relativistic fluids without coupling between matter and geometry (see Eq. (1)). This kind of relativistic models, where the base space is predetermined, is used in some applications of interest [33]. In Sect. 9 it will be discussed how to extend our techniques to general relativistic fluids. Remark 2. It is interesting to observe that we do not impose additional assumptions in order to characterize the structure of the level sets of the potential. In the literature additional hypotheses are usually considered: |∇ f 1 | is a function of f 1 [34], existence of a “reference spherical model” [3] or physical constraints, e.g. positivity of the density and pressure, asymptotic structure of the potential and existence of state equation [1, 2, 4]. 5. General Geometric Properties of the Equilibrium Shapes Theorem 2 reduces the original problem involving a difficult system of PDEs to a purely geometrical problem: the classification of equilibrium partitions on different spaces. For


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instance, in Sect.7 and 8, we will show examples of manifolds which do not admit equilibrium functions of certain types, thus obtaining by geometrical arguments an existence result: (P) cannot have these types of solutions on these spaces. Although the definition of equilibrium partition involves a particular function I the concept is mainly geometrical, as the following proposition shows. Proposition 1. Any analytic function representing an equilibrium partition is an equilibrium function. Proof. By hypothesis there exists an equilibrium function I representing . Consider another function Iˆ representing the same partition. By using the same argument as in Lemma 5 it is immediate to prove that Iˆ = F(I ) for a certain open set U , F being an analytic function. Since I is of equilibrium we have that locally (by the same argument) (∇ I )2 and I are functions of I . Now a straightforward computation yields that (∇ Iˆ)2 = F (I )2 (∇ I )2 and Iˆ = F

(I )(∇ I )2 + F (I )I . This implies that locally (∇ Iˆ)2 and Iˆ are functions of Iˆ, thus proving that Iˆ is a local equilibrium function. The globalization of this property follows from Lemma 1. Let us now prove a remarkable result which relates the equilibrium condition to the well known isoparametric property. Recall that a smooth function f : M → R is called isoparametric if (∇ f )2 = F( f ) and f = G( f ) in M, F and G being differentiable functions [35]. A regular level set of an isoparametric function is called isoparametric submanifold and the union of level sets is called isoparametric family. This concept was firstly introduced by Levi-Civita [36], Cartan [37, 38] and Segre [39] in a purely geometrical context. Two good surveys on this topic are the works of Nomizu [40] and Thorgbersson [41]. Note that in the literature it is sometimes considered another definition for an isoparametric family [42], which is not equivalent to the one considered in this paper. Before proving the theorem let us state some notation. Assume (without loss of generality) that the equilibrium function I has N different critical values {ci }1N (since I is analytic the set of critical values is discrete in R) and that I (M) = (−∞, +∞). The N +1 manifold M is therefore stratified as follows, M = i=1 Mi ∪ C( f ), C( f ) standing for the critical set of I and Mi = I −1 (ci−1 , ci ) (c0 = −∞ and c N +1 = +∞). Each set j Mi is possibly made up by several connected components Mi . j

Theorem 3. An equilibrium function is isoparametric on each Mi and hence its regular level sets are isoparametric submanifolds. j

Proof. In the open regions Mi the equilibrium function I is submersive and by Lemma 2 the partition is globally trivial. In Proposition 1 it was proved that (∇ I )2 and I are j functions of I in a certain open subset of Mi . The globalization of this property to the j whole Mi stems from the existence of a global transversal (non-closed) curve to the fibres of I , which is a consequence of the triviality of the partition (note that I can be j adapted to a coordinate system in Mi [18]). It is interesting to observe that the isoparametric character of an equilibrium function is generally only local, the following example illustrating this fact. Example 3. The analytic function I (x, y) = cos x 2 + y 2 in (R2 , δ) is of equilibrium type (its partition is formed by concentric circles). On the contrary it is not a global

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isoparametric function because (∇ I )2 = 1 − I 2 but I cannot be globally expressed as a function of I due to the existence of the critical fibres r = iπ , i ∈ N ∪ {0}. Anyway, as proved in Theorem 3, I is a well defined function of I in the domains M√ i = {iπ
0. This implies [61] that no stable curves exist. Other similar examples in dimension 2 can be found in the work of Ritoré. Another interesting example is given by the symmetric spaces of rank 1. The geodesic spheres are transitivity hypersurfaces of the group of isometries and therefore they are equilibrium submanifolds (Theorem 5). However not all the geodesic spheres are stable [60]. 9. Equilibrium Shapes of Relativistic Fluids The free-boundary problem (P) includes, as particular cases, the equations ruling Newtonian and relativistic fluids on Riemannian manifolds. In the relativistic case Einstein’s equations give rise to the additional constraint Rab = f 1−1 f 1;ab + 4π( f 2 − f 3 )gab ,

(24)

which expresses the coupling between the geometry of (M, g) and the potential f 1 . The metric tensor g can be proved to be analytic in M − ∂ [62] and Ct2 on ∂ (Synge’s junction condition [22]). If Eq. (24) is not taken into account (relativistic fluid model on a fixed space) then all the results obtained in this paper apply. When Eq. (24) is considered, Theorem 1 does not hold (its proof makes use of the analyticity of the metric on ∂), and therefore it is not possible to provide a full classification of the equilibrium shapes. Even in the case in which ARP could be proved, the proof of Theorem 2 would fail in general. This obstacle can be overcome when the metric is assumed to be conformally flat, i.e. g = e2φ δ, and the conformal factor φ agrees fibrewise with f 1 (this assumption is common in the literature, see e.g. [21]). In this case it is not difficult to show, proceeding as in Sect. 4, that ARP implies Theorem 2. Under mild physical assumptions it can be proved that the manifold M is diffeomorphic to Rn [63], and therefore Corollary 1 would imply (without imposing other physical constraints) that the equipotential sets are concentric spheres S n−1 , parallel hyperplanes Rn−1 or parallel coaxial cylinders S n−1−k × Rk (1 ≤ k ≤ n − 2). This would generalize, up to ARP, a theorem of Lindblom asserting that in dimension 3, and bounded domain , conformal flatness implies spherical symmetry [64]. Note that the hypothesis of conformal flatness is proved in [3, 4] under several physical assumptions. It would be interesting to prove ARP and Theorem 2 for general relativistic fluids. This would yield a complete classification of the equilibrium shapes without taking into account physical restrictions. Furthermore it would allow to detect the spaces on which ∂ is a round sphere or not. Note that the approaches of Beig & Simon and Lindblom&Masood-ul-Alam are adapted to prove the spherical symmetry of the equipotential sets, thus failing in more general situations. Our conjecture is that Theorems 1 and 2 remain true in the relativistic setting, without additional hypotheses. A possible proof may involve the concept of analytic representation of a metric. This requirement is rather natural since g, as f 1 , is an unknown of Eqs. (P) extending across the free-boundary. The question is how to define the analytic representation of a metric and how to prove that the metrics which are solutions to Eqs. (P) and (24) are analytically representable. If f 1 is shown to induce equilibrium partitions


113

then Kunzle’s work [34] (where his strong assumption (∇ f 1 )2 = F( f 1 ) would arise as a consequence of the equilibrium property) would imply the spherical symmetry of the fluid, just by taking into account some mild physical hypotheses. 10. Final Remarks and Open Problems This work has shown a connection between the level sets of the solutions to certain free-boundary problems and the equilibrium (isoparametric) condition. A remarkable consequence of this result has been the classification of the shapes of static fluids on manifolds. This suggests the interest, both mathematical and physical, of studying the structure of isoparametric submanifolds on Riemannian spaces. In this line many problems remain open, e.g. a complete classification in constant curvature or symmetric manifolds, and a better understanding of the interplay between geometry, topology and isoparametricity. An interesting consequence of our work is that we have provided a technique for characterizing the shapes of fluids on different spaces which is independent of whether ∂ is a round sphere or not. Up to now all the techniques available in the literature are adapted to the spherical symmetry situation. Consequently a deeper relationship between the geometrical and topological properties of (M, g) and the shapes of the fluid region has been established. We have shown that recovering the physical intuition that we have in the Euclidean space, e.g. the existence of contractible fluid domains, the connection with the isometries of (M, g) and the stability of the fluid regions, requires to restrict the base manifold. It would be interesting to ascertain whether techniques similar to the ones developed in this paper could be useful in other situations where the most interesting properties are of geometrical type, e.g. shapes of self-gravitating fluids in rotation, propagating interfaces, burning flames or computer vision [65]. Acknowledgements. A.P. thanks Maxim Kazarian for many helpful discussions and Ralf Spatzier for suggestions for improvement on a very preliminary version. A.P.’s research was partially supported by an EPSRC grant at the University of Warwick (UK) during the academic year 2001-2002. The latter part of his research was partially supported by Professor Peter Scott’s NSF grant at the University of Michigan. D.P.-S. is very grateful to Robert Beig, Rolando Magnanini, Mario Micallef, Stefano Montaldo, Renato Pedrosa, Cesar Rosales, Urs Schaudt and Janos Szenthe, for their interesting comments on different parts of this paper. He is also indebted to Alberto Enciso for his encouragement during the course of this work and the careful reading of the manuscript. Finally he is also grateful to Antti Kupiainen and the referee of the paper for their useful criticisms on previous versions of the article. D.P.-S.’s research was supported by FPI and FPU grants from UCM and MEC (Spain).

References 1. Lichtenstein, L.: Gleichgewichtsfiguren Rotiender Flüssigkeiten. Berlin: Springer, 1933 2. Lindblom, L.: Mirror planes in Newtonian stars with stratified flows. J. Math. Phys. 18, 2352 (1977) 3. Beig, R., Simon, W.: On the uniqueness of static perfect fluid solutions in general relativity. Commun. Math. Phys. 144, 373 (1992) 4. Lindblom, L., Masood-ul-Alam, A.K.M.: On the spherical symmetry of static stellar models. Commun. Math. Phys. 162, 123 (1994) 5. Bar, C.: Zero sets of solutions to semilinear elliptic systems of first order.Invent. Math. 138, 183 (1999) 6. Hardt, R. et al.: Critical sets of solutions to elliptic equations. J. Diff. Geom. 51, 359 (1999) 7. Caffarelli, L.A., Friedman, A.: Convexity of solutions of semilinear elliptic equations. Duke Math. J. 52, 431 (1985) 8. Cosner, C., Schmitt, K.: On the geometry of level sets of positive solutions of semilinear elliptic equations. Rocky Mount. J. Math. 18, 277 (1988)

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9. Lin, F.H.: Nodal sets of solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 44, 287 (1991) 10. Kukavica, I.: Level sets for the stationary solutions of the Ginzburg-Landau equation. Calc. Var. Part. Diff. Eqs. 5, 511 (1997) 11. Serrin, J.: A symmetry problem in potential theory. Arch. Rat. Mech. Anal. 43, 304 (1971) 12. Weinberger, H.F.: Remark on the preceding paper of Serrin. Arch. Rat. Mech. Anal. 43, 319 (1971) 13. Garofalo, N., Lewis, J.L.: A symmetry result related to some overdetermined boundary value problems. Amer. J. Math. 111, 9 (1989) 14. Henrot, A., Philippin, G.A.: Some overdetermined boundary value problems with elliptical free boundaries. SIAM J. Math. Anal. 29, 309 (1998) 15. Schaefer, P.W.: On nonstandard overdetermined boundary value problems. Nonlinear Anal. 47, 2203 (2001) 16. Newns, W.F.: Functional dependence. Amer. Math. Month. 74, 911 (1967) 17. Spivak, M.: A comprehensive introduction to differential geometry 5 vols. Berkeley: Publish or Perish, 1979 18. Gascon, F.G., Peralta-Salas, D.: On the construction of global coordinate systems in Euclidean spaces. Nonlinear Anal. 57, 723 (2004) 19. Gascon, F.G.: Non-wandering points of vector fields and invariant sets of functions. Phys. Lett. A 240, 147 (1998) 20. Avez, A.: Differential Calculus. Chichester: Wiley, 1986 21. Lindblom, L.: On the symmetries of equilibrium stellar models. Phil. Trans. R. Soc. Lond. A 340, 353 (1992) 22. Synge, J.L.: Relativity, the general theory. Amsterdam: North-Holland, 1966 23. Kinderlehrer, D., Nirenberg, L., Spruck, J.: Regularity in elliptic free-boundary problems. J. Anal. Math. 34, 86 (1978) 24. Karp, L., Margulis, A.S.: Newtonian potential theory for unbounded sources and applications to freeboundary problems. J. Anal. Math. 70, 1 (1996) 25. Morrey, C.B., Nirenberg, L.: On the analyticity of the solutions of linear elliptic systems of partial differential equations. Comm. Pure Appl. Math. 10, 271 (1957) 26. Morrey, C.B.: Multiple Integrals and the Calculus of Variations. Berlin: Springer, 1966 27. Browder, F.E.: The zeros of solutions of elliptic partial differential equations with analytic coefficients. Arch. Math. 19, 183 (1968) 28. Acquistapace, F., Broglia, F.: More about signatures and approximation. Geom. Dedicata 50, 107 (1994) 29. Lindblom, L.: Stationary stars are axisymmetric. Astrophys. J. 208, 873 (1976) 30. Morrey, C.B.: On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations: analyticity at the boundary. Amer. J. Math. 80, 219 (1958) 31. Hormander, L.: Linear Partial Differential Operators. Berlin: Springer, 1964 32. Sussmann, H.J.: Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180, 171 (1973) 33. Noble, S.C., Choptuik, M.W.: Collapse of relativistic fluids. Work in progress, available at http:laplace. physics.ubc.ca/∼ scn/fluad, 2002 34. Kunzle, H.P.: On the spherical symmetry of a static perfect fluid. Commun. Math. Phys. 20, 85 (1971) 35. Wang, Q.: Isoparametric functions on Riemannian manifolds. Math. Ann. 277, 639 (1987) 36. Levi-Civita, T.: Famiglie di superficie isoparametriche nell ordinario spazio euclideo. Rend. Accad. Naz. Lincei 26, 355 (1937) 37. Cartan, E.: Familles de surfaces isoparametriques dans les espaces a courbure constante. Ann. Mat. Pura Appl. 17, 177 (1938) 38. Cartan, E.: Sur quelques familles remarquables d’hypersurfaces. C. R. Congres Math. Liege 1, 30 (1939) 39. Segre, B.: Famiglie di ipersuperfie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni. Rend. Acc. Naz. Lincei 27, 203 (1938) 40. Nomizu, K.: Elie Cartan’s work on isoparametric families of hypersurfaces. Proc. Symp. Pure Math. 27, 191 (1975) 41. Thorbergsson, G.: Handbook of Differential Geometry, Vol. I, Amsterdam: North-Holland, 2000, pp. 963-995 42. Baird, P.: Harmonic maps with symmetry, harmonic morphisms and deformations of metrics. Res. Notes Math. 87, 1 (1983) 43. Serrin, J.: The form of interfacial surfaces in Korteweg’s theory of phase equilibria. Quart. Appl. Math. 41, 357 (1983) 44. Sakaguchi, S.: When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable. J. Anal. Math. 78, 219 (1999) 45. Alessandrini, G., Magnanini, R.: Symmetry and non-symmetry for the overdetermined Stekloff eigenvalue problem II. In Nonlinear Problems in Applied Mathematics. Philadelphia: SIAM, 1995


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46. Shklover, V.E.: Schiffer problem and isoparametric hypersurfaces. Rev. Mat. Iberoamericana 16, 529 (2000) 47. Tondeur, P.: Foliations on Riemannian Manifolds. New York: Springer, 1988 48. Molino, P.: Riemannian Foliations. Boston: Birkhauser, 1988 49. Hector, G., Hirsch, U.: Introduction to the geometry of foliations. Braunschweig: Friedr. Vieweg and Sons, 1986 50. Bochner, S.: Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52, 776 (1946) 51. Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of non-negative Ricci curvature. J. Diff. Geom. 6, 119 (1971) 52. Scott, P.: The geometries of 3-manifolds. Bull. London Math. Soc. 15, 401 (1983) 53. Sullivan, D.: Combinatorial invariants of analytic spaces. Proceedings of Liverpool Singularities Symposium I, Berlin: Springer, 1971, p.165 54. Besse, A.L.: Manifolds all of whose Geodesics are Closed. Berlin: Springer, 1978 55. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Providence: Amer. Math. Soc., 2001 56. Cao, J.: The existence of generalized isothermal coordinates for higher dimensional Riemannian manifolds. Trans. Amer. Math. Soc. 324, 901 (1991) 57. Nomizu, K.: On local and global existence of Killing vector fields. Ann. Math. 72, 105 (1960) 58. Palais, R.S.: On the existence of slices for actions of non-compact Lie groups. Ann. Math. 73, 295 (1961) 59. Szenthe, J.: On generalization of Birkhoff’s theorem. Preprint (2004) 60. Barbosa, J.L., DoCarmo, M., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197, 123 (1988) 61. Ritoré, M.: Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces. Commun. Anal. Geom. 9, 1093 (2001) 62. Muller zum Hagen, H.: On the analyticity of static vacuum solutions of Einstein’s equations. Proc. Cambridge Phil. Soc. 67, 415 (1970) 63. Masood-ul-Alam, A.K.M.: The topology of asymptotically Euclidean static perfect fluid space-time. Commun. Math. Phys. 108, 193 (1987) 64. Lindblom, L.: Some properties of static general relativistic stellar models. J. Math. Phys. 21, 1455 (1980) 65. Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge: Cambridge University Press, 1999 Communicated by A. Kupiainen


Communications in


On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour Boris Dubrovin SISSA, Via Beirut 2–4, 34014 Trieste, Italy. E-mail: [email protected] Received: 12 October 2005 / Accepted: 10 November 2005 Published online: 28 April 2006 – © Springer-Verlag 2006

Abstract: Hamiltonian perturbations of the simplest hyperbolic equation u t + a(u)u x = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.

1. Introduction In the present work we continue the study of Hamiltonian perturbations of hyperbolic PDEs initiated by the paper [10]. We consider here the simplest case of a single equation in one spatial dimension, u t + a(u)u x + b1 (u)u x x + b2 (u)u 2x + 2 b3 (u)u x x x + b4 (u)u x u x x + b5 (u)u 3x + · · · = 0. (1.1) Here is a small parameter; the coefficient of k is a graded homogeneous polynomial in the derivatives u x , u x x , …of the total degree (k + 1), deg u (n) = n, n > 0. The unperturbed equation u t + a(u)u x = 0

(1.2)

can be considered as the simplest example of a nonlinear hyperbolic system; the smooth functions b1 (u), b2 (u), etc. determine the structure of the perturbation.

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Such expansions arise, e.g., in the study of the long wave (also called dispersionless) approximations of evolutionary PDEs; see Sect. 5 below for other mechanisms that yield perturbed equations of the form (1.1). The unperturbed equation (1.2) admits a Hamiltonian description of the form δ H0 = 0, u t + {u(x), H0 } ≡ u t + ∂x δu(x) H0 = f (u) d x, f (u) = a(u), {u(x), u(y)} = δ (x − y).

(1.3)

(1.4)

The perturbed equations of the form (1.1) are considered up to equivalencies defined by Miura-type transformations [9] of the form (1.5) k Fk u; u x , . . . , u (k) , u → u + k≥1

where Fk (u; u x , . . . , u (k) ) is a graded homogeneous polynomial in the derivatives u x , u x x , . . . of the degree deg Fk = k. Using results of [15] (see also [6, 9]) one can show that any Hamiltonian perturbation of Eq. (1.2) can be reduced to the form δH = 0, H = H0 + H1 + 2 H2 + · · · , δu(x) Hk = h k u; u x , . . . , u (k) d x, deg h k u; u x , . . . , u (k) = k.

u t + ∂x

Recall that for H =

(1.6)

h(u; u x , u x x , . . . ) d x, δH = E h, δu(x)

where E=

∂ ∂ ∂ − ∂x + ∂x2 − ··· ∂u ∂u x ∂u x x

is the Euler – Lagrange operator. The following well known property of the Euler – Lagrange operator will be often used in this paper: E h = 0 iff there exists h 1 = h 1 (u; u x , . . . ) such that h = const + ∂x h 1 . Note that we do not specify here the class of functions u(x). The Hamiltonians H = H [u] can be ill defined (e.g., a divergent integral) but the evolutionary PDE (1.6) makes sense. The crucial point for the subsequent considerations is the following statement (see, e.g., [7]): for two commuting Hamiltonians δH δF ∂x =0 {H, F} = 0 ⇔ E δu(x) δu(x)

Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II

119

the evolutionary PDEs u t + ∂x

δH δF = 0 and u s + ∂x =0 δu(x) δu(x)

commute, (u t )s = (u s )t . For sufficiently small one expects to see no major differences in the behaviour of solutions to the perturbed and unperturbed Eqs. (1.1) and (1.2) within the regions where the x-derivatives are bounded. However the differences become quite serious near the critical point (also called the point of gradient catastrophe) where the derivatives of solution to the unperturbed equation tend to infinity. Although the case of small viscosity perturbations has been well studied and understood (see [3] and references therein), the critical behaviour of solutions to general conservative perturbations (1.6) to our best knowledge has not been investigated (see the papers [32, 12, 17–19, 23–25, 28] for the study of various particular cases). The main goal of this paper is to formulate the Universality Conjecture about the behaviour of a generic solution to the general perturbed Hamiltonian equation near the point of gradient catastrophe of the unperturbed solution. We argue that, up to shifts, Galilean transformations and rescalings this behaviour essentially does not depend on the choice of solution neither on the choice of the equation (provided certain genericity assumptions hold valid). Moreover, this behaviour near the point (x0 , t0 , u 0 ) is given by u u 0 + a 2/7 U b −6/7 (x − a0 (t − t0 ) − x0 ) ; c −4/7 (t − t0 ) + O 4/7 , (1.7) where U = U (X ; T ) is the unique real smooth for all X ∈ R solution to the fourth order ODE,

1 3 1 2 1 IV dU X=TU− U + , U = U + 2U U + U , etc., (1.8) 6 24 240 dX depending on the parameter T . Here a, b, c are some constants that depend on the choice of the equation and the solution, a0 = a(v0 ). Equation (1.8) appeared in [4] (for the particular value of the parameter T = 0) in the study of the double scaling limit for the matrix model with the multicritical index m = 3. It was observed that generic solutions to (1.8) blow up at some point of real line; the conjecture about existence of a unique smooth solution has been formulated. To our best knowledge, this conjecture remains open, although there are some supporting evidences [20]. The present paper is organized as follows. In Sect. 2 we classify all Hamiltonian perturbations up to the order 4 . They are parametrized by two arbitrary functions c(u), p(u). For the simplest example the perturbations of the Riemann wave equation u t + u u x = 0 read 2 2c u x x x + 4c u x u x x + c u 3x ut + u u x + 24 4 + 2 p u x x x x x + 2 p (5u x x u x x x + 3u x u x x x x ) (1.9) + p 7u x u 2x x + 6u 2x u x x x + 2 p u 3x u x x = 0.

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For c(u) = const, p(u) = 0 this is nothing but the Korteweg - de Vries (KdV) equation; for other choices of the functions c(u), p(u) it seems not to be an integrable PDE. Remarkably, for arbitrary choice of the functional parameters the perturbed equation possesses an infinite family of approximate symmetries (see [2, 9, 22, 30] for discussion of approximate symmetries). In principle our approach can be applied to classifying the Hamiltonian perturbations of higher orders. However, higher order terms do not affect the type of critical behaviour. In Sect. 3 we establish an important property of quasitriviality of all perturbations (cf. [9, 10, 27]). The quasitriviality is given by a substitution u → u + 2 K 2 (u; u x , u x x , u x x x ) + 4 K 4 u; u x , . . . , u (6) (1.10) that transforms, modulo O( 6 ) the unperturbed equation (1.2) to (1.6). Here the functions K 2 and K 4 depend rationally on the x-derivatives. We also formulate the first part of our Main Conjecture that says that, for sufficiently small the solution to the perturbed system exists at least on the same domain of the (x, t)-plane where the unperturbed solution is defined. In Sect. 4 we briefly discuss existence of a bihamiltonian structure compatible with the perturbation (see also Appendix below). Some examples of perturbed Hamiltonian equations are described in Sect. 5. In Sect. 6 we recollect some properties of the ODE (1.8) and we formulate the second part of the Main Conjecture describing the special function U (X ; T ) in (1.7) as a particular solution to (1.8). Finally, in Sect. 7 we give the precise formulation of the Universality Conjecture (Main Conjecture, Part 3) and give some evidences supporting it1 . Because of lack of space we do not consider the numerical evidences supporting the idea of Universality; they will be given in a subsequent publication (see also [16]). In the last section we outline the programme of further researches towards understanding of universality phenomena of critical behaviour in general Hamiltonian perturbations of hyperbolic systems. 2. Hamiltonian Perturbations of the Riemann Wave Equation Let us start with the simplest case of Hamiltonian perturbations of the equation vt + v vx = 0 ⇔ vt + {v(x), H0 } = 0, {v(x), v(y)} = δ (x − y), 3 v d x. H0 = 6

(2.1)

Lemma 2.1. Up to the order O( 4 ), all Hamiltonian perturbations of (2.1) can be reduced to the form δH = 0, δu(x) 3 u c(u) 2 − 2 u x + 4 p(u)u 2x x + s(u)u 4x d x, H= 6 24

u t + ∂x

(2.2)

where c(u), p(u), s(u) are arbitrary functions. Moreover, the function s(u) can be eliminated by a Miura-type transform. 1 Perhaps, only this Part 3 deserves the name of the Main Conjecture. However, the precise formulation of it depends on the first two parts.


121

Proof. The Hamiltonian must have the form H = H0 + H1 + · · · + 4 H4 , where the density of Hk is a graded homogeneous polynomial of the degree k. So, the density of H1 is a total derivative: H1 = α(u)u x d x, α(u)u x = ∂x A(u), A (u) = α(u). The density of the Hamiltonian H2 modulo total derivatives must have the form −

c(u) 2 u 24 x

for some function c(u). Similarly, H3 must have the form H3 = c1 (u)u 3x d x. Here c1 (u) is another arbitrary function. Let us show that H3 can be eliminated by a Miura-type transform. Let us look for it in the form u → u + {u(x), F} +

2 {{u(x), F}, F} + · · · , 2

(2.3)

choosing F = 2

α(u)u 2x d x.

Such a transformation preserves the Poisson bracket. The change of the Hamiltonian H will be given by δ H = {F, H } + O( 4 ). At the order 3 one has

1 3 3 2 α (u)u x − ∂x (α u x ) u u x d x = δH = α(u)u 3x d x. 2 2 So, choosing α(u) = −2c1 (u) we kill the terms cubic in . The rest of the proof is obvious: in order 4 all the Hamiltonians have the form H4 = p(u)u 2x x + s(u)u 4x d x for some functions p(u), s(u). The last term can be killed by the canonical transformation of the form (2.3) generated by the Hamiltonian 3 F =− s(u)u 3x d x. 2 The lemma is proved.

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Choosing s(u) = 0 one obtains the family (1.9) of Hamiltonian perturbations of the Riemann wave equation depending on two arbitrary functions c = c(u), p = p(u). We will now compare the symmetries of (2.1) and those of the perturbed system (2.2). It is easy to see that the Hamiltonian equation vs + a(v)vx = 0 ⇔ vs + {v(x), H 0f } = 0, H 0f = f (v) d x, f (v) = a(v)

(2.4)

is a symmetry of (2.1) for any a(v), (vt )s = (vs )t . Moreover, the Hamiltonians H 0f commute pairwise, {H 0f , Hg0 } = 0 ∀ f = f (u), ∀g = g(u). This family of commuting Hamiltonians is complete in the following sense. Lemma 2.2. The family of commuting Hamiltonians H 0f is maximal, i.e., if H = h(u; u x , u x x , . . . ) d x commutes with all functionals of the form H 0f then h(u; u x , u x x , . . . ) = g(u) + ∂x (. . . ) for some function g(u). We will now construct a perturbation of the Hamiltonians H 0f preserving the commutativity modulo O( 6 ). Like in Lemma 2.1 one can easily check that all the perturbations up to the order 4 must have the form c f (u) 2 f (u) − 2 u x + 4 p f (u)u 2x x + s f (u)u 4x dx Hf = 24 for some functions c f (u), p f (u), s f (u). To ensure commutativity one has to choose these functions as follows. Lemma 2.3. For any f = f (u) the Hamiltonian flow δHf = 0, H f = h f d x, u s + ∂x δu(x) 2 c2 f (4) 2 4 h f = f − c f ux + p f + u 2x x 24 480 c c f (4) c c f (5) c2 f (6) p f (4) p f (5) − u 4x + + + + −s f 1152 1152 3456 6 6

(2.5)

is a symmetry, modulo O( 6 ), of (2.2). Moreover, the Hamiltonians H f commute pairwise: {H f , Hg } = O( 6 ) for arbitrary two functions f (u) and g(u).


123

Proof. One has to check the identity δ Hg δHf ∂x = 0, E δu(x) δu(x) where E is the Euler – Lagrange operator. We leave this calculation as an exercise for the reader. 3 Observe that for f = u6 the Hamiltonian H f coincides with (2.2). Also for f = u (the Casimir of the Poisson bracket) and f = trivial,

u2 2

(the momentum) the perturbation is

H f = H 0f . We do not know under what conditions on the functional parameters c(u), p(u) higher order perturbations can be added to the Hamiltonians (2.5) preserving the commutativity. The examples of Sect. 5 show that this can be done at least for some particular choices of the functions. However, the remark at the end of Sect. 4 suggests that the answer is not always affirmative.

3. Solutions to the Perturbed Equations. Quasitriviality We address now the problem of existence of solutions to the perturbed equation for t < tC . We will construct a formal asymptotic solution to (2.2) (and also to all commuting flows (2.5)) valid on the entire interval t < tC . The basic idea is to find a substitution v → u = v + O() that transforms all solutions to all unperturbed equations of the form (2.4) to solutions to the corresponding perturbed equations (2.5). Quasitriviality Theorem. There exists a transformation v → u = v +

4

k Fk (u; u x , . . . , u (n k ) ),

(3.1)

k=1

where Fk are rational functions in the derivatives homogeneous of the degree k, independent of f = f (u), that transforms all monotone solutions of (2.4) to solutions, modulo O( 6 ), of (2.5) and vice versa. The general quasitriviality theorem for evolutionary PDEs admitting a bihamiltonian description was obtained in [10]2 . As we do not assume a priori existence of a bihamiltonian structure (see, however, the next section), we will give here a direct proof of quasitriviality for the family of commuting Hamiltonians (2.5). For convenience we chose s(u) =

c(u) c (u) . 3456

2 In a very recent paper [27] the quasitriviality result was proved, in all orders in , for an arbitrary perturbation of the Riemann wave equation vt + v vx = 0. It has also been shown that the same transformation trivializes also all symmetries of the perturbed equation.

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Theorem 3.1. Introduce the following Hamiltonian

K =

1 c(u) u x log u x + 3 24

c2 (u) u 3x x p(u) u 2x x − 5760 u 3x 4 ux

d x.

Then the canonical transformation u → v = u + {u(x), K } +

2 {{u(x), K }, K } + · · · 2

satisfies Hf =

f (v) d x + O( 6 ) ∀ f (u).

The inverse transformation is the needed quasitriviality. It is generated by the Hamiltonian 2

1 c (v) vx3x p(v) vx2x − c(v) vx log vx − 3 d x, − −K = 24 5760 vx3 4 vx that is 2 v → u = v − {v(x), K } + {{v(x), K }, K } + · · · 2

3 2 vx x vx x 7 vx x vx x x vx x x x = v + ∂x c + c vx + 4 ∂x c2 − + 24 vx 360 vx4 1920 vx3 1152 vx2 x vx x 2 47 vx x 3 37 vx x vx x x 5 vx x x x 2 vx x x − +c +c c − + 5760 vx 3 2880 vx 2 1152 vx 384 5760 vx 2 v v xxx xx +c c − 144 360 vx 1 2 7 c c vx vx x + c vx 3 + 6 c c vx vx x + c c vx 3 + c c(4) vx 3 + 1152 vx x 3 vx x vx x x vx x x x vx vx x +p + p . (3.2) − + v + p xxx 2 vx 3 vx 2 2 vx 2 In this formula c = c(v), p = p(v). Main Conjecture, Part 1. Let v = v(x, t) be a smooth solution to the unperturbed equation vt + a(v) vx = 0 defined for all x ∈ R and 0 ≤ t < t0 monotone in x for any t. Then there exists a solution u = u(x, t; ) to the perturbed equation u t + ∂x

δHf = 0, δu(x)

f (u) = a(u)

defined on the same domain in the (x, t)-plane with the asymptotic at → 0 of the form (3.2).


125

4. Are All Hamiltonian Perturbations Also Bihamiltonian? All unperturbed equations vs + a(v) vx = 0 are bihamiltonian w.r.t. the Poisson pencil (see the definition in [9]) {v(x), v(y)}1 = δ (x − y), 1 {v(x), v(y)}2 = q(v(x))δ (x − y) + q (v)vx δ(x − y) 2

(4.1)

for an arbitrary function q(u), vs + {v(x), H1 }1 = vs + {v(x), H2 }2 = 0,

H1 =

f 1 (v) d x,

H2 =

f 2 (v) d x

1 f 1 (v) = a(v) = q(v) f 2 (v) + q (v) f 2 (v). 2 To show that (4.1) is a Poisson pencil it suffices to observe that the linear combination {v(x), v(y)}2 − λ {v(x), v(y)}1 = (q(v(x)) − λ) δ (x − y) 1 + q (v)vx δ(x − y) 2

(4.2)

is the Poisson bracket associated [11] with the flat metric ds 2 =

dv 2 . q(v) − λ

Theorem 4.1. For c(u) = 0 the commuting Hamiltonians (2.5) admit a unique bihamiltonian structure obtained by a deformation of (4.1) with q(u) satisfying

c2 c q p(u) = (4.3) 5 − , s(u) = 0. 960 c q The proof of this result along with the explicit formula for the deformed bihamiltonian structure is sketched in the Appendix below. The assumption c = 0 is essential: one can check that for c(u) ≡ 0 the Hamiltonians (2.5) commute, modulo O( 6 ), only w.r.t. the standard Poisson bracket (1.4). On the other hand it turns out that for this particular choice of the functional parameters the deformation of commuting Hamiltonians cannot be extended to the order O( 8 ). 5. Examples Example 1. For c(u) = c0 = const, p(u) = s(u) = 0 one obtains from (2.2) the KdV equation u t + u u x + c0

2 u x x x = 0. 12

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B. Dubrovin

Choosing in (2.5) f (u) =

u k+2 (k + 2)!

one obtains the Hamiltonians of the KdV hierarchy ∂u δ Hk = 0, Hk = h k d x, k ≥ 0 + ∂x ∂tk δu(x) 2 u k−1 2 u k+2 − c0 u hk = (k + 2)! 24 (k − 1)! x 4 u k−2 u k−4 2 2 4 u − u + O( 6 ). + c0 96 5 (k − 2)! x x 36 (k − 4)! x The quasitriviality transformation (3.2) takes the form [2, 9]

v → u = v + ∂x2

2 c0 log vx + c0 2 4 24

vx3x 7 vx x vx x x vx x x x − + 4 3 360 vx 1920 vx 1152 vx2

+O( 6 ).

(5.1)

Example 2. The Volterra lattice q˙n = qn (qn+1 − qn−1 )

(5.2)

(also called difference KdV) has the following bihamiltonian structure [13] {qn , qm }1 = 2qn qm (δn+1,m − δn,m+1 ), 1 q˙n = {qn , H1 }1 , H1 = log qn , 2 q + q n m − 2 δn,m+1 − δn,m−1 2

1 1 + δn,m+2 − δn,m−2 , 2 2 q˙n = {qn , H2 }2 , H2 = qn .

(5.3)

{qn , qm }2 = qn qm

(5.4)

After substitution qn = ev(n) and division by 4 one arrives at the following bihamiltonian structure: 1 [δ(x − y + ) − δ(x − y − )] 4 2 = δ (x − y) + δ (x − y) + · · · , 3

{v(x), v(y)}1 =

(5.5)


1 {v(x), v(y)}2 = 1 − ev(x) δ (x − y) − ev vx δ(x − y) 2

1 5 (2 − 5 ev )δ (x − y) − ev vx δ (x − y) + 2 12 8 1 3 v − e (vx x + vx2 )δ (x−y) − ev (vx x x + 3vx vx x + vx3 )δ(x−y) + O( 4 ). 8 12

127

(5.6)

To compare this bihamiltonian structure with the one obtained in Theorem 4.1 the Poisson bracket (5.5) must be reduced to the standard form {u(x), u(y)}1 = δ (x − y)

(5.7)

by means of the transformation ∂x 2 4 vx x x x + O( 6 ). u= v = v − vx x + sinh ∂x 12 160 After the transformation the second bracket takes the form 1 {u(x), u(y)}2 = 1 − eu(x) δ (x − y) − eu u x δ(x − y) 2

3 1 2 u(x) 1 δ (x − y) + u x δ (x − y) + (7u x x + 5u 2x )δ (x − y) − e 4 8 24 1 3 + (2u x x x + 4u x u x x + u x )δ(x − y) + O( 4 ). 24

(5.8)

We leave as an exercise for the reader to compute the terms of order 4 and to verify that the Poisson bracket (5.8) is associated with the functional parameters chosen as follows c(u) = 2,

p(u) = −

1 1 , q(u) = 1 − eu , s(u) = . 240 4320

Example 3. The Camassa – Holm equation [5] (see also [14])

3 1 vt − 2 vx xt = v vx − 2 vx vx x + v vx x x 2 2

(5.9)

admits a bihamiltonian description (cf. [21]) after doing the following Miura-type transformation u = v − 2 vx x .

(5.10)

{u(x), u(y)}1 = δ (x − y) − 2 δ (x − y),

(5.11)

1 {u(x), u(y)}2 = u(x)δ (x − y) + u x δ(x − y). 2

(5.12)

The bihamiltonian structure reads

The Casimir H−1 of the first Poisson bracket analytic in has the form H−1 = h −1 d x, h −1 = u(x).

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B. Dubrovin

Applying the bihamiltonian recursion procedure one obtains a sequence of commuting Hamiltonians Hk = h k d x of the hierarchy, h0 =

1 1 u v, h 1 = [v 3 + u v 2 ], . . . . 2 8

The corresponding Hamiltonian flows u tk = {u(x), Hk }1 ≡ (1 − 2 ∂x2 )∂x read

δ Hk δu(x)

3 1 2 u t0 = u x , u t1 = v v x − v x v x x + v v x x x , . . . . 2 2

The last equation reduces to (5.9) after the substitution (5.10). To compare the commuting Hamiltonians with those given in (2.5) one must first reduce the first Poisson bracket to the standard form {u(x), ˜ u(y)} ˜ 1 = δ (x − y) by the transformation −1/2 1 3 u˜ = 1 − 2 ∂x2 u = u + 2u x x + 4u x x x x + · · · . 2 8 After the transformation the Camassa – Holm equation will read u˜ t =

3 u˜ u˜ x + 2 (2u˜ x u˜ x x + u˜ u˜ x x x ) + 4 (5 u˜ x x u˜ x x x + 3 u˜ x u˜ x x x x + u˜ u˜ x x x x x ) + · · · . 2

It is easy to see that the commuting Hamiltonians of Camassa – Holm hierarchy are obtained from (2.5) by the specialization c(u) = 8 u,

p(u) =

u , q(u) = u, s(u) = 0. 3

6. Introducing a Special Function Let us recall some properties of the differential equation

1 1 1 IV 2 X = T U − U 3 + (U + 2U U ) + U 6 24 240

(6.1)

often considered as a 4th order analogue of the classical Painlevé-I equation. First, it can be interpreted as a monodromy preserving deformation of the following linear differential operator with polynomial coefficients ∂ψ = W ψ, ∂z where the matrix W reads 1 12UU + 8zU + U W=− 2 w21 120

2(16z 2 + 8z U + 6U 2 + U − 60T ) , −12UU − 8zU − U

(6.2)


129

where w21 = 32 z 3 − 16z 2 U − 2z(2U 2 + U + 60 T ) + 8U 3 + 2U U − U + 120X. 2

Indeed, it coincides with the compatibility conditions W X − Uz + [W, U] = 0 of the linear system (6.2) with 

0

−1



∂ψ . = U ψ, U =  ∂X 2U − 2z 0

(6.3)

Moreover, the dependence of (6.2) on T is isomonodromic iff the function U (X ) depends also on the parameter T according to the KdV equation UT + U U +

1 U = 0. 12

(6.4)

This is the spelling of the compatibility condition of the linear system (6.2), (6.3) with ∂ψ 1 2U + 4z U . (6.5) = V ψ, V = −U ∂T 6 8z 2 − 4zU − 4U 2 − U The Painlevé property readily follows from the isomonodromicity: singularities in the complex (X, T )-plane of general solution to (6.1), (6.4) are poles [20]. Main Conjecture, Part 2 (cf. [4]). The ODE (6.1) has unique solution U = U (X ; T ) smooth for all real X ∈ R for all real values of the parameter T . Note that due to the uniqueness the solution in question satisfies the KdV equation (6.4). For T > 0. For T >> 0 the solution develops oscillations typical for dispersive waves [32] within a region around the origin; one can use the Whitham method to approximate U (X ; T ) by modulated elliptic functions within the oscillatory zone [18, 29]. Thus the solution in question interpolates between the two types of asymptotic behaviour (cf. [23] where the role of the special solution U (X ; T ) in the KdV theory was discussed).

130

B. Dubrovin

The solutions to the fourth order ODE (6.1) can be parametrized [20] by the monodromy data (i.e., the collection of Stokes multipliers) of the linear differential operator (6.3) with coefficients polynomial in z. The solution corresponding to given Stokes multipliers can be reconstructed by solving a certain Riemann – Hilbert problem. The particular values of the Stokes multipliers associated with the smooth solution in question have been conjectured in [20]. 7. Local Galilean Symmetry and Critical Behaviour We will now proceed to discussing the universality problem. Consider the perturbed PDE u t + {u(x), H f } = u t + a(u)u x + O( 2 ) = 0,

f (u) = a(u).

(7.1)

Let us apply the transformation (3.2) to the unperturbed solution v = v(x, t) of vt + a(v)vx = 0

(7.2)

obtained by the method of characteristics: x = a(v) t + b(v)

(7.3)

for some smooth function b(v). Let the solution arrive at the point of gradient catastrophe for some x = x0 , t = t0 , v = v0 . At this point one has x0 = a(v0 )t0 + b(v0 ), 0 = a (v0 )t0 + b (v0 ), 0 = a (v0 )t0 + b (v0 )

(7.4)

(inflection point). Let us assume the following genericity assumption: κ := −(a (v0 )t0 + b (v0 )) = 0.

(7.5)

Let us first recall the universality property for the critical behaviour of the unperturbed solutions: up to shifts, Galilean transformations and rescalings a generic solution to (7.2) near (x0 , t0 ) behaves like the cubic root function. We will present this well known statement in the following form. Introduce the new variables x¯ = x − a0 (t − t0 ) − x0 , t¯ = t − t0 , v¯ = v − v0 . Let us do the following scaling transformation x¯ → λ x, ¯ 2

t¯ → λ 3 t¯,

(7.6)

1 3

¯ v¯ → λ v. Lemma 7.1. After the rescaling (7.6) any generic solution to (7.2) at the limit λ → 0 for t < t0 goes to the solution of the cubic equation x¯ = a0 v¯ t¯ − κ

v¯ 3 . 6

(7.7)


131

In these formulae a0 = a(v0 ), a0 = a (v0 ). Note that the inequality κ a0 > 0

(7.8)

must hold true in order to have the solution well defined for t < t0 near the point of generic gradient catastrophe (7.4). To prove the lemma it suffices to observe that, after the rescaling (7.6) and division by λ Eq. (7.3) yields x¯ = a0 v¯ t¯ − κ

v¯ 3 + O λ1/3 . 6

The parameter κ can be eliminated from (7.7) by a rescaling. The resulting cubic function can be interpreted as the universal unfolding of the A2 singularity [1]. Our basic observation we are going to explain now is that, after a Hamiltonian perturbation the A2 singularity transforms to the special solution of (1.8) described above. Let us look for a solution to the perturbed PDE (7.1) in the form of a formal power series u = u(x, t; ) = v(x, t) + k vk (x, t) (7.9) k≥1

with v(x, t) given by (7.3) satisfying (7.1) modulo O( 5 ). We will say that such a solution is monotone at the point x = x0 , t = t0 if u x (x0 , t0 ; 0) ≡ vx (x0 , t0 ) = 0. According to the results of Sect. 3 all monotone solutions of the form (7.9) can be obtained by applying the transformation (3.2) to the nonperturbed solution (7.2) (more precisely, one has to allow -dependence of the function b(u)). Lemma 7.2. Let us perform the rescaling (7.6) along with → λ7/6

(7.10)

in the quasitriviality transformation (3.2). Then the resulting solution to the perturbed PDE will be equal to u = v0 + λ

1/3

2 2 v¯ + ∂x c0 log v¯ x + c0 2 4 24 3 7 v¯ x x v¯ x x x v¯ x x x x v¯ x x 2/3 + O λ × − + 360 v¯ x4 1920 v¯ x3 1152 v¯ x2

(7.11)

(cf. (5.1)) where c0 = c(v0 ), v¯ = v(x, ¯ t) is the solution to the cubic equation (7.7). Proof is straightforward.

(7.12)

132

B. Dubrovin

It remains to identify (7.11) with the formal asymptotic solution (6.6) to the ODE (6.1). This can be done by a direct substitution. An alternative way is to observe that, near the point of gradient catastrophe the perturbed PDE acquires an additional Galilean symmetry. Indeed, according to the previous lemma, locally one can replace the functions c(u), p(u) by constants c0 = c(v0 ), p0 = p(v0 ) (the constant p0 , however, does not enter in the leading term of the asymptotic expansion in powers of λ1/3 ). Let us show that in this situation any solution to the perturbed PDE of the form (7.9) satisfies also a fourth order ODE. Lemma 7.3. Let c(u) = c0 , p(u) = p0 . Then for any solution u(x, t; ) of the form (7.9) monotone at the point (x0 , t0 ) there exists a formal series g(u; ) = g0 (u) + k gk (u) k≥1

such that for arbitrary x, t sufficiently close to x0 , t0 the function u(x, t; ) satisfies, modulo O( 5 ), the following fourth order ODE: x =t

δHf δ Hg + . δu(x) δu(x)

(7.13)

Here g0 (u) = b(u). Proof. It is easy to see that the flow δHf (7.14) δu(x) is a symmetry of (7.1). Combining this symmetry with one of the commuting flows u τ = 1 − t ∂x

δ Hg =0 δu(x) one obtains another symmetry. The set of stationary points of this combination δ Hg δHf + −x =0 ∂x t δu(x) δu(x) u s + ∂x

is therefore invariant for the t-flow. Considering the limit → 0 it is easy to see that the integration constant vanishes on the solution (3.2), (7.2). The lemma is proved. The ODE for the function u(x) is closely related to the so-called string equation known in matrix models and topological field theory (see, e.g., [9]). Explicitly 2 x = t a(u) + b(u) + c0 t 2 a u x x + a u 2x + 2 b u x x + b u 2x 24

1 2 4 c0 t a + b u x x x x 2 p0 t a + b + + 240

1 c02 t a I V + b I V u x x x u x + 4 p0 t a + b + 120

11 2 V c0 t a + b V u x x u 2x + 4 p0 t a I V + b I V + 1440

1 2 VI 1 p0 t a V + b V + c0 t a + b V I u 4x + O( 5 ). + (7.15) 2 1152


133

Let us call the solution generic if, along with the condition κ := −(a (v0 )t0 + b (v0 )) = 0 it also satisfies c0 := c(v0 ) = 0.

(7.16)

Main Conjecture, Part 3. The generic solution described in the Main Conjecture, Part 1 can be extended up to t = t0 + δ for sufficiently small positive δ = δ(); near the point (x0 , t0 ) it behaves in the following way: 2 1/7 x − a0 (t − t0 ) − x0 a0 (t − t0 ) c0 4/7 . (7.17) u v0 + U ; + O κ2 (κ c03 6 )1/7 (κ 3 c02 4 )1/7 Here U = U (X ; T ) is the solution to the ODE (1.8) specified in the Main Conjecture, Part 2. To arrive at the asymptotic formula (7.17) we do in (7.15) the rescaling of the form (7.6) along with (7.10). After substitution to Eq. (7.15) and division by λ, one obtains

3 4 2 u¯ + c0 u¯ 2x + 2u¯ u¯ x x + c02 u¯ x x x x + O λ1/3 . x¯ = a0 u¯ t¯ − κ 6 24 240 In derivation of this formula we use that the monomial of the form k u ix1 u ix2x u ix3x x . . . after the rescaling will be multiplied by λ D with D=

1 1 k + (i 1 + i 2 + · · · ) 6 3

due to the degree condition i 1 + 2 i 2 + 3 i 3 + · · · = k. Adding the terms of higher order k > 4 will not change the leading term. Choosing 3/7

λ = 6/7 c0

we arrive at the needed asymptotic formula. Clearly the above arguments require existence and uniqueness of the solution to (1.8) smooth on the real line described in the Main Conjecture, Part 2.

8. Concluding Remarks We have presented arguments supporting the conjectural universality of critical behaviour of solutions to generic Hamiltonian perturbations of a hyperbolic equation of the form (1.2). In subsequent publications we will study the Main Conjecture in more details. The possibilities of using the idea of Universality in numerical algorithms to dealing with oscillatory behaviour of solutions to Hamiltonian PDEs will be explored. We will also proceed to the study of singularities of generic solutions to integrable Hamiltonian hyperbolic systems of conservation laws ∂h(u) u it + ∂x ηi j = 0, η ji = ηi j , det(ηi j ) = 0. (8.1) ∂u j

134

B. Dubrovin

Recall that, according to the results of [31] the system (8.1) is integrable if it diagonalizes in a system of curvilinear coordinates v k = v k (u), k = 1, . . . , n for the Euclidean/pseudo-Euclidean metric ds 2 = ηi j du i du j =

n

gk (v)(dv k )2 ,

−1 ηi j := ηi j ,

k=1

vtk + λk (v)vxk = 0, k = 1, . . . , n (in this formula there is no summation over repeated indices!). All Hamiltonian perturbations of the hyperbolic system (8.1) can be written in the form

δH h(u) + = 0, H = u it + ∂x ηi j j k h k (u; u x , . . . , u (k) ) d x, δu (x) k≥1

deg h k = k. We plan to study symmetries of the perturbed Hamiltonian hyperbolic systems. In particular, we will classify the perturbations preserving integrability and study the correspondence between the types of critical behaviour of the perturbed and unperturbed systems. The next step would be to extend our approach to Hamiltonian perturbations of spatially multidimensional hyperbolic systems (cf. [8]). Appendix: Bihamiltonian Structures Associated with the Perturbations of the Riemann Wave Hierarchy Theorem A.1. For arbitrary two functions c = c(u) = 0, q = q(u) the family of Hamiltonians (2.5) with

c2 c q p(u) = (A.1) 5 − , s(u) = 0 960 c q is commutative

{H f , Hg }1,2 = 0 mod O( 6 ) ∀ f = f (u), ∀g = g(u)

(A.2)

with respect to the Poisson pencil of the form {u(x), u(y)}1 = δ (x − y), {u(x), u(y)}2 = {u(x), u(y)}[0] + 2 {u(x), u(y)}[2] + 4 {u(x), u(y)}[4] + O( 6 ). Here the terms of order 0: 1 {u(x), u(y)}[0] 2 = q(u)δ (x − y) + q (u)u x δ(x − y). 2 All terms of higher orders are uniquely determined from the bicommutativity (A.2) provided validity of the constraint (A.1). Namely, the terms of order 2:


135

cq 3 δ (x − y) + cq u x δ (x − y) {u(x), u(y)}[2] 2 = 8 16

c q c q 5cq c q u x x 7cq u x x 2 + + + ux + + δ (x − y) 16 6 48 16 48 cq c q c q cq (4) 3 1 + + c q + cq u x u x x + u x x x δ(x−y). + ux + 48 24 48 12 24 The terms of order 4: 1 5 2 {u(x), u(y)}[4] 3cc q + c2 q δ V (x − y)+ 3cc q +c q u x δ I V (x − y) 2 = 192 384 3c c q cc q 3c 2 q 5cc q cc q 2 u x 2 + + + − + 32 32 32 48 240q c2 q 3 19cc q 3c2 q q c2 q (4) + + + − ux 2 192 640q 64 480q 2 c2 q 2 19c2 q 3c 2 q 3cc q 17cc q + + − + + u x x δ (x − y) 64 64 192 480q 960 3c 2 q c c q cc(4) q 19c c q 23cc q 5cc q (4) 7cc q + + + + + + + 128 32 128 128 384 64 64 c 2 q 2 cc q 2 cc q 3 c2 q 4 17cc q q c2 q (5) 3c 2 q + − − + − − 96 32 160q 160q 640q 80q 2 160q 3 3cc q 21c2 q 2 q 9c2 q 2 9c2 q q (4) 9c c q 3 + + − − + u x 1280q 1280q 64 64 1280q 2 +

3cc q 2 3c2 q 3 69cc q 11c 2 q 13cc q + − + + 64 64 160q 320 320q 2 13c2 q q 3c2 q (4) + − ux uxx 640q 80 c 2 q cc q 13cc q c2 q 2 c2 q + + − + + u x x x δ (x − y) 32 32 192 320q 60

+

+

c 2 q c c q cc(4) q c c q 2 cc q 2 c 2 q 3 cc q 3 + + − − + + 48 32 96 160q 480q 160q 2 160q 2 − + +

cc q 4 80q 3

+

c2 q 5 160q 4

11cc q 2 q 320q 2

−

+

9c 2 q q 35c c q 5cc q 9cc q q + − − 384 128 640q 640q

13c2 q 3 q 640q 3

−

cc q 2 19c2 q q 2 17c 2 q (4) + + 64q 384 1280q 2

cc q q (4) 17c2 q 2 q (4) 5cc q (4) 11c2 q q (4) 35cc q (5) − + − + 96 64q 1280q 1152 1920q 2

136

B. Dubrovin

11c2 q q (5) c2 q (6) − + 3840q 288 +

4

ux +

3c 2 q c c q cc(4) q 91c c q + + + 128 32 128 384

c 2 q 2 37cc q cc q 2 cc q 3 c2 q 4 59c 2 q 53cc q − + − + − + 384 60q 60q 320 240 30q 2 60q 3

47cc q q 173c2 q 2 q 77c2 q 2 169cc q (4) 77c2 q q (4) + − + − 640q 3840q 960 3840q 3840q 2 cc q 2 73c2 q (5) 3c c q cc q 5c 2 q cc q + + + − + u x 2u x x + 2880 128 128 96 16 80q 5c2 q q 31c2 q (4) c2 q 3 157cc q − + + + uxx 2 2 1920 384q 1920 160q 3c c q cc q c 2 q 3cc q cc q 2 + + + − + 64 64 12 32 60q 11c2 q q 11c2 q (4) c2 q 3 19cc q − + + + ux uxxx 160 640q 480 120q 2 c 2 q cc q 11cc q c2 q 2 17c2 q + + − + + u x x x x δ (x − y) 128 128 384 320q 1920 −

+

c c q q c 2 q c c q cc(4) q cc q q c 2 q 2 q + + − − + 192 128 384 640q 1920q 640q 2 +

cc q 2 q 640q 2

− +

c2 q 2 q 2 320q 3

cc q 2 q (4) 320q 2

− + −

cc q 3 q 320q 3 c2 q 3 1280q 2

+

c2 q 3 q (4) 640q 3

+

c2 q 4 q 640q 4

−

c 2 q 2 cc q 2 3cc q q 2 − + 640q 640q 640q 2

c 2 q q (4) 7c c q (4) cc q (4) cc q q (4) + − − 384 128 640q 640q 2

−

3cc q q (4) 13c2 q q q (4) c2 q (4) + − 2 640q 1280q 3840q

cc q q (5) c2 q 2 q (5) 17c 2 q (5) 5cc q (5) c2 q q (5) 5cc q (6) + − + − + 2304 576 640q 960q 1152 1280q 2 2 2 (6) 2 (7) (4) c q cc q cc q c c q 2 c q c q q 5 + + − + + − u x 3840q 2304 64 48 192 160q

+

−

cc q 2 c 2 q 3 cc q 3 cc q 4 c2 q 5 97c c q + + − + + 480q 960 160q 2 160q 2 80q 3 160q 4

+

c 2 q q 13cc q cc q q 19cc q 2 q 11c2 q 3 q − − + − 320 60q 60q 480q 2 480q 3

−

cc q q (4) 11c2 q 2 q (4) cc q 2 3c2 q q 2 19c 2 q (4) 67cc q (4) + − + + + 48q 320 960 48q 160q 2 960q 2


137

c2 q q (4) 131cc q (5) c2 q q (5) c2 q (6) − + + − u x 3u x x 80q 2880 240q 180 7c c q 7cc q 7c 2 q 2 7cc q 2 7cc q 3 7c2 q 4 + − + − + − 128 384 960q 960q 480q 2 960q 3 59c 2 q 23cc q 3c2 q 2 131cc q (4) cc q q 13c2 q 2 q + − + + − 960 320 30q 320q 1920 640q 2 3c c q cc q c 2 q 2 cc q 2 3c2 q q (4) 31c2 q (5) 2 + − + u + − − u x x x 320q 2880 64 64 160q 160q

+

+

cc q 3 80q 2

47c 2 q 13cc q 13cc q q c2 q 2 q + + − 960 240 480q 160q 3 60q 2 7c2 q q (4) 23c2 q (5) 49cc q (4) − + + u x 2u x x x 960 960q 2880

−

c2 q 4

+

7c2 q 2 960q cc q 2 5c 2 q 5cc q c2 q 3 3cc q + − + + + 2 192 192 96q 64 192q c 2 q cc q c2 q q c2 q (4) cc q 2 c2 q 3 − + − + u + + u x x x x x 96q 96 64 64 160q 320q 2 c2 q q c2 q (4) 9cc q − + + ux uxxxx 320 160q 160 cc q c2 q 2 c2 q − + + u x x x x x δ(x − y). 192 960q 480 −

To prove the theorem one has to analyze the commutativity conditions δ Hg δHf L =0 E δu(x) δu(x) for arbitrary two functions f (u), g(u). Here 1 2 L = q∂x + q u x − c q ∂x3 + · · · 2 8 is the Hamiltonian differential operator associated with the second Hamiltonian structure. To prove validity of Jacobi identity one has to check that the -terms in the second Hamiltonian structure can be eliminated by the quasitriviality transformation described in Sect. 3. We will omit the calculations. Observe that the family of bihamiltonian structures given in Theorem A.1 depends on two arbitrary functions c = c(u), q = q(u), in agreement with the results of [26]. It is understood that the Jacobi identity for the Poisson pencil holds true identically in λ modulo terms of the order O( 6 ). Acknowledgements. This work is partially supported by European Science Foundation Programme “Methods of Integrable Systems, Geometry, Applied Mathematics" (MISGAM), Marie Curie RTN “European Network

138

B. Dubrovin

in Geometry, Mathematical Physics and Applications" (ENIGMA), and by Italian Ministry of Universities and Researches (MIUR) research grant PRIN 2004 “Geometric methods in the theory of nonlinear waves and their applications".

References 1. Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts. Monographs in Mathematics 82. Boston, MA: Birkhäuser Boston, Inc., 1985 2. Baikov, V.A., Gazizov, R.K., Ibragimov, N.Kh.: Approximate symmetries and formal linearization. PMTF 2, 40–49 (1989) (In Russian) 3. Bressan, A.: One dimensional hyperbolic systems of conservation laws. In: Current developments in mathematics, 2002, Somerville, MA: Int. Press, 2003, pp. 1–37 4. Brézin, É., Marinari, E., Parisi, G.: A nonperturbative ambiguity free solution of a string model. Phys. Lett. B 242, 35–38 (1990) 5. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) 6. Degiovanni, L., Magri, F., Sciacca,V.: On deformation of Poisson manifolds of hydrodynamic type. Commun. Math. Phys. 253, no. 1, 1–24 (2005) 7. Dickey, L.A.: Soliton equations and Hamiltonian systems. Second edition. Advanced Series in Mathematical Physics 26. River Edge, NJ: World Scientific Publishing Co., Inc., 2003 8. Dobrokhotov, S., Pankrashkin, K., Semenov, E.: On Maslov’s conjecture on the structure of weak point singularities of the shallow water equations. Dokl. Akad. Nauk 379, no. 2, 173–176 (2001); English translation: Doklady Math. 64, 127–130 (2001) 9. Dubrovin, B., Zhang, Y.: Normal forms of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. http://arxiv.org/list/math.DG/0108160, 2001 10. Dubrovin, B., Liu, S.-Q., Zhang, Y.: On hamiltonian perturbations of hyperbolic systems of conservation laws, I: quasitriviality of bihamiltonian perturbations. Comm. Pure and Appl. Math. 59, 559–615 (2006) 11. Dubrovin, B., Novikov, S.P.: Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method. Dokl. Akad. Nauk SSSR 270, no. 4, 781–785 (1983); English translation: Soviet Math. Dokl. 27, 665–669 (1983) 12. El, G.A.: Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos 15, 037103 (2005) 13. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian methods in the theory of solitons. Springer Series in Soviet Mathematics, Berlin: Springer-Verlag, 1987 14. Fokas, A.S.: On a class of physically important integrable equations. Physica D 87, 145–150 (1995) 15. Getzler, E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111, 535–560 (2002) 16. Grava, T., Klein, C.: Numerical solution of the small disperion limit of the KdV equation and Whitham equations. http://arxiv.org/list/math-ph/0511011, 2005 17. Gurevich, A., Meshcherkin, A.: Expanding self-similar discontinuities and shock waves in dispersive hydrodynamics. Sov. Phys. JETP 60, 732–740 (1984) 18. Gurevich, A., Pitaevski, L.: Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP Lett. 38, 291–297 (1974) 19. Hou, T.Y., Lax, P.D.: Dispersive approximations in fluid dynamics. Comm. Pure Appl. Math. 44, 1–40 (1991) 20. Kapaev, A.A.:Weakly nonlinear solutions of the equation P12 . Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187 (1991), Differentsialnaya Geom. Gruppy Li i Mekh. 12, 88–109, 172–173, 175; translation in J. Math. Sci. 73, no. 4, 468–481 (1975) 21. Khesin, B., Misiołek, G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116–144 (2003) 22. Kodama, Y., Mikhailov, A.: Obstacles to asymptotic integrability. In: Algebraic aspects of integrable systems, Progr. Nonlinear Differential Equations Appl. 26, Boston, MA: Birkhäuser, 1997, pp. 173–204 23. Kudashev, V., Suleimanov, B.: A soft mechanism for the generation of dissipationless shock waves. Phys. Lett. A 221, 204–208 (1996) 24. Lax, P., Levermore, D.: The small dispersion limit of the Korteweg-de Vries equation. I, II, III. Comm. Pure Appl. Math. 36, 253–290, 571–593, 809–829 (1983) 25. Lax, P. D., Levermore, C.D., Venakides, S.: The generation and propagation of oscillations in dispersive initial value problems and their limiting behavior. In: Important developments in soliton theory, Springer Ser. Nonlinear Dynam., Berlin: Springer, 1993, pp. 205–241


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26. Liu, S.Q., Zhang, Y.: Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 54, 427–453 (2005) 27. Liu, S.Q., Zhang, Y.: On quasitriviality of a class of scalar evolutionary PDEs. J. Geom. Phys., 2006, to appear. http://arxiv.org/list/ nlin.SI/0510019, 2005 28. Lorenzoni, P.: Deformations of bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 44, 331– 375 (2002) 29. Potëmin, G.: Algebro-geometric construction of self-similar solutions of the Whitham equations. Uspekhi Mat. Nauk 43, no. 5(263), 211–212 (1988); translation in Russ. Math. Surv. 43, 252–253 (1988) 30. Strachan, I.A.B.: Deformations of the Monge/Riemann hierarchy and approximately integrable systems. J. Math. Phys. 44, 251–262 (2003) 31. Tsarëv, S.P.: The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Izv. Akad. Nauk SSSR Ser. Mat. 54, no. 5, 1048–1068 (1990); English translation in Math. USSR-Izv. 37, 397–419 (1991) 32. Zabusky, N.J., Kruskal, M.D.: Interaction of “solitons" in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965) Communicated by P. Constantin


Communications in


Dissipative Quasi-Geostrophic Equation for Large Initial Data in the Critical Sobolev Space Hideyuki Miura Mathematical Institute, Tohoku University Sendai, 980-8578, Japan. E-mail: [email protected] Received: 13 October 2005 / Accepted: 9 December 2005 Published online: 9 May 2006 – © Springer-Verlag 2006

Abstract: The critical and super-critical dissipative quasi-geostrophic equations are investigated in R2 . We prove local existence of a unique regular solution for arbitrary initial data in H 2−2α which corresponds to the scaling invariant space of the equation. We also consider the behavior of the solution near t = 0 in the Sobolev space. 1. Introduction Let us consider the two dimensional dissipative quasi-geostrophic equation:  ∂θ   + (−)α θ + u · ∇θ = 0 in R2 × (0, ∞),  ∂t u = (−R2 θ, R1 θ ) in R2 × (0, ∞),    θ| 2 t=0 = θ0 in R ,

(DQGα )

where the scalar function θ and the vector field u denote the potential temperature and the fluid velocity, respectively, and α is a non-negative constant. Ri = ∂∂xi (−)−1/2 (i = 1, 2) represents the Riesz transform. We are concerned with the initial value problem for this equation. It is known that (DQGα ) is an important model in geophysical fluid dynamics. Indeed, it is derived from general quasi-geostrophic equations in the special case of constant potential vorticity and buoyancy frequency. Since there are a number of applications to the theory of oceanography and meteorology, a lot of mathematical researches have been devoted to the equation. The case α = 1/2 is called critical since its structure is quite similar to that of the 3-dimensional Navier-Stokes equations. The case α > 1/2 is called sub-critical and α < 1/2 is called super-critical, respectively. In the sub-critical cases, Constantin and Wu [5], Wu [15] proved global existence of a unique regular solution. However, in the critical and super-critical cases, global well-posedness for large initial data is still open. In the critical case, Constantin, Cordoba and Wu [4] constructed a

142

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global regular solution for the initial data in H 1 with small L ∞ norm. In both critical and super-critical cases, Chae and Lee [2] and Ju [9] proved global existence of 2−2α and H 2−2α a unique regular solution for the initial data in the Besov space B2,1 under the smallness assumption of each homogeneous norm, respectively. For large initial data, Cordoba-Cordoba [6] proved local existence of a regular solution for the initial data in H s with s > 2 − α. Ju [9, 10] improved the admissible exponent up to s > 2 − 2α. In this paper we show local existence of a unique regular solution with initial data in H 2−2α for both critical and super-critical cases. In Ju [10], he conjectured the local H 1 solution in the critical case without smallness assumption on the initial data. Our theorem gives a positive answer to his question. Moreover, our theorem improves the class of initial data to construct the local regular solution. Indeed, H 2−2α is larger than H s (s > 2 − 2α). See Remark 1 below. Here the exponent 2 − 2α is important, because this is the borderline case with respect to the scaling. We observe that if θ (x, t) is the solution of (DQGα ), then θλ (x, t) ≡ λ2α−1 θ (λx, λ2α t) is also a solution of (DQGα ). Then the homogeneous space H˙ 2−2α is called scaling invariant, since θλ (·, 0) H˙ 2−2α = θ (·, 0) H˙ 2−2α holds for all λ > 0. The scaling invariant spaces play an important role for the theory of nonlinear partial differential equations. If the equation has a class of scaling invariance, then it coincides with the most suitable space to construct the solution which is expected unique and regular. (See, e.g. Danchin [7], Koch-Tataru [11].) We now sketch the idea of our proof. In contrast with other equations, it seems to be difficult to prove the local existence of regular solutions by the classical approach such as Fujita-Kato’s argument [8]. As is pointed out in [2], we have difficulty to find an appropriate space E which yields the following continuous bilinear estimate of the Duhamel term: · e−(·−s)(−)α (u · ∇θ )(s)ds ≤ Cθ 2 . E 0

E

For α ≤ 1/2, we see the linear part (−)α θ in (DQGα ) is too weak to control αthe nonlinear term u · ∇θ . In fact, the smoothing property of the semigroup e−t (−) is not enough to overcome the loss of derivatives in the nonlinear term. To avoid this difficulty, in [2, 9] they applied the cancellation property of the equation to construct the small global solution. However, it seems to be difficult to adopt their method to deal with the large initial data. So, in this paper we introduce a modified version of Fujita-Kato’s argument. To be precise, we derive a family of integral inequalities on the Littlewood-Paley decomposition of the solution, which makes it possible to utilize the cancellation property of the equation. In the usual Fujita-Kato argument, such cancellation property seems to be unavailable. In order to apply the cancellation property, we establish a new commutator estimate associated with the Littlewood-Paley operator in the Sobolev space. Such inequality plays an crucial role to estimate the nonlinear term. Combining with the cancellation property and the commutator estimate we obtain a priori estimates in the scaling invariant spaces. Thus we construct the local solution for large initial data in H 2−2α . As a byproduct of our approach, we can obtain weighted (in time) estimates of the solution near t = 0 in higher order Sobolev spaces. The paper is organized as follows. In Sect. 2, we define some function spaces and the precise statement of our theorem. Section 3 is devoted to establish some useful estimates such as the commutator estimate. Finally in Sect. 4 we prove the theorem.

Dissipative Quasi-Geostrophic Equation for Large Initial Data

143

2. Definitions and the Statement of the Theorem In this section we define some function spaces and then state the main theorem. Throughout this paper we deal with the two-dimensional space R2 . Let us first recall the definition of the Sobolev space. We define Z as the topological dual space of Z defined by Z ≡ { f ∈ S; x α f (x) d x = 0 for all α ∈ N2 }, where S denotes the space of Schwartz functions. ∞ 2 ˆ Let {φ j }∞ j=−∞ be the Littlewood-Paley decomposition of unity, i.e. φ ∈ C 0 (R \ ∞ 2 − j ˆ {0}), supp φˆ ⊂ {ξ ∈ R ; 1/2 ≤ |ξ | ≤ 2} and j=−∞ φ(2 ξ ) ≡ 1 except ξ = 0. We ˆ − j ξ ). define the Littlewood-Paley operator j as j = φ j ∗, where F(φ j )(ξ ) = φ(2 For 1 < p < ∞, we define the homogeneous and inhomogeneous Sobolev spaces H˙ s, p and H s, p by    1/2     for s ∈ R, H˙ s, p ≡ f ∈ Z ; f H˙ s, p ≡  (2s j | j f |)2  < ∞     j∈Z p

and

H s, p ≡ f ∈ S ; f H s, p ≡ f L p + f H˙ s, p < ∞ for s > 0,

respectively. We abbreviate H˙ s,2 = H˙ s and H s,2 = H s . Remark. Let P be the set of all polynomials. Then Z S /P holds. Since we cannot distinguish zero from other polynomials in S /P, H˙ s, p seems not to be appropriate as function spaces to treat equations. Fortunately, if the exponents s and p satisfy the condition s < 2/ p, then H˙ s, p can be regarded as a subspace of S . Indeed, for s < 2/ p, we have     H˙ s, p f ∈ S ; f H˙ s, p < ∞ and f = j f in S .   j∈Z

For the details, see, e.g. Kozono-Yamazaki [12]. Now we state the main theorem of this paper. Theorem 1. Let 0 < α ≤ 1/2. Suppose that the initial data θ0 ∈ H 2−2α . Then there exist a positive constant T and a unique solution θ of (DQGα ) in L ∞ (0, T ; H 2−2α ) ∩ L 2 (0, T ; H˙ 2−α ). Moreover such a solution θ belongs to C([0, T ); H 2−2α ) and it satisfies the following estimate: β

sup t 2α θ (t) H˙ 2−2α+β < ∞ f or 0 ≤ β < 2α.

(2.1)

0 2 − 2α. ii) In contrast with Chae-Lee [2] and Ju [9], we make use of the Fujita-Kato type argument to construct the solution. This approach provides us the weighted estimate (2.1) of the solution in higher order Sobolev space. iii) Ju [9] proved global existence of a solution for the initial data in H 2−2α with small homogeneous norm. Theorem 1 can be regarded as the local version of his result. In fact, by the argument of our proof, one can also prove the similar global existence theorem: Corollary 1. There exists a positive constant ε such that if the initial data θ0 ∈ H 2−2α satisfies θ0 H˙ 2−2α < ε, then one can take T = ∞ in Theorem 1. 3. Littlewood-Paley Operator and the Commutator Estimate In this section we recall several estimates related to the Littlewood-Paley operator. Throughout this paper we denote a positive constant by C (or C , etc.) the value of which may differ from one occasion to another. On the other hand, we denote Ci (i = 1, 2, · · · ) as the certain constants. We recall Bernstein’s inequality. Lemma 1. (i) Let s ∈ R, 1 ≤ p ≤ ∞. Then there exist positive constants C = C(s, p) and C = C (s, p) such that C2 js j f L p ≤ (−)s/2 j f L p ≤ C 2 js j f L p holds for all j ∈ Z. (ii) Let 1 ≤ p ≤ q ≤ ∞. Then there exists a positive constant C = C( p, q) such that j f L q ≤ C2(2/ p−2/q) j j f L p holds for all j ∈ Z. We prepare various product estimates in the Sobolev space. For this purpose we recall paraproduct formula introduced by Bony [1]. Paraproduct operators are defined by Tf g ≡ S j f j g, j∈Z

R( f, g) ≡ where S j f ≡

k≤ j−3 k

i f j g,

|i− j|≤2

f . Then we have the formal expression for the product: f g = T f g + Tg f + R( f, g).

The following estimates are fundamental properties for the paraproduct operators. For the proof see, e.g. Runst-Sickel [13]. Lemma 2. (i) Let s < 1, t ∈ R. Then there exists a positive constant C = C(s, t) such that T f g H˙ s+t−1 ≤ C f H˙ s g H˙ t holds for f ∈ H˙ s and g ∈ H˙ t .


145

(ii) Let s + t > 0. Then there exists a positive constant C = C(s, t) such that R( f, g) H˙ s+t−1 ≤ C f H˙ s g H˙ t holds for f ∈ H˙ s and g ∈ H˙ t A direct consequence is the following product estimate in the Sobolev space: Proposition 1. Let s, t < 1 and s + t > 0. Then there exists a positive constant C = C(s, t) such that f g H˙ s+t−1 ≤ C f H˙ s g H˙ t holds for f ∈ H˙ s and g ∈ H˙ t . Finally, we state the commutator estimate associated with the operator j , which plays an important role for the estimate of the nonlinear term. Proposition 2. Let 1 ≤ s < 2, t < 1 with s + t > 1. Then there exist positive constants C = C(s, t) such that [ f, j ]g L 2 ≤ C2−(s+t−1) j c j f H˙ s g H˙ t holds for j ∈ Z, f ∈ H˙ s and g ∈ H˙ t with

2 j∈Z c j

= 1. Here we denote

[ f, j ]g = f j g − j ( f g). Proof. Let us decompose the commutator [ f, j ]g by paraproduct formula as follows: [ f, j ]g = [T f , j ]g + R( f, j g) − j R( f, g) + T j g f − j Tg f. We estimate five terms on the right-hand side respectively. By the definition of paraproduct and localization in frequency, we have

[T f , j ]g =

[Sk f, j ]k g.

|k− j|≤3

Applying the mean value theorem, we see that the right-hand side is equal to |k− j|≤3

=2

−j

1

φ j (y)(y · (Sk ∇ f )(x − τ y))k g(x − y)dτ dy

0

|k− j|≤3

Since

1

φ(y)(y · (Sk ∇ f )(x − 2− j τ y))k g(x − 2− j y)dτ dy.

0

|y||φ(y)|dy < ∞, we have [T f , j ]g L 2 ≤ C2− j

|k− j|≤3

Sk ∇ f L p k g L p∗ ,

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H. Miura

where we have taken p < ∞ as s ≡ s + 2/ p < 2 and 1/ p + 1/ p ∗ = 1/2. We can choose such p by the assumption of s. Then Hölder’s inequality yields Sk ∇ f = 2(2−s )l 2(s −2)l l ∇ f l≤k−3

≤ C2

(2−s )k

(2(s −2)l l ∇ f )2

1/2 .

l∈Z

Hence we have Sk ∇ f L p ≤ C2

−(s −2)k

≤ C2

−(s −2)k

≤ C2

−(s −2)k

1/2 (s −2)l 2 (2 ∇ f ) l l∈Z

Lp

f H˙ s −1, p f H˙ s .

By finiteness of the number of the sum on k, we can estimate as follows: [T f , j ]g L 2 ≤ C2−(s −1) j f H˙ s k g L p∗ |k− j|≤3

≤ C2

−(s −1) j

≤ C2

−(s+t−1) j

f H˙ s j g L p∗

f H˙ s 2 j (s−s +t) j g L p∗

≤ C2−(s+t−1) j f H˙ s 2 jt j g L 2 ≤ C2−(s+t−1) j c j f H˙ s g H˙ t , where we define c j = (2 jt j g L 2 )/g H˙ t . Thus, we obtain the estimate for the first term. ˜ k = |k− j|≤2 j . Then we observe that Let ˜ k f k j g, R( f, j g) = |k− j|≤2

which yields the estimate of the second term: ˜ k f k j g L 2 R( f, j g) L 2 ≤ |k− j|≤2

≤

˜ k f L p k j g p∗ L

|k− j|≤2

≤ 2−(s+t−1) j

˜ k f L p 2(s−s +t)k k j g p∗ 2k(s −1) L

|k− j|≤2

≤ C2

−(s+t−1) j

c j f H˙ s g H˙ t ,

p, s

where and c j are chosen as above. Since s + t > 0, we can apply Lemma 2 to the third term: j R( f, g) L 2 ≤ Ccj 2−(s+t−1) j f H˙ s g H˙ t


147

with cj = (2(s+t−1) j j R( f, g) L 2 )/ f g H˙ s+t−1 . For the fourth term, we observe that T j g f = Sk j gk f, k≥ j−2

which yields

T j g f L 2 ≤ C

M( j g)|k f | L 2

k≥ j−2

≤ j g L p∗

|k f | L p

k≥ j−2

≤ C2−(s−s +t) j c j g H˙ t

|k f | L p .

k≥ j−2 ∗

In the above inequalities, we have used the L p -boundedness of the Hardy-Littlewood maximal operator M, where M f (x) ≡ supr >0 1/|B(x, r )| B(x,r ) | f (y)|dy. Since s = s + 2/ p > 1, we have

|k f | =

k≥ j−2

k≥ j−2

≤ C2

2−(s −1)k 2(s −1)k |k f | 

−(s −1) j



1/2 2

2(s −1)k

|k f |2 

.

k≥ j−2

Thus we can estimate the fourth term. Finally, since t < 1, Lemma 2 shows that j Tg f L 2 ≤ Ccj 2−(s+t−1) j f H˙ s g H˙ t with cj = (2(s+t−1) j j Tg f L 2 )/ f g H˙ s+t−1 . 4. Proof of Theorem 4.1. Linear estimates. In this subsection, we consider the linear dissipative equation. The following lemma is closely related to Chemin [3, Prop. 2.1], which characterizes the evolution of the solution to the linear equation. α

Lemma 3. Let e−t (−) a ≡ F −1 (e−t|·| a), ˆ where F −1 denotes the inverse Fourier transform. Then there exist positive constants λ and λ (λ < λ ) depending only on α > 0 such that e−2 for all t > 0.

2α j λ t

2α

α

j a L 2 ≤ e−t (−) j a L 2 ≤ e−2

2α j λt

j a L 2

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H. Miura α

Proof. Let u(t) ≡ e−t (−) j a. Then u satisfies   ∂u + (−)α u = 0 in R2 × (0, ∞), ∂t  u| 2 t=0 = j a in R . Taking the inner product in L 2 with the first equation and u, we have 1 d u2L 2 + (−)α/2 u2L 2 = 0. 2 dt By Lemma 1, there exist positive constants λ and λ (λ < λ ) such that 1 d u2L 2 + λ22α j u2L 2 ≤ 0, 2 dt and 1 d u2L 2 + λ 22α j u2L 2 ≥ 0. 2 dt Dividing the above inequalities by u L 2 and then integrating on the interval (0, t), we have e−2

2α j λ t

u(0) L 2 ≤ u(t) L 2 ≤ e−2

2α j λt

u(0) L 2 .

By definition of u, we obtain the desired result. Now we state the smoothing estimates. Proposition 3. For α > 0 and s ≥ 0, there exists a positive constant C = C(s, α) such that α

s

sup t 2α e−t (−) a H˙ s ≤ Ca L 2

(4.1)

t>0

for all a ∈ L 2 . In particular, we have α

s

lim t 2α e−t (−) a H˙ s = 0,

(4.2)

t→0

for all a ∈ L 2 . Moreover, if 0 ≤ s ≤ α, then we have α

e−t (−) a L 2α/s (0,∞; H˙ s ) ≤ Ca L 2

(4.3)

for all a ∈ L 2 . Proof. We have  e

−t (−)α

a H˙ s = 

j∈Z

1/2 2 α 22 js e−t (−) j a 2  . L


149

On the other hand, it follows from the previous lemma that α

e−t (−) j a2L 2 ≤ e−2

2α j+1 λt

j a2L 2 .

Here, we observe that sup 22 js e−λt2

2α j+1

j∈Z

s

≤ Ct − α ,

which yields  sup t

s 2α

e

−t (−)α

0 0 and j ∈ Z. 2α/s−1

and then integrating the above identity in time, Multiplying this inequality by v j we have ∞ λ22α j v j (t)2α/s dt = Cv j (0)2α/s , 0

that is, 2s j v j L 2α/s = Cv j (0). Taking l 2 -norm on both sides of this estimate, we obtain  

j∈Z

1/2 2s j v j L 2α/s 

 ≤C

j∈Z

1/2 v 2j (0)

.

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H. Miura

Since α/s ≥ 1, the left-hand side is estimated from below as follows: 1/2  1/2   2s j v j 2 2α/s  =  22s j v 2j L α/s  L

j∈Z

So we have

j∈Z

1/2 2s j 2 ≥ 2 v j j∈Z α/s L  1/2  2s j 2  = 2 vj j∈Z

 1/2  2s j 2  2 vj j∈Z

 ≤C L 2α/s

. L 2α/s

1/2 v 2j (0)

.

j∈Z

From Lemma 3, we obtain (4.3). 4.2 Proof of Theorem 1 Step 1. A priori estimates. We first show an a priori estimate in L 3 (0, T ; H˙ 2−4α/3 ). More precisely, we will prove that there exist a positive constant C1 and a bounded function I1 = I1 (T ) with I1 (T ) ≤ Cθ0 H˙ 2−2α and

lim I1 (T ) = 0

(4.4)

T →0

such that θ L 3 H˙ 2−4α/3 ≤ I1 (T ) + C1 θ 2L 3 H˙ 2−4α/3 T

(4.5)

T

holds for all solutions θ of (DQGα ). Here we write the space L p (0, T ; H˙ s ) as L T H˙ s . Applying the operator j to (DQGα ), we obtain p

∂t θ j + (−)α θ j = − j (u · ∇θ ), where we denote θ j ≡ j θ . Adding u · ∇ j θ on both sides, we have ∂t θ j + (−)α θ j + u · ∇ j θ = [u, j ]∇θ. Taking the inner product with the above inequality and θ j , and then applying Lemma 1, we obtain from the divergence free condition that 1 d θ j 2L 2 + λ22α j θ j 2L 2 ≤ [u, j ]∇θ L 2 θ j L 2 . 2 dt Dividing both sides by θ j L 2 , we have d θ j L 2 + λ22α j θ j L 2 ≤ [u, j ]∇θ L 2 . dt


151

Applying Proposition 2 with s = 2 − 4α/3 and t = 1 − 4α/3 and Calderón-Zygmund’s inequality, we obtain 1 d θ j L 2 + λ22α j θ j L 2 ≤ [u, j ]∇θ L 2 2 dt ≤ Cc j 2−(2−8α/3) j u H˙ 2−4α/3 ∇θ H˙ 1−4α/3 ≤ Cc j 2−(2−8α/3) j θ 2H˙ 2−4α/3 . Integrating both sides in time on the interval (0, t), we have t 2α j 2α j θ j (t) L 2 ≤ e−2 λt θ j (0) L 2 + Cc j 2−(2−8α/3) j e−2 λ(t−s) θ (s)2H˙ 2−α ds. 0

Multiplying the above inequality by 2(2−4α/3) j and then taking the l 2 -norm with respect to j, we can estimate the H˙ 2−4α/3 norm of θ as: 1/2  2α j+1 λt 22(2−4α/3) j e−2 θ j (0)2L 2  θ (t) H˙ 2−4α/3 ≤  j∈Z

 2 1/2 t 2α j c j 24α j/3 +C  e−2 λ(t−s) θ (s)2H˙ 2−4α/3 ds  

j∈Z

0

≡ I + I I. In order to show (4.5), we need to estimate L 3T norm of the right-hand side. According to Lemma 3 and (4.3), we see that the first term is estimated as  1/2 2 2α j  2(2−4α/3) j e−2 λt θ j (0) L 2  ≤ Cθ0 H˙ 2−2α . 3 j∈Z LT

Let

 1/2 2  (2−4α/3) j −22α j λt I1 (T ) ≡ 2 e θ j (0) L 2  j∈Z

. L 3T

Then absolute continuity of the integral yields (4.4). Since sup 24α j/3 e−2

2α j λ(t−s)

< C(t − s)−2/3 ,

j

we can estimate the second term as:   2 1/2 t 2α j I I L 3 = C  c j 24α j/3 e−2 λ(t−s) θ (s)2H˙ 2−4α/3 ds  T 0 j∈Z 3 LT t −2/3 ≤C θ (s)2H˙ 2−4α/3 ds (t − s) L 3T

0

≤ Cθ 2L 3 H˙ 2−4α/3 , T

152

H. Miura

where we used Hardy-Littlewood-Sobolev’s inequality in the last inequality. Therefore we obtain the a priori estimate (4.5). Similarly to the previous arguments, we can also show that there exists a bounded function I2 = I2 (T ) with I2 (T ) ≤ Cθ0 H˙ 2−2α and

lim I2 (T ) = 0

T →0

satisfying θ L 2 H˙ 2−α ≤ I2 (T ) + Cθ 2L 3 H˙ 2−4α/3 . T

(4.6)

T

Moreover, we have θ L ∞ H˙ 2−2α ≤ θ0 H˙ 2−2α + Cθ 2L 2 H˙ 2−α . T

T

Combining the above estimates with the maximum principle [6] θ (t) L 2 ≤ θ0 L 2 , we obtain the following estimate θ L ∞ H 2−2α ≤ θ0 H 2−2α + Cθ 2L 2 H˙ 2−α . T

(4.7)

T

Step 2. Convergence of approximation sequences. To construct the solution, we consider the following successive approximation: ∂t θ 0 + (−)α θ 0 = 0 in R2 × R+ , θ 0 |t=0 = θ0 in R2 and

 ∂t θ n+1 + (−)α θ n+1 + u n · ∇θ n+1 = 0 in R2 × R+ ,    u n = (−R2 θ n , R1 θ n ) in R2 × R+ ,    n+1 θ |t=0 = θ0 in R2 ,

for n = 0, 1, 2 . . . . We will establish uniform estimates on θ n . Similarly to the arguments in Step 1, we can show that there exists a bounded function I1 with lim T →0 I1 (T ) = 0 such that θ 0 L 3 H˙ 2−4α/3 T

≤ I1 (T ),

θ n+1 L 3 H˙ 2−4α/3 ≤ I1 (T ) + C1 θ n L 3 H˙ 2−4α/3 θ n+1 L 3 H˙ 2−4α/3 T

T

T

for n = 0, 1, 2 . . . . Taking T0 > 0 so small that I1 (T0 ) ≤ 1/(4C1 ), we have θ n L 3 H˙ 2−4α/3 ≤ 2I1 (T ) for T < T0 . T

By (4.6), we can also show that there exists a bounded function I2 with lim T →0 I2 (T ) = 0 such that θ n L 2 H˙ 2−α ≤ I2 (T ) + C(I1 (T ))2 for T < T0 . T

(4.8)


153

Moreover, (4.7) yields θ n L ∞ H 2−2α ≤ θ0 H 2−2α + C(I3 (T ))2 for T < T0 , T

(4.9)

where we write I3 (T ) ≡ I2 (T ) + C(I1 (T ))2 . Using (4.8), we will prove the convergence of the sequence θ n in L 4T H˙ 3/4 . Let δθ n+1 = θ n+1 − θ n , δu n+1 = u n+1 − u n , δθ 0 = θ 0 and δu 0 = u 0 , and we have following equations of the differences:  ∂t δθ n+1 + (−)α δθ n+1 + u n · ∇δθ n+1 + δu n · ∇θ n = 0 in R2 × R+ ,    δu n = (−R2 δθ n , R1 δθ n ) in R2 × R+ ,    δθ n+1 |t=0 = 0 in R2 , for n = 0, 1, 2 . . . . Similarly to the arguments in Step 1, we have 2 1 d n+1 2 n n+1 , ) + j (δu n · ∇θ n ), δθ n+1 δθ j 2 +λ22α j δθ n+1 j 2 ≤ − j (u · ∇δθ j L L 2 dt ≡ j θ n+1 − j θ n . Since divu n = 0, we have where δθ n+1 j n+1 u n · ∇δθ n+1 = 0. j , δθ j By Hölder’s inequality, we have d n+1 n n+1 L 2 + j (δu n · ∇θ n ) L 2 , δθ j 2 +λ22α j δθ n+1 j 2 ≤ [u , j ]∇δθ L L dt which yields t 2α j n+1 e−2 λ(t−s) u n , j ∇δθ n+1 L 2 + j (δu n · ∇θ n ) L 2 ds. δθ j (t) 2 ≤ C L

0

(4.10) By s = 2 − α and t = −1/4 in Proposition 2, we have n u , j ∇δθ n+1 2 ≤ C2−(3/4−α) j c j u n H˙ 2−α ∇δθ n+1 H˙ −1/4 L

≤ C2−(3/4−α) j c j θ n H˙ 2−α δθ n+1 H˙ 3/4 . On the other hand, by Proposition 1, we have j (δu n · ∇θ n ) 2 ≤ C2−(3/4−α) j c δu n · ∇θ n ˙ 3/4−α j H L ≤ C2−(3/4−α) j cj δu n H˙ 3/4 θ n H˙ 2−α , where j cj 2 = 1. Multiplying (4.10) by 23/4 j , and then taking the l 2 -norm with respect to j, we have δθ n+1 (t) H˙ 3/4   2 1/2 t 2α j 2α j e−2 λ(t−s) θ n H˙ 2−α (c j δθ n+1 H˙ 3/4 + cj δu n H˙ 3/4 )ds  , ≤C j∈Z

0

154

H. Miura

which yields

δθ n+1 L 4 H˙ 3/4 ≤ C θ n L 2 H˙ 2−α δθ n+1 L 4 H˙ 3/4 + δθ n L 4 H˙ 3/4 θ n L 2 H˙ 2−α T T T T T n n+1 n ≤ C2 θ L 2 H˙ 2−α δθ L 4 H˙ 3/4 + δθ L 4 H˙ 3/4 . T

T

By (4.8), there exists T1 > 0 such that Hence we have

θ n

L 2T H˙ 2−α

T

< 1/(3C2 ) for all n = 0, 1, 2 . . . .

1 δθ n L 4 H˙ 3/4 T1 T1 2 1 0 ≤ n+1 θ L 4 H˙ 3/4 T1 2 C ≤ n+1 θ0 H˙ 3/4−α/2 2 C ≤ n+1 θ0 H 2−2α . 2 This shows the existence of the function θ ∈ L 4T1 H˙ 3/4 satisfying limn→∞ θ n = θ in L 4T1 H˙ 3/4 . Furthermore, the uniform estimates (4.8) and (4.9) show that θ also belongs 2−2α ∩ L 2 H ˙ 2−α . We can also prove the uniqueness by similar arguments as to L ∞ T1 H T1 above. Here we can easily check that θ satisfies (DQGα ). We next prove continuity of the solution with values in H 2−2α . By the standard bootstrap argument, it suffices to show the right continuity at t = 0. For the purpose, firstly we prove continuity of the solution with values in H r for 0 ≤ r < 1 − α. Indeed, since u and θ satisfy δθ n+1 L 4

H˙ 3/4

≤

u, θ ∈ L 2 (0, T1 ; H 2−α ) and ∂t θ = −(−)α θ − u · ∇θ, we easily see that the right-hand side of the above identity belongs to L 1T1 H r for 0 ≤ r < 2−2α , 1 − α, which yields that θ ∈ C([0, T1 ); H r ). From the fact that θ belongs to L ∞ T1 H Lemma 1.4 in [14, Chap. 3] shows that θ ∈ Cw ([0, T ); H 2−2α ). By (4.7), we have θ (t) − θ0 2H 2−2α ≤ θ (t)2H 2−2α + θ0 2H 2−2α − 2θ (t), θ0 H 2−2α ≤ 2θ0 2H 2−2α − 2θ (t), θ0 H 2−2α + Cθ 2L 2 H 2−2α , t

H 2−2α . Since θ

is weakly continuous, the second where ·, · H 2−2α is the inner product of term converges to 2θ0 2H 2−2α as t tends to 0. On the other hand, the third term converges to 0 as t tends to 0 because of absolute continuity of the L 2t -norm on t > 0. This shows continuity of the solution at t = 0 with values in H 2−2α . Step 3. Weighted estimates. For the proof of (2.1) and (2.2), it suffices to show β

lim sup t 2α θ n (t) H˙ 2−2α+β = 0

t→0 n≥0

(4.11)

for 0 < β < 2α. We divide the proof into two cases 0 < β < α and α ≤ β < 2α.


155

Case 1. We prove (4.11) for 0 < β < α. For n = 0 (4.1) shows that β

sup t 2α θ 0 (t) H˙ 2−2α+β ≤ Cθ0 H˙ 2−2α .

(4.12)

0 and the L 2 -unitary product ( , ) L 2 of smooth half-densities τ = f · dens X and υ = g · dens X ' are related by X f · g dens X = (τ, υ) L 2 = τ, υ. Suppose then that u t , t > 0, is a family of smooth half-densities on X such that u t → u as t → 0, in the topology of the

Equivariant Asymptotics for Bohr-Sommerfeld Lagrangian Submanifolds

257

space of all generalized half-densities whose wave front is conormal to . In view of the self-adjointness of the orthogonal projector k, on H(X )k, , we obtain: = lim u t , vk, L 2 (X ) (u k, , vk, ) L 2 (X ) = lim k, (u t ), vk, ) 2 t→0 t→0 L (X ) = lim u t , vk, = u, vk, t→0 f λ · vk, dens , =

(75)

where dens is the Riemannian density on . 4.1. The transverse case. Consider the smooth map given by group action restricted to , ϒ : (h, g, x) ∈ S 1 × G × → (h, g) · x ∈ X. To fix ideas, suppose first that ϒ is transversal to = ∩ ( ◦ π )−1 (0) . In this case, ϒ −1 ( ) is a finite set: ϒ −1 ( ) = { y˜1 , . . . , y˜r }, where y˜ j = (h j , g j , y j ) for some h j ∈ S 1 , g j ∈ G and y j ∈ . Hence y)j =: ϒ( y˜ j ) = (h j , g j ) · y˜ j ∈ for every j. Now let U j ⊆ be some arbitrarily small neighbourhood of y j . Since vk, = O(k −∞ ) away from , in view of (75) we have r

(u k, , vk, ) L 2 (X ) ∼

j=1 U j

f λ · vk, dens .

(76)

Let us fix Heisenberg local coordinates ( p, q, θ ) for X centered at ) y j and adapted to , defined on an open neighbourhood V j ) y j . Thus, ∩ V j ⊆ V j is defined by conditions θ = f (q) and p = h(q), as described in §2.2. We may arrange, given our assumptions, that * ∂ ∂ T)y j = span ,..., . (77) ∂q1 )y j ∂qn−g )y j The following is left to the reader: Lemma 4.1. Given (77), we have y j ) = span g X ()

∂

∂ pn−g+1

for appropriate tn−g+1 , . . . , tn ∈ T)y j .

) yj

* ∂ + tn−g+1 , . . . , + tn , ∂ pn )y j

(78)

258

M. Debernardi, R. Paoletti

Let us now consider the Legendrian submanifold ) y j ∈ j =: ϒ {(h j , g j )} × ⊆ X, obtained by ‘translating’ by the action of (h j , g j ) ∈ S 1 × G. Given (77) and Lemma 4.1, the present transversality assumption implies: Lemma 4.2. In the above situation, ( p1 , . . . , pn−g ) restrict to local coordinates on j centered at ) y j , and ( p1 , . . . , pn−g , qn−g+1 , . . . , qn ) restrict to local coordinates on j centered at ) yj. Therefore ( p1 , . . . , pn−g , qn−g+1 , . . . , qn ) may be viewed in a natural manner as local coordinates on centered at y j , defined on some open neighbourhood U j ⊆ . In order to apply Theorem 1.1, we need to relate these coordinates on to the local Heisenberg coordinates on X . Given x = (x1 , . . . , xn ), to simplify our notation let us write x = (x , x ), where x = (x1 , . . . , xn−g ), x = (xn−g+1 , . . . , xn ). The following is left to the reader: We have: Lemma 4.3. There exists an R-linear map A j : Rn → Cn ∼ = {0} ⊕ Cn ⊆ R ⊕ Cn such that if y ∈ U j ⊆ has local coordinates Heisenberg coordinates

√1 k

Aj

( p , q ) +

O(k −1 ).

√1 ( p , q ) k

on , then it has local

Let y( √1 ( p , q )) denote the point in U j having local coordinates √1 ( p , q ). By k k Theorem 1.1 and Lemma 4.3, passing to rescaled coordinates on U j we may then write the j th summand in (76) as: −n/2 f λ · vk, dens = k f λ (k −1/2 ( p , q ) vk, k −1/2 A j ( p , q )+ O(k −1 ) Rn

Uj

×D k −1/2 ( p , q ) dp dq .

(79)

Inserting the asymptotic expansion of Theorem 1.1 in (79), we conclude Proposition 4.1. If ϒ : S 1 × G × → X is transversal to , the j th summand in (83) is ( j) ( j) f λ · vk, dens ∼ k −g/2 ρ0 + k −(g+ f )/2 ρ f , Uj

f ≥1

where ( j) ρ0

( (2π )n+g k dim(V ) 1 = h j χ (g −1 )j ) f λ (y j ) f σ ( y)j ) j ) ( y |G π(y j ) | π n 2g −S y j A j ( p ,q ),A j ( p ,q ) −i T+ y j A j ( p ,q ),A j ( p ,q ) · e + dp dq . Rn


259

In the action-free case, the present transversality assumption means that S 1 × → X is transverse to . For every j = 1, . . . , r , T)y j j ⊆ Tπ()y j ) M is a Lagrangian subspace transversal to T)y j . Thus, in the given Heisenberg local coordinates adapted to at ) yj, we have T)y j j = {( p, Z j p) : p ∈ Rn } ⊆ Tπ()y j ) M ∼ = Rn ⊕ Rn ,

(80)

where Z j is a symmetric matrix. Therefore, the p’s restrict to a system of local coordinates on j (whence on ), and A j ( p) = p + i Z j p. Let ı Jπ()y j ) : Grlag (Tπ()y j ) M) × Grlag (Tπ()y j ) M) → R be the invariant introduced in Sect. 2.4; let us write J j = Jπ()y j ) . Applying the asymptotic expansion of Corollary 1.1, Corollary 4.1. Suppose that the two projections → M and → M are transversal. Let ϒ : S 1 × → be the map induced by the action, and suppose ϒ −1 () = { y˜1 , . . . , y˜r }, where y˜ j = (h j , y j ). Set ) y j =: h j · y j and j =: h j · for every j. Then k − f /2 ρ f , (u k , vk ) ∼ ρ0 + f ≥1

where n r −1 (2π ) 2 k 2 t ρ0 = h j f λ (y j ) f σ ( y)j ) ı J j T)y j j , T)y j e− p +i p Z j p dp. n πn R

j=1

4.2. The clean case. Now we shall make the following more general hypothesis: i) and are both transversal to X ; let us set =: ∩ X , =: ∩ X . ii) The smooth map given by group action restricted to , ϒ : (h, g, x) ∈ S 1 × G × → (h, g) · x ∈ X, meets nicely; by this, we mean that every connected component of ϒ −1 ( ) is a manifold, and that for every ς = (h, g, x) ∈ ϒ −1 ( ) we have Tς ϒ −1 ( ) = (dς ϒ)−1 Tϒ(ς) . iii) there exist integers r , r ≥ 1 such that for every x ∈ and y ∈ one has | ∩ (G · x)| = r and | ∩ (G · y)| = r .

(81)

iv) G acts freely on M . Definition 4.1. Let us set Y˜ =: ϒ −1 ( ) ⊆ S 1 × G × . Let π : S 1 × G × → be the projection onto the third summand, and let us set Y =: π (Y˜ ) ⊆ . Lemma 4.4. Let = ∩ X . Then there exists an open neighbourhood V ⊆ of such that ϒ is immersive on S 1 × G × V . Proof. This follows from the horizontality of and of the G-action on X , and from Corollary 2.1.

260


Proposition 4.2. Suppose that the hypotheses i), ii) and iii) above are satisfied. Let Y˜1 , . . . , Y˜r ⊆ S 1 × G × be the connected components of Y˜ , and let Y j =: π (Y˜ j ) ⊆ . Then: i) for every j = 1, . . . , r , there exists h j ∈ S 1 such that Y˜ j ⊆ {h j } × G × ; ii) every Y j is a submanifold, and the induced map π j : Y˜ j → Y j is an unramified covering; iii) the Y j ’s, with possible repetitions, are the connected components of Y . j . Proof. i) Suppose that (h, g, x) ∈ Y˜ j for some j, and consider (a, v, w) ∈ T(h,g,x) Y Since ϒ(Y j ) ⊆ and is Legendrian, we conclude that ∂ + d(h,g,x) ϒ(0, v, w) 0 = αϒ(h,g,x) d(h,g,x) ϒ(a, v, w) = αϒ(h,g,x) a ∂θ = a + αϒ(h,g,x) d(h,g,x) ϒ(0, v, w) = a. The latter equality follows from the horizontality of and of the G-action on X . Since Y˜ j is connected, the statement follows. π j is not an immersion, by part i) ii) and iii) Let π j : Y˜ j → be the projection. If there exists (h j , g, x) ∈ Y˜ j and a tangent vector of the form (0, v, 0) ∈ T(h j ,g,x) Y˜ j , for some 0 = v ∈ Tg G. By Lemma 4.4, 0 = d(h j ,g,x) ϒ (0, v, 0) ∈ Tϒ(h j ,g,x) ∩ Tϒ(h j ,g,x) G · ϒ(h j , g, x) , against Corollary 2.1. Suppose now that h ∈ {h 1 , . . . , h r } ⊆ S 1 . Let , Y (h) =: Y˜ j . h j =h

Suppose that y ∈ π (Y (h) ); there are as many inverse images of y in Y (h) as there are group elements g ∈ G such that (h, g) · y = g · (h · y) ∈ ; in other words, (h) −1 (y) = ∩ G · h · y) = d . (82) Y ∩ π On the other hand, in an immersion with compact domain the number of points in a inverse image can only jump up. Therefore, given (82) the cardinality of a fibre has to component Y˜ j of Y (h) . If on the be constant for each map Y˜ j → Y j , for every −1connected other f : X → Y is an immersion, and f (y) is constant for every y ∈ f (X ), then f (X ) is a manifold and the induced map X → f (X ) is an unramified covering. We note in passing that by the same argument, and given the symmetry of our hypothesis on and , we also have: Proposition 4.3. For every j = 1, . . . , r , let j = ϒ(Y˜ j ). Then the j ’s are disjoint manifolds, and are the connected components of ϒ(Y˜ ). The induced map Y˜ → ϒ(Y˜ ) is an unramified covering.


261

Definition 4.2. For every j = 1, . . . , r , set c j =: n − dim(Y j ) and let d j be the degree of the unramified cover π j : Y˜ j → Y j . Thus, for every y ∈ Y j there exist distinct s1 j (y), . . . , sd j , j (y) ∈ G such that (h j , si j (y), y) ∈ Y˜ j , and therefore (h j , si j (y)) · y ∈ , i = 1, . . . , d j . Locally on Y j near y we may think of si j ’s as G-valued smooth maps. The si j ’s are not globally well-defined as smooth maps Y j → G; nonetheless, collectively they do define a smooth map from Y j to the appropriate symmetrized product of G. Now let U j ⊆ be some arbitrarily small tubular neighbourhood of the submanifold Y j ⊆ . Since vk, = O(k −∞ ) away from , in view of (75) we have (u k, , vk, ) L 2 (X ) ∼

r j=1

Uj

f λ · vk, dens .

(83)

Remark 4.1. Since the Y j ’s are not necessarily all distinct, (83) is not literally true. However, to avoid making our exposition too heavy, we shall be slightly vague on this; we shall thus act as the Y j were all disjoint. In the following computations, each summand in (83) will split as the sum of various other contributions, and we shall not sum the same contribution twice. Suppose 1 ≤ j ≤ r . For every y ∈ Y j , we may find an open neighbourhood d j ˜ y ∈ S ⊆ Y j which is uniformly covered by π˜ j , meaning that π˜ −1 j (S) = i=1 Si ⊆ Y˜ j , a disjoint union where each S˜i projects diffeomorphically onto S under π˜ j , and (h j , si j (y), y) ∈ S˜i . Perhaps after restricting S, by Lemma 4.4 we may further assume that for each i the map induced by ϒ, (h j , si j (y), y) → (h j , si j (y)) · y is a diffeomorphism onto its image, Si =: ϒ( S˜i ) ⊆ . S˜i ∼ =)

(84)

We may then find a finite open cover {S ja }a∈A of Y j with the following properties: i) each S ja is the domain of a coordinate chart, say R ja = (r1 , . . . , rn−c j ) : S ja → Bn−c j (0, ) ⊆ Rn−c j , for some > 0; d j ˜ ii) each S ja is uniformly covered by π˜ j , and π˜ −1 j (S ja ) = i=1 Si ja is a disjoint union, where for each i, a, (85) S˜i ja =: (h j , si j (y), y) : y ∈ S ja ⊆ Y˜ j ; Si ja =: ϒ( S˜i ja ) ⊆ for every i, a; iii) ϒ induces a diffeomorphism S˜i ja ∼ =) iv) for every i, a there exist an open neighbourhood Ti ja ⊆ X of ) Si ja , and a smooth ) Si ja the map κ = κi ja : Si ja × Ti ja → B2n+1 (0, ), such that for every y ∈ ) partial function κ y : Ti ja → B2n+1 (0, ) is a Heisenberg chart adapted to at y (Lemma 2.3).

262


Now recall that for every j we have fixed a tubular neighbourhood U j ⊆ of Y j ; let p j : U j → Y j be the projection, and set U ja =: p −1 U j . Thus, {U ja }a∈A j (S ja ) ⊆ is a finite open cover of U j . By introducing a partition of unity a ϕ ja = 1, we may decompose the j th summand in (83) as f λ · vk, dens = ϕ ja f λ · vk, dens . (86) Uj

U ja

a

We are thus reduced to considering the asymptotics of each summand in (86). Given (85) we may the apply a relative version of the argument in §4.1; rescaling will now be in the coordinates in U ja which are transversal to Y j . We now leave it to the reader to verify that, using the local coordinates R ja = (r1 , . . . , rn−c j ) on S ja , one obtains an asymptotic expansion ( ja) (n−g−c j )/2 ϕ ja f λ · vk, dens ∼ k ρ0 (r ) dr U ja

+

Rn−c j

( ja)

k (n−g−c j − f )/2 ρ f

,

(87)

f ≥1

where, in view of the asymptotic expansion of Theorem 1.1, ( 1 (2π )n+g k dim(V ) ( ja) hj χ sl j (y(r )) ρ0 (r ) = n g Veff [π(y(r ))] π 2 l e−Sr (z)−i Pr (z) dz, · ϕ ja y(r ) f λ y(r ) f σ h j · sl j (y(r )) Rc j

for quadratic forms Sr , Pr on Rc j , with Sr positive definite. In the action-free case, this becomes: ( ja) ϕ ja f λ · vk dens ∼ k (n−c j )/2 ρ0 (r ) dr U ja

+

Rn−c j

( ja)

k (n−c j − f )/2 ρ f

,

(88)

f ≥1

where n

( ja) ρ0 (r )

dim(V ) (2π ) 2 k = hj Veff [π(y(r ))] π n · ϕ ja y(r ) f λ y(r ) f σ h j · y(r ) T ja y(r ) ,

T ja y(r ) =: ı Jh j ·y(r ) Th j ·y(r ) · j , Th j ·y(r )

−1

·

R

cj

e

− p2 +i p t Z h j ·y( p) p

dp,

Z r being an appropriate c j × c j symmetric matrix. Acknowledgements. We are very grateful to the referee for suggesting various improvements in presentation, and to Steve Zelditch for some interesting remarks.


263

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Communications in


Self-Organized Forest-Fires near the Critical Time J. van den Berg , R. Brouwer CWI, Kruislaan 413, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. E-mail: [email protected]; [email protected] Received: 29 October 2005 / Accepted: 6 December 2005 Published online: 22 April 2006 – © Springer-Verlag 2006

Abstract: We consider a forest-fire model which, somewhat informally, is described as follows: Each site (vertex) of the square lattice is either vacant or occupied by a tree. Vacant sites become occupied at rate 1. Further, each site is hit by lightning at rate λ. This lightning instantaneously destroys (makes vacant) the occupied cluster of the site. This model is closely related to the Drossel-Schwabl forest-fire model, which has received much attention in the physics literature. The most interesting behaviour seems to occur when the lightning rate goes to zero. In the physics literature it is believed that then the system has so-called self-organized critical behaviour. We let the system start with all sites vacant and study, for positive but small λ, the behaviour near the ‘critical time’ tc , defined by the relation 1 − exp(−tc ) = pc , the critical probability for site percolation. Intuitively one might expect that if, for fixed t > tc , we let simultaneously λ tend to 0 and m to ∞, the probability that some tree at distance smaller than m from O is burnt before time t goes to 1. However, we show that under a percolation-like assumption (which we can not prove but believe to be true) this intuition is false. We compare with the case where the square lattice is replaced by the directed binary tree, and pose some natural open problems. 1. Introduction 1.1. Background and motivation. Consider the following, informally described, forestfire model. (A precise description follows later in this section). Each site of the lattice Zd is either vacant or occupied by a tree. Vacant sites become occupied at rate 1, independently of anything else. Further, sites are hit by lightning at rate λ, the parameter of the model. When a site is hit by lightning, its entire occupied cluster instantaneously burns down (that is, becomes vacant). Part of vdB’s research is supported by BRICKS project AFM 2.2.

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J. van den Berg, R. Brouwer

This is a continuous-time version of the Drossel-Schwabl model which has received much attention in the physics literature. See e.g. [1, 3, 5, 9] and sections in the book by Jensen [7]. For comparison with real forest-fires see [8]. The most interesting questions are related to the asymptotic behaviour when the lightning rate tends to 0. It is believed that this behaviour resembles that of ‘ordinary’ statistical mechanics systems at criticality. In particular, it is believed that, asymptotically, the cluster size distribution has a power-law behaviour. Heuristic results confirming such behaviour have been given in the literature, but the validity of some of these results is debatable (see [5]) and almost nothing is known rigorously (except for the one-dimensional case). Our goal is more modest, and we address some basic problems which, surprisingly, have so far been practically ignored, although their solution is crucial for a beginning of rigorous understanding of these models. We restrict to the 2-dimensional case. That is, the forest is represented by the square lattice. It seems to be taken for granted in the literature that, informally speaking, as we let λ tend to 0, the steady-state probability that a given site, say the origin O, is vacant stays away from 0. But is this really obvious? (Even, is it true?) The intuitive reasoning seems to be roughly as follows: “If the limit of the probability to be occupied would be 1, then the system would have an ‘infinite occupied cluster’. But that cluster would be immediately destroyed, bringing the occupation density away from 1: contradiction”. Of course this reasoning is, mildly speaking, quite shaky and we believe that a rigorous solution of this problem is necessary for a clear understanding of the forest-fire model. The problems investigated in this paper are, although not the same as the one just described, of the same spirit. Instead of looking at the steady-state distribution, we start with all sites vacant and look at the time tc at which, in the modified model where there is only growth but no ignition, an infinite cluster starts to form. Intuitive reasoning similar to that above makes plausible that, informally speaking, for every t > tc , the probability that O burns before time t stays away from 0 as λ tends to 0. Continuing such intuitive reasoning then leads to the ‘conclusion’ that, again informally speaking, if we take m sufficiently large and replace the above event by the event {Some vertex at distance ≤ m from O burns before time t}, the corresponding probability will be, as λ tends to 0, as close to 1 as we want. We relate this to problems which are closer to ordinary percolation. In particular we show that, under a percolation-like assumption (which we believe to be true), the above ‘conclusion’ is false. We hope our results will lead to further research and clarification of the above problems.

1.2. Formal statement of the problems. So far, we have not defined our model precisely yet. We now give this more precise description, formulate some of the above mentioned problems more formally, and introduce much of the terminology used in the rest of this paper. We work on the square lattice, i.e. the graph of which the set of sites (vertices) is Z2 , and where two vertices (i, j) and (k, l) share an edge if |i − k| + | j − l| = 1. To each site we assign two Poisson clocks: one (which we call the ‘growth clock’) having rate 1, and the other (the ‘ignition clock’) having rate λ. All Poisson clocks behave independently of each other. A site can be occupied by a tree or vacant. These states are denoted by 1 and 0 respectively. Initially all sites are vacant. We restrict ourselves to a finite box B(n) := [−n, n]2 . (In our theorems we consider the behaviour as n → ∞). The dynamics is as follows: when the growth clock of a site v rings, that site becomes

Self-Organized Forest-Fires near the Critical Time

267

occupied (unless it already was occupied, in which case the clock is ignored); when the ignition clock of a site v rings, each site that has an occupied path in B(n) to v, becomes vacant instantaneously. (Note that this means that if v was already vacant, nothing happens.) Now let ηvn (t) = ηv (t) ∈ {0, 1} denote the state of site v at time t, and define η(t) = ηn (t) := (ηvn (t), v ∈ B(n)). Note that, for each n, (ηn (t), t ≥ 0) is a finite-state (continuous-time) irreducible Markov chain with state space {0, 1} B(n) . The assignment of Poisson clocks to every site of the square lattice provides a natural coupling of the processes ηn (·), n ≥ 1 with each other, and with other processes (see below). For m ≤ n, we often use the informal phrase “ηn has a fire in B(m) before time t” for the event {∃v ∈ B(m) and ∃s ≤ t such that ηvn (s − ) = 1 and ηvn (s) = 0}. Similarly, we use “ηn has at least two fires in B(m) before time t” for the event {∃v, w ∈ n (u − ) = 1 and ηn (s) = ηn (u) = 0}. Note that B(m) and ∃s < u ≤ t s.t. ηvn (s − ) = ηw v w we allow v and w to be equal. Let Pλ be the measure that governs all the underlying Poisson processes mentioned above (and hence, for all n simultaneously, the processes ηn (·)). Often, when there is no need to explicitly indicate the dependence on λ, or when we consider events involving the growth clocks only, we will omit this subscript. It is trivial that for all times t and all n, m the probability that ηn has a fire in B(m) before time t goes to 0 as λ ↓ 0, and hence lim lim Pλ (ηn has a fire in B(m) before time t) = 0.

n→∞ λ↓0

A much more natural (and difficult!) question is what happens when we reverse the order of the limits. For the investigation of such questions it turns out to be very useful to consider the modified process σ (t) on the infinite lattice, which we obtain, loosely speaking, if we obey the above mentioned growth clocks but ignore the ignition clocks: σv (t) = I{The growth clock at v rings in [0,t]} , where I denotes the indicator function. It is clear that, for each time t, the σv (t), v ∈ Z2 , are Bernoulli random variables with parameter 1 − exp(−t). So, if we define tc by the relation pc = 1 − exp(−tc ), where pc is the critical value for ordinary site percolation on the square lattice, we see that σ (t) has no infinite occupied cluster for t ≤ tc but does have an infinite cluster for t > tc . To illustrate the usefulness of comparison of η with σ (and as introduction to more subtle comparison arguments), we show that lim sup lim sup Pλ (ηn has a fire in O before time t) ≤ θ (1 − e−t ), λ↓0

(1)

n→∞

where θ (.) is the percolation function for ordinary site percolation. The argument is as t (O) denote the occupied cluster of 0 in the configuration σ (t). It is easy follows: Let C to see from the process descriptions above that in order to have, for the process ηn , a fire in 0 before time t, it is necessary (but not sufficient) that at least one of the ignition t (O) has rung before time t. Using the independence of the different clocks in the set C Poisson clocks, we have

268


Pλ (ηn has a fire in O before time t) ∞ t (O)| = k and ∃v ∈ C t (O) that has ignition before time t) Pλ (|C ≤ k=1

t (O)| = ∞) + P(|C ∞ t (O)| = k)(1 − e−λtk ) + θ (1 − e−t ). P(|C = k=1

Note that, in the r.h.s. above, the first term does not depend on n and, as λ → 0, clearly goes to 0 (by bounded convergence). The desired result follows. In particular, we have for each m and each t ≤ tc , lim lim Pλ (ηn has a fire in B(m) before time t) ≤ |B(m)|θ (1 − e−t ) = 0, λ↓0 n→∞

(2)

where |B(m)| denotes the number of sites in B(m). But what happens right after tc ? Intuitively one might argue as follows: “If the l.h.s. of (2) is 0 for some t > tc , then roughly speaking, the system at time t looks as in ordinary percolation with parameter 1 − exp(−t), so that an infinite occupied cluster has built up, and this cluster intersects B(m) with positive probability. But an infinite cluster has an infinite total ignition rate and hence catches fire immediately: contradiction. Hence for each t > tc the l.h.s. of (2) is strictly positive.” As we said before, such reasoning is very shaky. Its conclusion is correct for the directed binary tree (see Lemma 4.5). We have some inclination to believe that the conclusion also holds for the square lattice, but prefer to formulate this as an open problem, rather than a conjecture: Open Problem 1.1. Is, for all t > tc , lim sup lim sup Pλ (ηn has a fire in O before time t) > 0 ? λ↓0

(3)

n→∞

Believing the answer to the above problem is affirmative, it is intuitively very tempting to go further and ‘conclude’ that also the answer to the following problem is affirmative: Open Problem 1.2. Is it true that for all t > tc and each ε > 0 there exists an m such that lim sup lim sup Pλ (ηn has a fire in B(m) before time t) > 1 − ε ? λ↓0

(4)

n→∞

The intuitive (and again shaky) reasoning here is, roughly speaking, that if the answer to Problem 1.1 is affirmative, there will be a positive density of sites that burn before time t, and hence the probability of having such a site in B(m) will tend to 1 as m → ∞. Our main result, Theorem 2.2, indicates that the behaviour of the process may be considerably different from what the above intuition suggests. At this point, one could wonder whether it is really necessary to first restrict to finite n, so that we have the annoying ‘extra’ limit n → ∞ in our theorems and problem formulations: is, for each λ > 0, the model well-defined on the infinite lattice? For sufficiently large λ one can easily see that this is true. (Using domination by suitable


269

Bernoulli processes one can, for such λ, make a standard graphical construction.) M. Dürre (see [4]) has recently shown, by more abstract means, that an infinite-volume forest-fire process exists for every λ > 0, but for small λ the uniqueness of such process is still open. In Sect. 4 we consider a slightly modified process that is obviously, by a graphical construction, well-defined on the infinite lattice. In this modified process occupied clusters with size larger than or equal to L (the parameter of the model) are instantaneously removed. For that model we have results very similar to those for the original one. In this paper we will assume knowledge of some ‘classical’ results in 2-dimensional percolation, in particular the standard RSW-type results (see [6], Chap. 11). 2. Statement of the Main Results 2.1. A percolation-like critical value. In this subsection we define a percolation-like critical value, denoted by δˆc , which plays a major role in the statement of our main results. First some notational remarks. Recall that pc denotes the critical probability for site percolation on the square lattice. The product measure with density p will be denoted by Pp . The event that there is an occupied path from a set V to a set W is denoted by {V ↔ W }. Let n be a positive integer, and consider the box B = [0, 4n] × [0, 3n]. By the boundary of B, denoted by ∂ B, we mean the set of those sites in B that have a neighbour in the complement of B. We are now ready to define δˆc . Let δ ∈ [0, 1]. Suppose the sites of B are, independently of each other, occupied with probability pc and vacant with probability 1 − pc . Next, informally, we destroy the occupied cluster of the boundary. That is, each vertex in B that initially had an occupied path to the boundary of B is made vacant. Finally, in the resulting configuration, each vacant site (that is, each site that initially was vacant, or that was initially occupied but made vacant by the above destruction step) is, independently, made occupied with probability δ. It is straightforward to see that in the final configuration a site v ∈ B is occupied with probability pc − Ppc (v ↔ ∂ B) + (1 − pc + Ppc (v ↔ ∂ B)) δ. If we let n grow and choose v further and further away from ∂ B, this clearly converges to pc + (1 − pc )δ. Although this is larger than pc , the final configuration has complicated spatial dependencies and therefore it is not clear whether, in the bulk, it is ‘essentially supercritical’. In particular, let A be the box [n, 3n] × [n, 2n], and consider the probability pn (δ) that the final configuration has an occupied vertical crossing of A. (As is well-known, in ordinary supercritical percolation the probability of such event goes to 1 as n → ∞.) It is clear that pn (δ) is increasing in δ, and we define δˆc = sup {δ : pn (δ) is bounded away from 1, uniformly in n}.

(5)

Conjecture 2.1. δˆc > 0. In spite of serious attempts no proof (or disproof) of this conjecture has been found yet. It is supported by simulation results but, since the box size our simulations could handle is limited, one has to be careful with interpreting such results. Conjecture 2.1 is very similar in nature to, and ‘somewhat’ weaker than (see the discussion below), Conjecture 3.2 in [2]. There we proved, among other results, that

270


assumption of that conjecture yields, informally speaking, the non-existence of a process on the square lattice, starting with all sites vacant, where (as in our model) vacant sites always become occupied at rate 1, and where infinite occupied clusters instantaneously become vacant. Such a non-existence result, although theoretically interesting, looks somewhat esoteric. In the present paper we show that the conjecture also has remarkable consequences for the ‘concrete’ and natural forest-fire models η(·). Conjecture 2.1 is weaker than the above mentioned conjecture in [2], in the sense that we can prove that the correctness of the latter implies that of the former but we don’t know how to prove the reverse implication. Since the weaker form is sufficient for our purposes (here as well as in [2]), we decided to present that form. 2.2. The main results. Recall the definition of δˆc in (5). We are now ready to state our main results: Theorem 2.2. If δˆc > 0, there exists a t > tc such that for all m, lim inf lim inf Pλ (ηn has a fire in B(m) before time t) ≤ 1/2. λ↓0

n→∞

(6)

The key to Theorem 2.2 is the following proposition (which is also interesting in itself): Proposition 2.3. If δˆc > 0, there exists a t > tc such that for all m, lim lim sup Pλ (ηn has at least 2 fires in B(m) before time t) = 0. λ↓0

(7)

n→∞

The proofs of the above proposition and theorem are given in Sect. 3. 3. Proofs The proof of our main theorem (Theorem 2.2) depends heavily on Proposition 2.3. For the proof of the proposition we need two auxiliary models. One of these, the ‘pure growth’ model σ (t), was already introduced in Section 1. The other, which has the same growth mechanism but where removal of trees takes place at time tc only, is described below. 3.1. Removal at tc only. Let I denote the set of all positive even integers i and consider the annuli Ai := B(5 · 3i )\B(3i ), i ∈ I . Note that these annuli are pairwise disjoint. In the process we are going to describe, again every site can be vacant (have value 0) or occupied (value 1). By a ‘surrounding i cluster’ we will mean an occupied circuit C around 0 in the annulus Ai , together with all occupied paths in Ai that contain a site in C. The process is completely determined by the Poisson growth clocks introduced in Sect. 1, in the following way. Initially each site is vacant. Whenever the growth clock of a site rings, the site becomes occupied. (When it already is occupied, the clock is ignored.) Destruction (1 → 0 transitions) only takes place at time tc : at that time, for each positive even integer i, each ‘surrounding i cluster’ is instantaneously made vacant. After tc the growth mechanism proceeds as before. Let ξv (t) denote the value of site v at time t. Earlier in this paper we mentioned an obvious but useful relation (comparison) between the pure growth process σ (·) and the forest-fire process η(·). There is also a


271

useful relation between ξ(·) and η(·), but its statement and proof are less straightforward (see Lemma 3.2 in Sect. 3.2). Another lemma involving the process ξ(·) that will be important for us is the following. Lemma 3.1. If δˆc > 0 there exist γ < 1 and ε > 0 such that for all i ∈ I , P(∂ B(3i ) → ∂ B(3 · 3i ) in the configuration ξ(tc + ε)) < γ .

(8)

The proof of this lemma is very similar to that of Lemma 3.4 of [2]. (The pn ’s we defined a few lines before (5) differ from the ‘corresponding’ an ’s in [2], but the modifications in the proof arising from this difference are straightforward.) 3.2. Proof of Proposition 2.3. Fix m. Since the probability in the statement of the proposition is monotone in m, we may assume that m is of the form 3l for some even positive integer l. (So each annulus Ai , i ∈ I , defined in the previous subsection, is either contained in B(m) or disjoint from B(m)). Let τ = τ (n, m) be the first time that ηn has a fire in B(m); more precisely, τ := inf{t : ∃v ∈ B(m) s.t. ηvn (t) = 0 and ηvn (t − ) = 1}. Next, define, for 1 > λ > 0, 1 K (λ) := √ , 3 λ 1 , k(λ) := √ 4 λ A(k(λ), K (λ)) := B(K (λ))\B(k(λ)).

(9)

Further, define the following events: B1 = B1 (λ) := { no ignitions in B(K (λ)) before or at time τ }, B2 = B2 (λ) := { σ (tc ) has a vacant *-circuit in A(k(λ), K (λ))}, where by ‘*-circuit’ we mean a circuit (surrounding 0) in the matching lattice (i.e. the lattice obtained from the square lattice by adding the two ‘diagonal edges’ in each face of the square lattice). We will use the following relation between the forest-fire process η(·) and the auxiliary process ξ(·) described in the previous subsection. Lemma 3.2. Let λ ∈ (0, 1). On B1 ∩ B2 we have, for all t > τ , all v ∈ B(k(λ)) \ B(m) and all n, that ηvn (t) ≤ ξv (t).

(10)

Proof. (of Lemma 3.2). Suppose B1 ∩ B2 holds. Take n, t and v as in the statement of the lemma. Obviously, we may assume that k(λ) > m. To simplify notation we will, during the proof of this lemma, omit the superscript n from η, and the argument λ from k and K . Suppose ξv (t) = 0. We have to show that then also ηv (t) = 0. Since ξv (t) = 0, the growth clock of v does not ring in the interval (tc , t], and we may assume that just before tc the occupied ξ cluster of v surrounds B(m). (Otherwise the desired conclusion follows trivially.) From the definitions of the processes it then follows that at time tc

272


the occupied σ cluster of v, which we will denote by C, surrounds B(m). By B2 we have that C is in the interior of a vacant (that is, having σ (tc ) = 0) *-circuit in A(k, K ). Clearly, η ≡ 0 on this circuit during the time interval (0, tc ], which prevents fires starting in its exterior to reach its interior. From this, and the event B1 , we conclude that τ > tc , and that η(tc ) and σ (tc ) agree in the interior of this circuit. In particular, the occupied η cluster of v at time tc equals the above mentioned set C. From B1 it follows that at time τ a connected set is burnt which contains sites in B(m) as well as in the complement of B(K ). But then it also contains a site in C (because C surrounds B(m) and lies inside B(K )). So the whole set C, and in particular v, burns at some time s ∈ (tc , τ ]. Since the growth clock of v does not ring between time tc and t, it follows that indeed ηv (t) = 0. This completes the proof of Lemma 3.2.

Now we go back to the proof of the proposition. Assume δˆc > 0. Choose ε and γ as in Lemma 3.1. By (2) it is sufficient to prove that lim lim sup Pλ (ηn has at least 2 fires in B(m) in (tc , tc + ε)) = 0. λ↓0

(11)

n→∞

Define, in addition to B1 and B2 above, the event B˜ 1 = { no ignitions in B(K (λ)) in the time interval (0, tc + ε)}. We have P(at least 2 fires in B(m) in (tc , tc + ε)) ≤ P({at least 2 fires in B(m) in (tc , tc + ε)} ∩ B˜ 1 ∩ B2 ) +P( B˜ 1c ) + P(B2c ).

(12)

Now note that B˜ 1 does not depend on n, and that P( B˜ 1c ) ≤ λ |B(K (λ))| (tc + ε) → 0, as λ ↓ 0,

(13)

by the definition of K (λ) (see (9)). Next, note that the probability of B2 does not depend on n either, and that the domination of η by σ gives: P(B2c ) ≤ P{∂ B(k(λ)) ↔ ∂ B(K (λ)) in σ (tc )} → 0, as λ ↓ 0,

(14)

by a well-known result from ordinary percolation and the fact that K (λ)/k(λ) → ∞ as λ ↓ 0. Finally, we handle the event in the first term on the right hand side of (12). Since we will take limits as λ ↓ 0, we may restrict to λ’s for which k(λ) > m. Then we have the following relation between events: {at least 2 fires in B(m) in (tc , tc + ε)} ∩ B˜ 1 ∩ B2 = {τ ∈ (tc , tc + ε) and at least 1 fire in B(m) in (τ, tc + ε)} ∩ B˜ 1 ∩ B2 ⊂ {∂ B(m) ↔ ∂ B(k(λ)) in ηn (s) for some s ∈ (τ, tc + ε)} ∩ B1 ∩ B2 ⊂ {∂ B(m) ↔ ∂ B(k(λ)) in ξ(tc + ε)}, (15) where the second inclusion follows from Lemma 3.2 (and the monotonicity of ξ(t) for t > tc ), and the first inclusion holds because, by the event B˜ 1 , fires in B(m) before time tc + ε, can only arrive from outside B(K (λ)). To handle the probability of the


273

last event in (15), first observe that, for each i, the random variables ξv (t), t ≥ 0, v ∈ A(i) are completely determined by Poisson clocks assigned to the sites inside the annulus A(i). We use the notation I (λ) for the set of all positive even integers j with m < 3 j < 5 · 3 j ≤ k(λ). Since the annuli A(i), i ∈ I are disjoint, we get from Lemma 3.1 that P{∂ B(m) ↔ ∂ B(k(λ)) in ξ(tc + ε)} ≤ γ |I (λ)| .

(16)

Combining (15) and (16), and using that k(λ), and hence also |I (λ)| goes to ∞ as λ ↓ 0, we get lim lim sup P({at least 2 fires in B(m) before time (tc + ε)} ∩ B˜ 1 ∩ B2 ) = 0. (17) λ↓0 n→∞

Combining (12), (13), (14) and (17) completes the proof of Proposition 2.3

3.3. Proof of Theorem 2.2. Proof. Suppose δˆc > 0 and that for all t > tc there exists an m = m(t) such that lim inf lim inf Pλ ( fire in B(m) before time t) > 1/2. λ↓0

n→∞

(18)

We will show that this leads to a contradiction. Choose t as in Proposition 2.3. Now take u ∈ (tc , t). By (18) there exist m 0 and α(u) > 0 such that lim inf lim inf Pλ ( fire in B(m 0 ) before time u) > 1/2 + α(u). λ↓0

n→∞

(19)

By (1) (and the continuity of θ ) we can choose an s ∈ (tc , u) with lim inf lim inf Pλ ( fire in B(m 0 ) before time s) ≤ α(u)/2. λ↓0

n→∞

(20)

By (18) there exists an m 1 > m 0 such that lim inf lim inf Pλ ( fire in B(m 1 ) before time s) > 1/2. λ↓0

n→∞

(21)

Clearly, P( fire in B(m 0 ) before time u) ≤ P( fire in B(m 1 ) before time s and fire in B(m 0 ) between times s and u) +P( fire in B(m 0 ) before time s) +P(no fire in B(m 1 ) before time s). (22) Now for each term in (22) we take lim inf λ↓0 lim inf n→∞ . Then, by Proposition 2.3 the first term on the r.h.s. will vanish. Using this, and applying (20) and (21) to the second and the third term respectively, yields lim inf lim inf P( fire in B(m 0 ) before time u) ≤ 1/2 + α(u)/2, λ↓0

n→∞

which contradicts (19). This completes the proof of Theorem 2.2.

274


4. Discussion and Modified Models In the model above it was the square lattice which played the role of space. Completely analogous results can be proved, in the same way, for the triangular or the honeycomb lattice. In the following subsections we discuss some less obvious modifications of the model (different ignition mechanism; binary tree instead of square lattice). 4.1. Ignition of sufficiently large clusters. Again we work on the square lattice. In this model the growth mechanism is the same as before (that is, vacant sites become occupied at rate 1), but the ignition mechanism is different: Instead of the ignition rate λ we have an (integer) parameter L. The ignition rule now is that whenever a cluster of size ≥ L occurs, it is instantaneously ignited and burnt down (that is, each of its sites becomes vacant). A very pleasant feature of this model is that, since the interactions now have finite range, it can be defined on the infinite lattice using a standard graphical construction. This frees us from the necessity to first work on B(n) and later take limits as n → ∞, and thus from the annoying double limits we had in our main results. As before, we start at time 0 with all sites vacant. Let ηv[L] (t) denote the value (0 or 1) of site v at time t. The analog of Open Problem 1.1 is: Open Problem 4.1. Is, for all t > tc , lim sup P(η[L] has a fire in O before time t) > 0 ?

(23)

L→∞

Similarly, there is a straightforward analog of Open Problem 1.2. Although this modified model is seemingly simpler than the original one, we think the problems are, essentially, as hard as before. We have, with δˆc as before (see (5)), analogs of Theorem 2.2 and Proposition 2.3. Theorem 4.2. If δˆc > 0, there exists a t > tc such that for all m, lim inf P(η[L] has a fire in B(m) before time t) ≤ 1/2. L→∞

(24)

Proposition 4.3. If δˆc > 0, there exists t > tc such that for all m, lim P(η[L] has at least 2 fires in B(m) before time t) = 0.

L→∞

(25)

Theorem 4.2 follows from Proposition 4.3 in the same way as Theorem 2.2 from Proposition 2.3. The proof of Proposition 4.3 is very similar to that of Proposition 2.3 and we only indicate the main modifications: Instead of (9) we define K L := L 1/3 , k L := L 1/4 . Next, the events B1 , B2 are replaced by the single event B3 := {σ (tc ) has a vacant *-circuit in A(k L , K L )}, and Lemma 3.2 is replaced by the following lemma, whose proof is a straightforward modification of that of the former. (Of course we take m as before, and τ = τ (L , m) is now defined as the first time that η[L] has a fire in B(m).)


275

Lemma 4.4. On B3 we have, for all t > τ and all v ∈ B(k L ) \ B(m) that ηv[L] (t) ≤ ξv (t).

(26)

The proof of Proposition 4.3 now proceeds as before.

4.2. The binary tree. In this subsection we consider the same dynamics as for the process η in Sects. 1–3, but now we take the directed binary tree instead of the square lattice. By the infinite binary directed tree, denoted by T , we mean the tree where one vertex (called the root) has two edges, each other vertex has three edges, and where all edges are oriented in the direction of the root. The root will be denoted by O. By the children of a site v we mean the two sites from which there is an edge to v. (And we say that v is the parent of these sites.) By the first generation of v we mean the set of children of v, by the second generation the children of the children of v, etc. The subgraph of T containing O and its first n generations will be denoted by T (n). Let us now describe the model in detail. We work on T (n). Initially all sites are vacant. As in the original (Sect. 1) model vacant sites become occupied at rate 1 and occupied sites are ignited at rate λ. When a site v is ignited, instantaneously each site on the occupied path from v in the direction of the root is made vacant. The forest-fire interpretation is not very natural here. More natural is the interpretation in terms of a nervous system: Replace the word ‘site’ by ‘node’, ‘occupied’ by ‘alert’, vacant by ‘recovering’, ‘ignition’ by ‘arrival of a signal from outside the system’. Then the above description says that whenever an alert node v receives a signal (either from a child, or from outside the system), it immediately transmits it to its parent (except when v = O, in which case it ‘handles’ the signal itself), after which it needs an exponentially distributed recovering time to become alert again. As before we use 1 to represent an occupied (‘alert’) and a 0 to represent a vacant (‘recovering’) vertex. Let ζv (t) ∈ {0, 1} denote the state of vertex v at time t. If we want to stress dependence on n we write ζvn (t). As in Sect. 1, the processes ζ n (·) can be completely described in terms of independent Poisson growth and ignition clocks, assigned to the sites of T . Recall that site percolation on the binary tree has critical probability 1/2, and percolation probability function θ ( p) = (2 p − 1)/ p, for p ≥ 1/2. Combining this with the same arguments that led to (1) shows that, if we first let n go to ∞ and then λ to 0, the probability that the root burns before time log 2 goes to 0, and, moreover, that for t > log 2, lim sup lim sup Pλ (O burns before time t) ≤ λ↓0

n→∞

1 − 2e−t . 1 − e−t

(27)

A nice feature of the binary tree is that we can (quite simply in fact) also prove a lower bound (compare with Open Problem 1.1 for the square lattice): Lemma 4.5. For all t > log 2, lim inf lim sup Pλ (ζ n has a fire in O before time t) ≥ λ↓0

n→∞

Note that this lower bound is half the upper bound (27).

1 1 − 2e−t . 2 1 − e−t

(28)

276


Proof. Define the functions f nλ (t) := Pλ (ζ n has a fire in O before time t), t > 0, and gnλ (s, t) := f nλ (t) − f nλ (s), 0 < s < t, i.e. the probability that the first time that O burns is between s and t. Fix a t > log 2 and take t˜ ∈ (log 2, t). Suppose that lim inf lim sup f nλ (t) < λ↓0

n→∞

1 1 − 2e−t˜ . 2 1 − e−t˜

(29)

We will show that this leads to a contradiction. By (29) there exists an α > 0 and a sequence (λi , i = 1, 2, · · · ), which is decreasing, converges to 0 and has, for all i, lim sup f nλi (t) < n→∞

1 1 − 2e−t˜ − α. 2 1 − e−t˜

(30)

Fix j large enough such that e−λ j t˜(1 + 2α(1 − e−t˜)) > 1.

(31)

The reason for this choice will become clear later. Observe that, if v and w are the children of O, the processes ζvn+1 (·) and ζwn+1 (·) are independent copies of ζ On (·) (and are also independent of the Poisson clocks at O). Also observe that, to ensure that the first fire at the root occurs between times t˜ and t, it is sufficient that the growth clock of O rings before time t˜, no ignition occurs at the root before time t˜, at least one of its children burns between times t˜ and t and none of its children burns before time t˜. Hence, by these observations, λ gn+1 (t˜, t) ≥ (1 − e−t˜) e−λt˜ gnλ (t˜, t)2 + 2gnλ (t˜, t) (1 − f nλ (t˜) . (32) Now we take λ equal to λ j in (32), and apply (30) (noting that f nλ (t) ≥ f nλ (t˜)). λ This gives that (with the abbreviation gk for gk j (t˜, t), k = 1, 2, · · · ) for all sufficiently large n, λ

gn+1 ≥ (1 − e−t˜) e−λ j t˜ 2gn (1 − f n j (t˜)) ≥ gn × e−λ j t˜(1 + 2α(1 − e−t˜)) .

(33)

However, the factor behind gn in the r.h.s. of (33) does not depend on n and is, by (31), strictly larger than 1, so that the sequence of gn ’s ‘explodes’: a contradiction. Hence lim inf lim sup f nλ (t) ≥ λ↓0

n→∞

1 1 − 2e−t˜ . 2 1 − e−t˜

(34)

This holds for each t˜ ∈ (tc , t). Letting t˜ ↑ t in (34) completes the proof of Lemma 4.5.


277

By a similar ‘independent copies’ observation as used a few lines above (32) (now for all sites in the m th generation of the root), Lemma 4.5 immediately gives the following corollary (compare with Theorem 2.2 and Proposition 2.3): Corollary 4.6. For all t > log 2, all ε > 0 and all k, there exists m such that lim inf lim sup Pλ (ζ n has at least k fires in T (m) before time t) > 1 − ε. λ↓0

(35)

n→∞

Acknowledgements. We thank Antal Járai, Ronald Meester and Vladas Sidoravicius for stimulating discussions.

References 1. van den Berg, J., Járai, A.A.: On the asymptotic density in a one-dimensional self-organized critical forest-fire model. Commun. Math. Phys. 253, 633–644 (2004) 2. van den Berg, J., Brouwer, R.: Self-destructive percolation. Random Structures and Algorithms 24, Issue 4, 480–501 (2004) 3. Drossel, B., Schwabl, F.: Self-organized critical forest-fire model. Phys. Rev. Lett. 69, 1629–1632 (1992) 4. Dürre, M.: Existence of multi-dimensional infinite volume self-organized critical forest-fire models. Preprint (2005) 5. Grassberger, P.: Critical behaviour of the Drossel-Schwabl forest fire model. New J. Phys. 4, 17.1–17.15 (2002) 6. Grimmett, G.R.: Percolation, Berlin-Heidelberg-New York: Springer-Verlag (1999) 7. Jensen, H.J.: Self-Organized Criticality, Cambridge Lecture Notes in Physics, Cambridge: Cambridge Univ. Press (1998) 8. Malamud, B.D., Morein, G., Turcotte, D.L.: Forest Fires: An example of self-organized critical behaviour, Science 281, 1840–1841 (1998) 9. Schenk, K., Drossel, B., Schwabl, F.: Self-organised critical forest-fire model on large scales. Phys. Revi. E 65, 026135-1-8 (2002) Communicated by H. Spohn


Communications in


Charges and Fluxes in Maxwell Theory on Compact Manifolds with Boundary Marcos Alvarez1 , David I Olive2 1 Centre for Mathematical Science, City University, London Northampton Square, London EC1V 0HB, UK.

E-mail: [email protected]

2 Physics Department, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK.

E-mail: [email protected] Received: 27 March 2003 / Accepted: 17 February 2006 Published online: 5 August 2006 – © Springer-Verlag 2006

Abstract: We investigate the charges and fluxes that can occur in higher-order Abelian gauge theories defined on compact space-time manifolds with boundary. The boundary is necessary to supply a destination to the electric lines of force emanating from brane sources, thus allowing non-zero net electric charges, but it also introduces new types of electric and magnetic flux. The resulting structure of currents, charges, and fluxes is studied and expressed in the language of relative homology and de Rham cohomology and the corresponding abelian groups. These can be organised in terms of a pair of exact sequences related by the Poincaré-Lefschetz isomorphism and by a weaker flip symmetry exchanging the ends of the sequences. It is shown how all this structure is brought into play by the imposition of the appropriately generalised Maxwell’s equations. The requirement that these equations be integrable restricts the world-volume of a permitted brane (assumed closed) to be homologous to a cycle on the boundary of space-time. All electric charges and magnetic fluxes are quantised and satisfy the Dirac quantisation condition. But through some boundary cycles there may be unquantised electric fluxes associated with quantised magnetic fluxes and so dyonic in nature. 1. Introduction In the search for a unified theory of particle interactions encompassing both the standard model and Einstein’s theory of gravity the most promising candidate seems to be the superstring and M-theories which require space-time to have dimensions 10 and 11 respectively for internal consistency. A common feature of these is the presence of states known as “ p-branes”, objects which, classically at least, can be pictured as extended objects resembling p-dimensional surfaces (or volumes) in space. As time evolves these sweep out surfaces (or volumes) of one dimension higher, p + 1. When p = 0, the object is simply a point particle tracing out a world-line, w, in space-time. It has a geometrically natural interaction with Maxwell’s electromagnetic field specified by the addition of a term in the action taking the schematic form

280

M. Alvarez, D. I. Olive

“q

w

A”.

(1.1)

The same can be done for any positive value of p less than m, the dimension of space-time, with the proviso that the gauge potential A now has degree ( p + 1), matching the fact that the world-volume, w, has ( p + 1)-dimensions [O, N, T1]. When p equals one, so that the brane is a string, A is the familiar Kalb-Ramond [KR] gauge potential (see also [CS]). Naively Stokes’ theorem implies that expression (1.1) is unchanged when A is altered according to A → A + dχ ,

(1.2)

where χ is of degree p and arbitrary, if it is assumed that w is closed, i.e. a ( p + 1)-cycle. This generalised gauge invariance suggests that an important physical role would be played by the following quantity which is invariant with respect to (1.2): F = d A.

(1.3)

This is the ( p + 2)-form field strength, reducing to the familiar one of Maxwell when p = 0. The most natural equations of motion for F take the form: d F = 0,

(1.4)

d ∗ (h F) = ∗ j,

(1.5)

in exterior calculus notation, although there are more elaborate possibilities. In order to include a common feature of supergravity/superstring theories we have admitted the presence of a positive scalar function of scalar fields, h(φ), in (1.5), that equals unity in vacuo. Apart from this feature, these are Maxwell’s equations generalised in the form envisaged by Hodge, and ∗ denotes his duality operation, constructed by means of a metric on space-time, here assumed to be a fixed background [H, F]. We shall henceforth refer to them as Maxwell’s equations. The inhomogeneous Maxwell equation (1.5) will play an important role in what follows irrespective of the detailed form of the quantity h as long as it reduces to unity in vacuo and we shall not have to consider equations of motion for the scalar fields. The quantity j is the “electric current” and is a ( p + 1)-form, so possessing the same degree as A. By virtue of (1.5) and the nilpotency of the exterior derivative, d, it has to be “conserved” so that d ∗ j = 0,

(1.6)

and we shall always suppose this. Such field theories are indeed part of modern superstring/M-theory and it is therefore important to understand their properties by answering the questions listed below, particularly when the background space-time is taken to be topologically complicated. But these equations are only a part of the larger theory and not the whole. In this subtheory, no account need be taken of supersymmetry and the values of p and m, the dimension of space-time M, can treated as arbitrary. Many special features of these subtheories have become familiar [N, T1, T2, HT, DGHT], but our aim is to uncover yet more general structure as will be seen.

Charges and Fluxes in Maxwell Theory on Compact Manifolds with Boundary

281

When p vanishes and m equals 4 there are three notions familiar since the times of Faraday, electric charge, electric flux and magnetic flux. (Magnetic charge is excluded by (1.4) since magnetic current is). These three quantities are all conserved, that is unchanged by various sorts of evolution, including that in time. It will be seen to be important to distinguish the notions before examining the possibility of any relations between them. Allowing p and m to be arbitrary, the physical questions considered in this paper concern: (1) the classification and enumeration of the independent charges and fluxes, (2) the understanding of how the Maxwell equations (1.4) and (1.5) relate the notions of electric flux and charge, (3) the determination of the possible numerical values of these charges and fluxes, (4) the understanding of how quantum theory can relate the values of electric and magnetic conserved quantities (yielding the generalised Dirac quantisation condition). The answers turn out to be more subtle and interesting than we had expected and it is this that motivates this presentation. It is found important to resist the common temptation to simplify by taking the space-time manifold, M, to be closed as this results in oversimplification. When account is taken of the generalised Maxwell equations, (1.4) and (1.5), all electric charges and fluxes then vanish, leaving magnetic fluxes as the only available conserved quantities, as Henneaux and Teitelboim [HT] emphasised. In particular this applies to the situation with m = 4 considered by Misner and Wheeler [MW1] who were amongst the earliest to advocate the application of homology theory to unified field theories. Thus it is essential to allow space-time, M, to possess a boundary, B, a manifold of dimension one less, interpreted as corresponding to the “points at spatial infinity” through which “electric field lines” may escape, thereby furnishing a potentially nontrivial flux. When this is done, the answers to the physical questions above are provided by a set of results in pure mathematics whose physical relevance is, we believe, hitherto unappreciated by physicists. Once we have established the appropriate definitions we find the connections, made more precise in the text: electric charges ⇔ relative homology of space-time, electric fluxes ⇔ absolute homology of the boundary of space-time, magnetic fluxes ⇔ absolute homology of space time. All these charges and fluxes are expressed as integrals over some sort of cycle in space-time and homology deals with the classification of these cycles in the way that is appropriate to the physics. There are precisely three types of homology in the situation just described and all three play a physical role according to the connections just listed. Moreover the relationships between the different sorts of conserved quantity correspond to relationships between these different sorts of homology. All of the aforementioned types of homology class form elements of an abelian group, the appropriate homology group, H∗ , say. Taking into account all values of p that are possible in the given fixed background space-time, M, these abelian groups can be arranged in a certain order such that there is a natural homomorphism acting between successive members. This provides a sequence with the property of being exact, that is, at each stage, the homology group, H∗ , possesses a subgroup, K ∗ , say, that is at the same time the kernel of the succeeding homomorphism and the image of the preceding one. This is the exact sequence of relative homology (of space-time). A more refined classification of the physical notions of charge and flux will depend on the distinction

282


between the subgroup K ∗ and the coset group H∗ /K ∗ within each homology group H∗ . This structure is explained in more detail in the text as it becomes relevant to the development of the physical arguments and particularly in Sects. 7 and 10, as well as the Appendix. Relevant mathematical background together with more detail can be found in [S1] and [M]. Each homology group, H∗ , is abelian, and usually of infinite order. For reasons explained they are essentially discrete lattices of finite dimension, b∗ which is known as the Betti number. The number of linearly independent charges and fluxes will be expressible in terms of Betti numbers in a surprisingly complicated way that we shall determine. These results will answer the first two of the physical questions listed above. An important subtlety is that although the definition of the conserved electric charge as an integral over the conserved current, j works irrespective of whether or not the generalised Maxwell equations (1.4) and (1.5) are assumed to hold, the counting of the charges does depend on this choice, being more complicated when they do hold, as they should when account is taken of physical relevance. For example, when spacetime is closed, all possible non-trivial electric charges are forced to vanish by Eqs. (1.4) and (1.5). The point is that there exist conserved currents on space-time for which it is impossible to integrate (1.5) to obtain a field strength, F. Consequently these currents will be forbidden on the physical grounds that the field strengths must exist. It is the aforementioned exact sequence of relative homology that clarifies the occurrence of this phenomenon as explained in Sect. 4 and amplified later. Answering the third of the physical questions listed above requires an explicit form of the conserved current, j (w), due to a p-brane with world-volume w as implied by (1.1) and (1.5) together. This is provided by a singular differential form involving Dirac δ-functions whose support is w. Then the electric charge associated with integrating over a relative cycle S is q times the intersection number of S and the absolute cycle, w. The coefficient q is defined by (1.1) and the intersection number is well defined as w and S have dimensions summing to m, that of space-time. As this intersection number is unchanged by homologies of both w and S, it is defined on their homology classes. Since the groups formed by these classes are essentially lattices whose dimension is the relevant Betti number, it follows that the intersection data is encoded in the intersection matrix, I , formed of the intersection numbers between elements of bases of the two lattices. This matrix, I , has integer entries, is square and unimodular (that is, has determinant equal to ±1), the latter two properties being consequences of “Poincaré-Lefschetz duality”, another feature of the exact sequence of relative homology. So far this analysis does not use the “Maxwell equations”, (1.4) and (1.5), and hence applies whether or not they are chosen to hold. If not, the electric charges take values equal to an integer times q. Conversely the unimodularity of the intersection matrix means that there exist brane configurations realising all possible values of these sets of values. If Maxwell’s equations are chosen to hold, as they should, the situation is more complicated as many potential electric charges are forced to vanish, apparently contradicting the unimodular property of the intersection matrix. The resolution of this paradox depends on the recognition that some brane configurations are forbidden as they yield conserved electric currents for which Maxwell’s equations (1.4) and (1.5) cannot be integrated to yield a field strength. This is explained in more detail in Sect. 8 and requires the intersection matrix to have a more detailed structure than so far apparent. This is revealed by writing it in block form according to the kernel subgroup, K ∗ , of each H∗ , and the coset H∗ /K ∗ . One block has to vanish identically and this is


283

verified explicitly in Sect. 10 and the Appendix. This leaves square matrices on the block diagonal each of which have to be unimodular. The upshot is that the only brane configurations that are allowed by the integrability requirement are those that are homologous to cycles in the boundary, B, of space-time, M. The surviving electric charges again take values that are integral multiples of q. Conversely there are allowed brane configurations that realise all possible sets of these values. Another phenomenon quantified by the exact sequence of homology is the existence of electric fluxes which are not equal to electric charges and hence not quantised. Through the same cycles there may flow quantised magnetic fluxes of the field coupling to the brane dual to that coupling to the electric field so that the overall effect is suggestive of something dyonic. All results so far are “classical”, invoking no quantum theory. Taking the latter into account requires that the schematic term (1.1) in the action be unambiguous when suitably exponentiated. This constrains the values of the magnetic fluxes to satisfy a generalisation of Dirac’s celebrated quantisation when compared with any of the electric charges [D][WY][AO1]. The resulting picture is beautifully consistent yet unexpectly rich. Nevertheless our analysis made a number of implicit simplifications compared with the full superstring theory that are so far unavoidable. Some of these are listed in the conclusion, Sect. 11, and it is hoped that a subsequent elaboration of our present methods will lead to answers removing these assumptions. A technical Appendix extends the idea of a distribution valued form associated with a bulk cycle (such as the brane world-volume) to chains both in the bulk and on the boundary. These constructions are used to derive the weak form of Poincaré-Lefshetz duality used in establishing the vanishing of an off-diagonal block of the previously mentioned intersection matrix. Relative topology has been used previously to discuss certain aspects of branes in M-theory. A partial description of the role of relative cohomology in the classification of charges in generalised Maxwell theory was sketched in Sect. 2 of [MW2]. In [KS] relative cohomology is used to present a geometric description of certain brane intersections in M-theory. A similar analysis of D2-branes in Wess-Zumino-Witten theory can be found in [FS].

2. First Notions Taken as given is a fixed background space-time M, assumed oriented and compact, but possibly of complicated topology. It has dimension m and initially it is assumed to be closed. On it is defined a field strength F that is a ( p + 2)-form satisfing the generalised Maxwell equations (1.4) and (1.5). According to the first of these F is closed so that locally there is defined a ( p + 1)-form gauge potential A, (1.3), with a gauge ambiguity with respect to the gauge transformations (1.2), where χ is a p-form also defined locally. The quantity j is a p + 1-form denoting the electric current due to the matter degrees of freedom. For the time being it does not have to be assumed that it has the form that (1.1) would imply. Electric current conservation is the statement that ∗ j is a closed form on M, (1.6). This follows from the above Maxwell equation (1.5) as d 2 vanishes, but we shall assume its validity even when Maxwell’s equations are disregarded.

284


The first notion of an electric charge is associated with the current j without any reference to the field strength, F. Hence Maxwell’s equations can be temporarily disregarded. It is formulated by considering an oriented region S that is a (m − p − 1)-chain over which it is possible to integrate the matching form ∗ j: Q(S) = ∗ j. (2.1) S

Conventionally the region S would be thought of as “ space-like” but this is not essential. The virtue of the definition (2.1) is that it is insensitive to alterations of S by homologies that preserve its boundary, ∂ S. Thus, if S = S + ∂C, so ∂ S = ∂ S, Q(S ) = Q(S) as Q(∂C) = ∂C ∗ j = C d ∗ j = 0, using Stokes’ theorem and current conservation (1.6). This establishes a good sense in which the charge Q is conserved. The disadvantage of this definition is that the regions S for which the charge is defined lack any real homological significance unless it is assumed that S is closed, ∂ S = 0. Now the result means that each electric charge, Q(S), is preserved by homologies of S, that is, unchanged by the kinds of evolution associated with these homologies. Homologous surfaces form absolute homology classes which themselves form an abelian group under addition of surfaces, in this case the absolute homology group of M, denoted Hm− p−1 (M; ZZ ). Without any field strengths satisfying the Maxwell equations this would be the end of the story as there would be no fluxes to consider. Since field strengths are included, it is necessary to consider the effect of applying Maxwell’s equation (1.5): Q(S) = ∗ j = d ∗ (h F) = ∗(h F) = 0 S

S

∂S

as ∂ S vanishes. Thus all electric charges vanish when Maxwell’s equations hold on a closed space-time, M. In physical terms, the problem is that the Maxwell equation (1.5) attaches electric lines of force to the electric charge distribution and these lines have nowhere to go. Mathematically the point is that when the conserved electric current, j, is such that any Q(S) fails to vanish, it is impossible to integrate (1.5) to obtain the field strength F on M. This is unaceptable on physical grounds. An obvious remedy is to provide a destination for the lines of force by allowing space-time, M, to be non-compact, and this will be considered next. But it will remain necessary to check the integrability of Maxwell’s equations in the sense just described. 3. Electric Charges and Relative Homology Instead of allowing space-time, M, to be non-compact, as just suggested, we shall do something slightly different and keep it compact but allow it to have a non-trivial boundary, B = ∂M, of one dimension less. This can be thought of as comprising those points at spatial infinity through which electric lines of force may escape. On the other hand, electric current, j, is not allowed to escape, that is its Hodge dual, ∗ j, is assumed to be localised and this is expressed by the boundary condition: ∗ j B = 0. (3.1)


285

More precisely this means that the restriction of the differential form ∗ j to B vanishes. Thus the normal components of j vanish on B. In addition, it is assumed that the scalar function, h, occurring in (1.5), takes its vacuum value on B: h B = 1. (3.2) Of course Maxwell’s equations, (1.4) and (1.5) and also current conservation, (1.6) still hold on M, or as we shall say, in the bulk. The same expression (2.1) for the electric charge holds good except that now, instead of assuming ∂ S vanishes, we assume that it lies in B, and so S has become what is called a relative cycle. Suppose that S is altered by a relative homology: S → S = S + ∂C + β,

C ∈ M,

β ∈ B.

Then Q(∂C) = ∂C ∗ j = C d ∗ j = 0, by Stokes’ theorem and (1.6), while Q(β) = β ∗ j = 0 by (3.1). So Q(S) = Q(S )

if S ∼ S

(3.3)

in relative homology M mod B. In particular Q(S) vanishes if S ∼ 0. Thus the electric charge is well defined as an integral over relative homology classes, denoted [S] and forming an abelian group, Hm− p−1 (M, B; ZZ ). This is one sense in which the electric charges are conserved. The abelian group structure arises because two like relative cycles can be added to form a third. According to (2.1) this addition law is respected by the electric charges: Q([S]) + Q([S ]) = Q([S + S ]) = Q([S] + [S ]),

(3.4)

and this furnishes another sense in which they are conserved. Some elements of this homology group have finite order and are called torsion elements. Thus if S is not trivial, that is not relatively homologous to 0, yet has the property that there exists an integer n such that n[S] = [nS] is trivial, then, by the above, Q([S]) = Q([nS])/n = 0. Altogether such elements form a finite abelian subgroup T , (the torsion group), which can be divided out of Hm− p−1 (M, B; ZZ ) to form a free group Fm− p−1 (M, B; ZZ ) = Hm− p−1 (M, B; ZZ )/T

(3.5)

which can be regarded as a lattice of finite dimension bm− p−1 (M, B). This dimension is the corresponding Betti number. Because there are no contributions from torsion elements, electric charges are only defined on Fm− p−1 (M, B; ZZ ). Hence the integer bm− p−1 (M, B) counts what appears to be the number of linearly independent electric charges that can be defined on the space-time M. This conclusion is an overestimate for reasons to be explained in the next section. From now on, the conventions of this section will be adopted, and absolute chains will be denoted by lower case Roman letters (a,b,c … s,t,u,v,w..), relative chains by upper case Roman letters (A,B,C … S,T,U,V,W..) and chains in the boundary by Greek letters (α, β, γ . . . φ, χ , ψ..). The letters later in the alphabet will denote the corresponding cycles.

286


4. Electric Fluxes and Electric Charges The above derivation of the topological classification of electric charges by relative homology Fm− p−1 (M, B; ZZ ), used only the properties (1.6) and (3.1) of the current j, and not the Maxwell equations d F = 0 and (1.5). Current conservation (1.6) can be regarded as a necessary local condition for the integrability of the field strength F, given the current, j, but it is not sufficient, as already has been seen when space-time, M, has no boundary, nor will it be so when it does have a boundary. Assuming the Maxwell equation (1.5) does hold, the electric charge Q(S) can be rewritten as an electric flux: Q(S) = ∗ j = d ∗ (h F) = ∗(h F) = ∗F, (4.1) S

S

∂S

∂S

by Stokes’ theorem and (3.2). Of course ∂ S is in the space-time boundary, B, and is a cycle though not necessarily a boundary of a chain there, even though it is in the bulk, M. But it is possible to provide a more general definition of electric flux than this by considering any cycle in B, not just one that is a boundary of a relative cycle: E (φ) = ∗F, φ ∈ B, ∂φ = 0. (4.2) φ

This extended definition works on all the absolute homology classes of the boundary, Hm− p−2 (B; ZZ ), or, more precisely, the free parts, Fm− p−2 (B; ZZ ), defined as before. To check, consider the absolute homology in the boundary, φ → φ + ∂γ , γ ∈ B. Then E (∂γ ) = ∂γ ∗F = γ d ∗ (h F) = γ ∗ j = 0, using Stokes’ theorem and Eqs. (1.5) and (3.1). So indeed E (φ) = E (φ ) if φ ∼ φ in absolute homology in B, in distinction to the electric charges that appeared to correspond to relative homology. So, since their classifications differ, electric charges and electric fluxes must be distinguished. This distinction manifests itself in two different physical ways. First, not all electric fluxes are expressible as electric charges because not all cycles on the boundary, B, are boundaries of chains on M. The electric fluxes that are equal to charges are associated with cycles on the boundary, B, that are also boundaries of chains on M, as in (4.1). These classes of cycles form a subgroup of the absolute homology group of the boundary, B, Hm− p−2 (B; ZZ ), that we shall denote as follows: K m− p−2 (B; ZZ ) = {classes of boundary cycle φ satisfying φ = ∂ S for some bulk chain S}.

(4.3)

Secondly there are apparently non-trivial electric charges, Q(S), which must vanish if they are expressible as fluxes. This happens precisely when ∂ S is a boundary in B, as well as in M, according to Stoke’s theorem applied to (4.1). The classes of these cycles form a subgroup of the relative homology group that we shall denote as follows: K m− p−1 (M, B; ZZ ) = {classes of relative cycle, R, satisfying ∂ R = ∂α, α ∈ B}. (4.4) It follows that it is the vanishing of the charges associated with these cycles that is the extra integrability condition on Maxwell’s equation (1.5) in order to obtain a field strength, F, given a conserved current, j.


287

To recap, electric charges defined on K m− p−1 (M, B; ZZ ) all vanish, leaving nontrivial charges associated with each coset element of this subgroup. Furthermore only those electric fluxes defined on K m− p−2 (B; ZZ ) are expressible as electric charges. What is happening mathematically is that the boundary operation ∂ mapping relative cycles to boundary cycles induces a map ∂∗ :

Hm− p−1 (M, B; ZZ ) −→ Hm− p−2 (B; ZZ ).

(4.5)

In fact this is a group homomorphism with kernel K m− p−1 (M, B; ZZ ), (4.4), and image K m− p−2 (B; ZZ ), (4.3). So, by Lagrange’s theorem, Hm− p−1 (M, B; ZZ )/K m− p−1 (M, B; ZZ ) ≡ K m− p−2 (B; ZZ ), and it is this group (or more precisely the free part) that classifies the non-trivial electric charges. Applying this to the free parts that are lattices classifying the corresponding charges and fluxes, the number of linearly independent electric charges is given by bm− p−1 (M, B) − sm− p−1 (M, B) = sm− p−2 (B),

(4.6)

explaining the overestimate mentioned previously. The integers s∗ (X ) are the dimensions of the lattices specifying the free part of K ∗ (X ; ZZ ). As there are bm− p−2 (B) linearly independent fluxes, sm− p−2 (B) of which are expressible as electric charges the difference, the number bm− p−2 (B) − sm− p−2 (B), specifies the number of linearly independent electric fluxes that cannot be equated to electric charges of the form (2.1). 5. Relation Between the Preliminary and Final Versions of Electric Charge For reasons that become clear later, it is worth asking a question that seems rather ridiculous from a physical point of view, namely how to relate the class of electric charge obtained by integrating ∗ j over a bulk cycle to the class obtained by integrating over a relative cycle. The first class, considered in our preliminary discussion still makes sense when space-time has a boundary since a bulk cycle can be considered as a special case of a relative cycle. The reason the question is apparently ridiculous from a physical point of view is that these preliminary charges do all vanish when account is taken of Maxwell’s equations as already seen. Consider an absolute bulk (m − p − 1)-cycle, r , and decompose it into the sum of a (m − p − 1)-chain in B and a remainder that contains no such chain: r = R + α. As ∂r = 0, ∂ R = −∂α ∈ B, so that R is a relative cycle. Furthermore, if r is trivial as a bulk cycle, so r = ∂a, then R = ∂a − α and so is trivial as a relative cycle. Hence the projection map j : r → R induces a map, j∗ , of absolute bulk homology classes to relative homology classes: j∗ :

Hm− p−1 (M; ZZ ) −→ Hm− p−1 (M, B; ZZ ).

(5.1)

This is actually a homomorphism. Its kernel consists of bulk cycles, r for which R is relatively trivial, so r = (∂C + β) + α = ∂C + γ , where γ ∈ B. Since r is closed, so is γ . Thus γ is a cycle in the boundary and the kernel can be denoted K m− p−1 (M, ZZ ) = {classes of bulk cycle homologous to cycles in B}.

(5.2)

288


On the other hand the image of j∗ consists of classes of relative cycle whose boundary is the boundary of a chain within B. This coincides with the subgroup K m− p−1 (M, B; ZZ ) already defined as being the kernel of ∂∗ in the previous section, (4.3). Putting together j∗ and ∂∗ as two successive homomorphisms: ∂∗

j∗

Hm− p−1 (M; ZZ ) −→ Hm− p−1 (M, B; ZZ ) −→ Hm− p−2 (B; ZZ ), we see that this sequence is exact at Hm− p−1 (M, B; ZZ ) as K m− p−1 (M, B; ZZ ) is both the the image of j∗ and the kernel of ∂∗ . This is a short segment of the exact sequence of relative homology alluded to in the introduction and more segments will be seen when magnetic fluxes are considered next. The complete exact sequence will be presented in later sections. A textbook presentation can be found in [M]. Of course, as we saw at the start, all electric charges vanish that are integrals over cycles in Hm− p−1 (M; ZZ ). This agrees with the fact already found above that they also vanish on K m− p−1 (M, B; ZZ ), which is the image of the former group under the action of j∗ . 6. Action Principle for p-Branes and Magnetic Flux Quantisation The standard (naive) expression for the term in the action describing the interaction of the field strength, F with the current, j, that is its source, according to (1.4) and (1.5), is A ∧ ∗ j. (6.1) M

Naively, this term is gauge invariant on its own with respect to the transformation (1.2) given that the electric current, j, is conserved, (1.6), and localised, (3.1). Ideally the current should be expressible in terms of quantum mechanical wave functions for the matter but it is only really understood how to do this when p = 0 so that the branes are point particles. By default, the only accepted way to proceed is to adopt the classical geometric picture described in the introduction. The evolution of the p-brane in space-time is specified by its world-volume, w, an absolute bulk ( p + 1)-cycle on M. Then the action term (6.1) takes the form (1.1) mentioned at the start. Because we already know the classical equations of motion in the Maxwell form (1.4) and (1.5), the detailed form of the action is only really relevant in the quantum theory. In that context, the expressions (1.1) and (6.1) are equally problematical (which explains the use of the words “schematic or naive”) as they involve the gauge potential, A, which is only defined locally, whilst the integration extends globally over all of spacetime, M. Consequently, in a topologically complicated space-time such as the one being imagined, there are problems in patching together this expression in overlapping neigh iq A/ w bourhoods of space-time. Fortunately it is the exponentiated action e that enters the Feynman action principle and this is more amenable. One needs to know how this phase alters when w is altered by a boundary. That is tantamount to requiring that the phase has a meaning when w is a boundary of a bulk chain. This can be done provided the background field strength F satisfies the Dirac quantisation conditions [D] for all magnetic fluxes through bulk ( p + 2)-cycles [N, T2]: 2π M (v) = F ∈ ZZ , ∂v = 0. (6.2) q v


289

As d F = 0 these fluxes are defined on the classes of the absolute homology of spacetime M, forming the group H p+2 (M, ZZ ), or more precisely the free part of this, F p+2 (M; ZZ ), a lattice of dimension b p+2 (M). A parenthetic remark concerning this quantisation condition (6.2) is that it is known not really to be correct when wave functions are considered, as is, so far, only possible when p vanishes and the brane is therefore a point particle. Then there is a possibility of fractional quantisation conditions when the wave function is of a spinor nature (involving half-integers instead of integers). The precise rule is easy to state when m = 4 [AO1]. By this stage of the argument it has become established that, as claimed in the introduction, there is a connection between the physical notions of electric charge, electric flux and magnetic flux of a p-brane and mathematical notions of relative homology, absolute boundary homology and absolute bulk homology and more precisely, with the free parts of the abelian homology groups, Hm− p−1 (M, B; ZZ ), Hm− p−2 (B; ZZ ) and H p+2 (M; ZZ ), respectively. But there is a more detailed structure connected to the subgroups K ∗ of H∗ , for short, that plays a role in the exact sequence of relative homology and moreover possesses a physical relevance. Let us illustrate this last point by investigating magnetic fluxes through B cycles with a view to comparing electric and magnetic fluxes. Later on we shall see how this comparison will indicate a generalised dyonic phenomenon that is possibly related to the Zwanziger-Schwinger quantisation condition [Z2, S2]. Magnetic fluxes can already be defined for cycles in the boundary, B, rather than in the bulk, M, but nothing appears to be gained by this as cycles in B are automatically cycles in M but may become boundaries of bulk chains when regarded as M-cycles and hence homologically trivial in the bulk. Associated with this idea is the inclusion map, i, which induces the homomorphism: i∗ :

H p+2 (B; ZZ ) −→ H p+2 (M; ZZ ),

(6.3)

with kernel consisting of the classes of cycle just mentioned that become boundaries. This is precisely the subgroup K p+2 (B; ZZ ) of the type met before, (4.3), (with p + 2 replaced by m − p − 2), as the image of the homomorphism, ∂∗ , (4.5), induced by the boundary operator and met before in the comparison of electric charges and fluxes. The image of this homomorphism is clearly given by classes of bulk cycle homologous to a cycle in the boundary and these precisely form the subgroup K p+2 (M; ZZ ), (5.2), already met as the kernel of the homomorphism j∗ , (5.1), (again with p + 2 replaced by m − p − 2). All magnetic fluxes on cycles of K p+2 (B; ZZ ) vanish, as F= F = d F = 0, φ

∂S

S

by (4.3), Stokes’ theorem and (1.4), corresponding to the fact that these cycles are trivial as bulk cycles. So the only non-trivial magnetic fluxes through boundary cycles correspond to the b p+2 (B) − s p+2 (B) cosets of K p+2 (B; ZZ ) in H p+2 (B; ZZ ). These observations will become more interesting when we are able to compare them with the corresponding properties of electric fluxes through boundary cycles later on. Thus we have two more examples of a coincidence between images and kernels of different homomorphisms. This phenomenon is part of the exact sequence of relative homology mentioned in the introduction, an important pattern that has been emerging gradually and will be elaborated now.

290


7. The Exact Sequence of Relative Homology of Space-Time Our study within a general setting of the physical concepts of electric charge, electric flux and magnetic flux has revealed how these are described as integrals over cycles in space-time that are respectively relative, boundary and bulk type and unchanged by the appropriate homologies. So they are certainly classified by the corresponding homology groups Hm− p−1 (M, B; ZZ ), Hm− p−2 (B; ZZ ) and H p+2 (M, ZZ ), when p-branes are considered. We have also met three different types of homomorphism between the three types of homology group, denoted i ∗ , j∗ and ∂∗ , and illustrated by (6.3), (5.1) and (4.5). Associated with all of these is an image and kernel which is always a very specific subgroup of the relevant homology group, illustrated by (4.4), (4.3) and (5.2). If p is allowed to run over all the values compatible with possible p-branes in the given background space-time M, the set of all possible homology groups can be arranged as an ordered sequence with homomorphisms of one or other of the above three types relating each successive pair: ∂∗

i∗

j∗

∂∗

i∗

. . .−→Hm− p−1 (B) −→ Hm− p−1 (M) −→ Hm− p−1 (M, B) −→ Hm− p−2 (B) −→ . . . . (7.1) This is the exact sequence of relative homology well known to pure mathematicians in the context of algebraic topology, and more careful and detailed treatments can be found in various textbooks. The notation has been compressed by omitting reference to the integers ZZ . Assuming space-time, M, is connected, this exact sequence of abelian groups starts and finishes with the trivial group, written as 1 in multiplicative notation: 1 → Hm (M, B) → Hm−1 (B) → Hm−1 (M) → Hm−1 (M, B) → . . . and . . . H1 (B) → H1 (M) → H1 (M, B) → H0 (B) → H0 (M) → 1. Thus, besides the two trivial terms terminating the exact sequence, there are 3m terms. From the sequence it is now possible to evaluate in terms of the Betti numbers the numbers sq (B), sq (M) and sq (M.B) that are the dimensions of the free parts of the kernels (4.3), (5.2) and (4.4) and entered the counts of the various charges and fluxes. The exact sequence (7.1) implies a similar but simpler exact sequence for the free parts of the homology groups (obtained by dividing out the torsion subgroup). Working over real coefficients rather than integers yields an exact sequence of vector spaces with dimensions given by the Betti numbers and linked by linear maps replacing the group homomorphisms. To understand what happens consider such a sequence in simplified notation: 1 → V0 → V1 → V2 → V3 → . . . VN → 1.

(7.2)

If K n ⊂ Vn is the kernel/image, then, by exactness K n ≡ Vn−1 /K n−1 (retaining multiplicative notation). So repeating K n = Vn−1 /Vn−2 /Vn−3 / . . . /V1 /V0 /1,

(7.3)


291

and taking dimensions, sn = dimK n = bn−1 − bn−2 . . . (−1)n+1 b0 = bn − bn+1 . . . (−1) N −n b N ,

(7.4)

using the fact that s N +1 , which equals the alternating sum of all the Betti numbers, vanishes. Applying these formulae to the exact sequence of relative homology, (7.1), yields sn (M) = bn (M) − bn (M, B) + bn−1 (B) − bn−1 (M) . . . , sn (M, B) = bn (M, B) − bn−1 (B) + bn−1 (M) − bn−1 (M, B) . . . , sn (B) = bn (B) − bn (M) + bn (M, B) − bn−1 (B) . . . , showing how the count of electric charges, (4.6), depends on the topology of space-time, M. It is familiar that in the understanding of electro-magnetic duality on closed spacetime manifolds, M, a property known as Poincaré duality is important. There is an analogous property for manifolds with boundary that will play an important role in the present context. This is known as Poincaré-Lefschetz duality and a short explanation follows. Corresponding to the integer homology groups already defined it is possible to define integer cohomology groups denoted H q (M; ZZ ) and so on. There is also an exact sequence of homomorphisms linking these in the sense of ascending superscript: . . . → H p (B) → H p+1 (M, B) → H p+1 (M) → H p+1 (B) → . . . .

(7.5)

The statement of Poincaré-Lefschetz duality is that the corresponding terms in the two exact sequences (7.1) and (7.5) are isomorphic as groups. So H p+1 (M, B; ZZ ) ≡ Hm− p−1 (M; ZZ ), H p+1 (M; ZZ ) ≡ Hm− p−1 (M, B; ZZ ),

(7.6)

and H p (B; ZZ ) ≡ Hm− p−1 (B; ZZ ).

(7.7)

The last isomorphism is simply Poincaré duality for the boundary, B, which is automatically a closed manifold of dimension m − 1. Notice how the superscripts and subscripts in an isomorphism are always complementary in the sense of summing to the dimension of the relevant manifold and how (7.6) relates relative topology to absolute topology in the bulk. There is yet another relation between homology and cohomology that results from the universal coefficient theorem by considering the coefficients to be real numbers rather than integers. The resultant groups are simply the vector spaces, with dimension equal to the Betti number, spanned by the lattices given by the free parts of the integer groups as previously mentioned. Then a homology group of given suffix and type is the dual of the cohomology group of corresponding superscript and type: H q (M; IR) = Hq (M; IR)∗ , H q (B; IR) = Hq (B; IR)∗ .

H q (M, B; IR) = Hq (M, B; IR)∗ , (7.8)

292


By means of these and the Poincaré-Lefschetz duality relations (7.6) and (7.7), the cohomology groups can be eliminated to yield the following relations between homology groups: Hq (M; IR) = Hm−q (M, B; IR)∗

and

Hq (B; IR) = Hm−q−1 (B; IR)∗ . (7.9)

Because the Betti numbers are the dimensions of these real vector spaces, particular consequences are the following equalities: bq (M) = bm−q (M, B)

and

bq (B) = bm−q−1 (B).

(7.10)

The corresponding duality relations for the dimensions on the image/kernels of the exact sequence, the numbers sq (M), sq (B) and sq (M, B), will be important and are easily obtained by recognising that in the simplified notation for the exact sequence, (7.2), Vn = VN∗ −n . So bn = b N −n and hence by (7.4), sn = s N +1−n . As a consequence dim Vn = bn = sn + sn+1 = dimK n + dim(V /K )n and b N −n = dim VN −n = s N +1−n + s N −n = dim(V /K ) N −n + dim K N −n . This means the dimensions of the two complementary subspaces of V , namely K and V /K interchange under duality, N ↔ N − n. In particular sm− p−1 (M, B) = b p+1 (M) − s p+1 (M) and s p+1 (M) = bm− p−1 (M, B) − sm− p−1 (M, B).

(7.11)

In fact, by (7.10) these two equations are the same as each other. 8. Electric Charges as Intersection Numbers With this information we are now well prepared to consider the physical question as to the possible numerical values of the generalised electric charges (2.1). Given a suitable expression for the conserved, localised electric current, j, the charges are evidently determined without recourse to Maxwell’s equations (1.4) and (1.5). Hence in this calculation these equations can be temporarily renounced, provided it is remembered that their reinstatement will reduce the number of independent electric charges, as explained in Sect. 3. We shall defer this reinstatement and the detailed understanding of the issues it raises until the following section. Just as in the discussion of magnetic fluxes and their quantisation in Sect. 6, we shall have to resort to the geometrical picture of a brane world-volume, as this will give us tractable form for the current. This is found by equating (1.1) and (6.1), the two versions of the term in the action responsible for the brane coupling to the gauge potential: q A= A ∧ ∗ j. (8.1) w

M


293

Now A is taken to be an arbitrary ( p + 1)-form on M, so it follows that ∗ j = qµ(w),

(8.2)

where µ(w) is a singular (m − p − 1)-form involving a product of the same number of Dirac δ-functions with support on the absolute ( p + 1)-cycle w and differentials in the variables transverse to it. It follows that its restriction to B vanishes, as it should (3.1). In the Appendix it will be shown to be closed as well. Inserting the p-brane current (8.2) into the electric charge (2.1) yields Q(w; S) = q µ(w). S

This is invariant under relative homologies of S according to the work of Sect. 3. Now consider a bulk homology of the world volume, w, w → w = w + ∂a. By linearity µ(w ) = µ(w) + µ(∂a). As discussed in the Appendix, µ(a) exists for a bulk chain, a (and now involves step functions as well as Dirac-delta functions) and, moreover, obeys dµ(a) = µ(∂a), up to a sign. Hence the change in the electric charge, Q(w; S), due to this homology is µ(a) = µ(a)B = 0. Q(w − w; S) = q µ(∂a) = q dµ(a) = q S

∂S

S

∂S

So Q(w; S) is defined on the homology classes Hm− p−1 (M, B; ZZ ) × H p+1 (M; ZZ ), or rather on the corresponding product of free parts. So it can be assumed that the relative cycle S intersects the absolute bulk cycle of complementary dimension, w, at discrete points. Then the integral for the electric charge is recognised as [HT] Q(w; S) = q I (w, S), where I (w, S) denotes the intersection number of the absolute bulk cycle w with the relative cycle S, being the algebraic sum of the number of these points, taking into account signs due to relative orientation. This intersection number possesses a number of mathematical properties that are important for the physical interpretation of this result that we shall now describe. Choose bases S j and wi in the lattices Fm− p−1 (M, B; ZZ ) and F p+1 (M; ZZ ) that are the free parts of the two relevant homology groups. Then all intersection numbers are specified by knowledge of the matrix I (wi , S j ) = Ii j

∈ ZZ .

(8.3)

This intersection matrix, I , has b p+1 (M) rows and bm− p−1 (M, B) columns and hence is square, by (7.10). Yet another consequence of Poincaré-Lefschetz duality is that this matrix I is unimodular: det I = ±1.

(8.4)

Putting these results together it follows that all electric charges are quantised: Q(S) ∈ q ZZ

(8.5)

as integral multiples of the coupling constant, q, that enters the action. Thus any electric charge paired with any magnetic flux satisfies the Dirac quantisation condition: Q(S) M (v) ∈ 2π ZZ .

(8.6)

294


In deriving this quantisation condition it was implicitly assumed that there is only one species of p-brane and that it had a definite coupling constant, q, as defined above. Classically it is possible to imagine several distinct species of p-brane, distinguished by different coupling constants, q1 , q2 , . . . , q N whose ratios may be irrational. Then the total electric charge contained in the relative cycle, S, would now be Q(S) =

N

qi I (wi , S),

(8.7)

i=1

where wi is the world-volume of the brane of species i. This electric charge is not quantised if the ratios of coupling constants are irrational. However, in order to make sense of the exponentiated quantum action (6.1) for each species of brane the magnetic flux quantisation (6.2) has to hold separately with q replaced in turn by each species of coupling constant q1 , q2 . . . q N . It is this that forces their ratios to be rational, as we now show. Quantum mechanical consistency requires any given flux M (v) to be quantised separately for each of the N coupling constants, M (v) =

2π 2π 2π m1 = m2 = · · · mN, q1 q2 qn

in which m i ∈ ZZ for all i = 1, . . . , N . Therefore qi /q j = m i /m j for all i and j, so that the ratios of the coupling constants must be rational, as claimed. It follows that Q(S) M (v) =

N

qi I (wi , S)

i=1

= 2π

N i=1

2π mj qj qi m j = 2π I (wi , S)m i , qj N

I (wi , S)

(8.8)

i=1

which is in 2π ZZ . That is, (8.6) must remain true in the presence of several distinct species of p-branes carrying different charges. If Maxwell’s equations are now taken into account, then it follows that certain of the electric fluxes are indeed quantised, namely those obtained by integrating over boundary cycles within K m− p−2 (B; ZZ ), as these fluxes are equal to electric charges. On the other hand, there is no reason to believe that the remaining electric fluxes are quantised, and we shall return to some comments on this later. All that can be said as a result of (8.5) is that, for the latter fluxes, the quantities exp( 2πi q E (v) ) are well defined on the cosets (H/K )m− p−2 (B; ZZ ). The physical consequence of I being unimodular is that a configuration of the braneworld-volume can be found that realises any assignment of charges satisfying (8.5). Finally let us comment on the connection between the physical arguments of this section and the mathematical arguments of the preceding one, outlining Poincaré-Lefschetz duality. By current conservation, (1.6), the dual current, ∗ j, is a closed (m − p − 1)form, that, in addition, has vanishing restriction on the boundary of space-time, (3.1). This means that it is a relative (m − p − 1)-cocycle in the sense of de Rham cohomology, m− p−1 and so defines a class of Hde Rham (M, B; IR). The same is therefore true of µ(w) by (8.2), which therefore provides a map from the homology class of the world-volume, m− p−1 w, H p+1 (M; ZZ ) to Hde Rham (M, B; IR). This is part of the Poincaré-Lefschetz isomorphism, (7.6). It is possible to develop this line of thought and this is done in the Appendix.


295

9. Maxwell’s Equations and the Intersection Matrix Two (correct) arguments have been developed in this paper that apparently lead to a contradiction. We shall now explain what this is, and how it is resolved by finding that the intersection matrix, I , (8.3), has further detailed properties, hitherto unexpected. In Sect. 4 it was shown that the effect of Maxwell’s equations is to force all electric charges, Q(S), to vanish when the relative cycle over which they are integrated, S, belongs to K m− p−1 (M, B; ZZ ). Yet according to the preceding section, irrespective of Maxwell’s equations, the electric charge due to a brane configuration with worldvolume, w was seen to be proportional to the intersection number of w with S. The apparent contradiction arises from the fact that the intersection matrix is non-singular, as a consequence of its being unimodular (8.4). To make this clearer, it is natural to partition the intersection matrix in a way that distinguishes each kernel K within each H from the cosets H/K . This is done by choosing the basis {S j } so that the first sm− p−1 (M, B) elements form a basis of K m− p−1 (M, B) while the remainder refer to the cosets (H/K )m− p−1 (M, B). The basis {wi } is chosen so that the last s p+1 (M) elements form a basis for K p+1 (M) while the remainder refer to the cosets. Corresponding to this, the intersection matrix, (8.3), is written in the block form

(H/K )(M)



K (M,B)

(H/K )(M,B)

A

Y

 

I (w, S) = K (M)

  .

X

(9.1)

B

That electric charges associated with K m− p−1 (M, B) all vanish seems to imply that the submatrices A and X vanish, apparently contradicting the fact that the overall matrix has determinant equal to ±1. But this is not a correct interpretation of what has been shown. The correct interpretation is that the absolute bulk homology classes of the brane world-volume, w, that yield non-zero charges associated with relative cycles of homology belonging to the subgroup K m− p−1 (M, B; ZZ ) are all forbidden because Maxwell’s equations cannot then be integrated to yield field strengths, given the corresponding currents (8.2). Thus the only permitted homology classes of world-volume are those whose intersection number with all elements of K m− p−1 (M, B; ZZ ) vanish. These classes should form a subgroup and it is natural to anticipate that this be provided by the kernel K p+1 (M; ZZ ). The condition for this is that the submatrix X in (9.1 vanish. This is perfectly consistent with the unimodularity of I, (8.4), since, by (7.11), the consequences of Poincaré-Lefschetz duality for the kernels, the block diagonal submatrices A and B are both square. Consequently ±1 = det I = det A det B, implying that the block diagonal submatrices A and B, possessing integer entries, are both unimodular too. Thus it is the submatrix B that gives the physical charges, Q(S), for S in the coset (H/K )m− p−1 (M, B; ZZ ) as it determines the intersection numbers between these relative classes and the permitted homology classes of brane world-volume. According to (5.2), (with m − p − 1 replaced by p + 1) these permitted world-volumes are those homologous to cycles in the boundary of space-time, B. It is remarkable that such a

296


selection rule on brane configurations can be derived without recourse to any equations of motion for the brane degrees of freedom. Since the submatrix, B, that determines the physical charges, is unimodular, the previous conclusion that there exist brane configurations realising any assignment of quantised charges, (8.5), holds good even when the selection rule is taken into account. What remains is to provide an independent check that the block submatrix X in (9.1) vanishes. This is a geometrical condition that should hold for any background space-time, M, with boundary B, and it can be rewritten as: I (K p+1 (M), K m− p−1 (M, B)) = 0.

(9.2)

This vanishing theorem will be demonstrated in the next section using some results developed in the Appendix. 10. De Rham Cohomology, Field Strengths and Currents The argument will be interesting as it brings into play further parallels between physical and mathematical concepts and sheds light on the more abstract ideas involving the two related exact sequences mentioned previously and to be elaborated below. We can no longer avoid describing de Rham cohomology which deals with the exterior derivative, d, of differential forms (such as the field strengths and currents we have been talking about). We have to explain the three types of cohomology group that arise, given a manifold with boundary; how they can be arranged in an exact sequence, and how that exact sequence is related to the one for homology groups already explained. A real q-form, ω, on M is an absolute bulk cocycle if it is coclosed, dω = 0. Two such cocycles are absolutely cohomologous in the bulk if H q (M; IR) :

ω ∼ ω

⇐⇒

ω = ω + dα.

(10.1)

As indicated, these form equivalence classes which constitute elements of the absolute bulk homology group H q (M) (in the sense of de Rham). The groups are abelian since composition is by addition. Actually these groups are real vector spaces since they are closed under multiplication by real numbers. Essentially the same concepts can be applied to q-forms, φ, on the boundary, B. φ is a boundary cocycle if it exists on B and is coclosed there. Two such cocycles are absolutely cohomologous on the boundary if H q (B; IR) :

φ ∼ φ

⇐⇒

φ = φ + dβ.

(10.2)

Again these are equivalence relations whose classes form the group indicated. The third and last concept is that of a relative cocycle, η, which is defined in the bulk, on M, is coclosed there, dη = 0 and has vanishing restriction to the boundary, ηB = 0. Two such cocycles are relatively cohomologous if they differ by a coexact form dα with the property that the restriction of α to the boundary is coexact there. H q (M, B; IR) : η ∼ η ⇐⇒ η = η + dα, α B = dβ. (10.3) Again these are equivalence relations whose classes form the group indicated. In each case the cohomology relation preserves the appropriate coclosure property. Physical


297

examples are provided by the field strength, F, which is an absolute bulk ( p + 2)-cocycle, ∗F B , which is an absolute boundary (m − p − 2)-cocycle and the dual current, ∗ j, which is a relative (m − p − 1)-cocycle. Thus there are three types of cohomology matching the three types of homology already explained. Furthermore they both exist for a range of values of the integer q specifying the dimension of the cycle or the degree of the form as appropriate. When taken over real numbers, homology and cohomology groups of matching type and integer q are related in a nice way, as dual vector spaces, see (7.8). To understand the first example of these relations, let ω be a q-cocycle and v a q-cycle, both in the absolute bulk sense, and consider ω ∈ IR. v

This integral enjoys a number of properties: 1) It is invariant under the appropriate homologies of v, v → v = v + ∂a, and cohomologies of ω, (10.1). 2) It is linear in v and ω separately and hence provides a real bilinear form. 3) It is nonsingular; that is there is no nontrivial class of either type such that the integral vanishes for all classes of the other type. The first two properties are easy to check but the third, nonsingularity, is quoted as a known theorem (of de Rham). Of course the magnetic flux (6.2) already defined is an example of such an integral. Precisely analogous constructions work for the other two types of homology/cohomology and yield the remaining duality relations (7.8). Physical examples of these integrals are provided by electric charge, (2.1), and electric flux, (4.2), involving relative and boundary homology/cohomology respectively. Space-time, M, is itself a relative m-cycle and hence it is appropriate to integrate relative m-cocycles over it. The wedge product η ∧ ω is such a cycle if η and ω are respectively relative and absolute bulk cocycles of complementary degree (summing to m). So it is natural to consider η ∧ ω ∈ IR. M

This integral is (1) invariant under the appropriate cohomologies of ω and η, (10.1) and (10.3), (2) bilinear in ω and η, (3) nonsingular. As a consequence there results the duality relation H q (M; IR) = H m−q (M, B; IR)∗ which, when combined with the previous duality relations (7.8), implies H q (M; IR) = Hm−q (M, B; IR) and H q (M, B; IR) = Hm−q (M; IR), a weak version of Poincaré-Lefschetz duality, (7.6) (weak because it is over the reals rather than the integers). A weak version (over the reals) of the similar relation for the boundary, (7.7), can likewise be checked.

298


These results are sufficient to show that there exists an exact sequence of de Rham cohomology groups but it is worth demonstrating this explicitly in order to find precise definitions of the common kernel/image subgroups of these groups. Relative cocycles are automatically absolute cocycles in the bulk too and this leads to the homomorphism j∗ :

H q (M, B) → H q (M).

The kernel of j ∗ is made up of the elements that are trivial in H q (M): K q (M, B) = {classes of H q (M, B) satisfying η = dα, dα B = 0},

(10.4)

(10.5)

while the image appears to consist of elements of H q (M) with ωB vanishing. Absolute cocycles in the bulk automatically yield cocycles in the boundary when restricted to it. So ω → ωB yields the homomorphism i∗ :

H q (M) → H q (B)

(10.6)

with kernel K q (M) = {classes of H q (M) with ωB coexact}.

(10.7)

This obviously includes the image of j ∗ and tallies after applying bulk cohomologies (10.1). The image of i ∗ will be specified below. Given a coclosed form, β0 , on the boundary, dβ0 = 0 on B, there is a way to find a closed form ηβ , of one degree higher on the bulk whose restriction to the boundary automatically vanishes so that it is relatively coclosed. Although the procedure is not unique, the degree of ambiguity lies in a single relative cohomology class and so the procedure leads to a homomorphism, known as the Bockstein homomomorphism: d∗ :

H q (B)

→

H q+1 (M, B).

(10.8)

Let β denote an extension of β0 from the boundary, that is, a formon M, not necessarily closed, satisfying β B = β0 . Then, if ηβ = dβ, dηβ = 0 and ηβ B = dβ B = dβ0 = 0 and so ηβ is a relative cocycle. Consider now β0 and β0 , forms which are cohomologous in H q (B), (10.2), so β0 = β0 = dα, (on B). If they have extensions β and β , respectively to the bulk ηβ − ηβ = d(β − β) and (β − β)B = dα, which means ηβ and ηβ are relatively cohomologous, (10.3), as desired. In particular, this applies to the ambiguity arising when β and β are different extensions of the same β0 . The image of d ∗ is obviously given by increasing q by unity in (10.5), originally the kernel of j ∗ but the kernel of d ∗ is trickier. Obviously ηβ is trivial in relative cohomology (10.3) whenever β0 is coexact but this means it is trivial in H q (B). But ηβ is also trivial if it vanishes, that is if β0 extends to a form β in the bulk which is still coclosed. Thus K q (B) = {classes of H q (B) extending to coclosed forms on M}.

(10.9)


299

This is also the image of i ∗ . Thus we have a series of identifications of images and kernels and the results can be all assembled in the following grand diagram: i∗

.. → ∂∗

d∗

j∗

i∗

j∗

H p (B) →H p+1 (M, B)→

H p+1 (M)

i∗

d∗

∂∗

i∗

→ H p+1 (B) → .. (10.10)

.. →Hm− p−1 (B)→Hm− p−1 (M)→Hm− p−1 (M, B)→Hm− p−2 (B)→ .. The upper sequence is composed of the homomorphisms of the de Rham cohomology groups just described. It is exact because at each stage the kernels and images coincide as was just explained. The lower sequence is the exact sequence of homology (7.1) explained in previous sections whilst the vertical arrows indicate the PoincaréLefschetz isomorphisms (7.6) and (7.7). The most powerful version of this diagram refers to groups taken over the integers, ZZ , but for some parts of the diagram we have only given arguments establishing a weaker version, over the reals, IR. By (7.3), a consequence of exactness, the kernel subgroups of the pairs of groups related by the Poincaré-Lefschetz isomorphism are themselves isomorphic. This suggests that the result we wish to prove, (9.2), is equivalent to its cohomological counterpart: η ∧ ω=0 if η ∈ K m−q (M, B; IR) and ω ∈ K q (M; IR). M

(10.11) But this is quite easy to prove using the results above, as we now see. By (10.5) the integral equals M dα ∧ ω = M d(α ∧ ω) theorem asdω vanishes by (10.7). By Stokes’ on M the integral equals B α ∧ ω = B α B ∧ ωB = B α ∧ dγ as ωB is coexact by (10.7). But, by (10.5), α is coclosed so the integral equals B d(α ∧ γ ) = ∂ B α ∧ γ = 0, by Stokes’ for the boundary, B, and the fact that the latter is automatically closed. Vanishing theorems analogous to (10.11) also apply to the integrals like w ω coupling a pair of like homology and cohomology groups. For example, the electric charge Q(S) = S ∗ j couples the relative homology of the integration domain, S, Hm− p−1 (M, B) to the relative de Rham cohomology of the dual current, ∗ j, H m− p−1 (M, B) and vanishes when S ∈ K m− p−1 (M, B), (4.4) and ∗ j ∈ K m− p−1 (M, B), (10.5). The latter condition certainly hold when Maxwell’s equation, (1.5), for the field strength, F, holds. This vanishing theorem is then precisely what was proven in our earlier discussion of electric charges, and that is now seen to be part of a more general pattern. The last step is the derivation of the vanishing theorem (9.2) for the intersection matrix from the vanishing theorem for cohomology, (10.11), proven above, using the upward arrow in the Poincaré-Lefschetz isomorphism, (10.10). A convenient concrete version of this map is provided by the quantity µ(w) that enters the expression (8.2) for the dual current ∗ j due to a brane whose world-volume is the absolute cycle w, and generalisations of this to be explained in the Appendix. These maps will provide homomorphisms between the groups indicated in (10.10) mapping the appropriate kernel subgroups into each other. The desired result follows by combining these results with the fact that the intersection number can be written I (w, S) = µ(w) ∧ µ(S). M

300


11. Discussion Motivated by the physical questions of elucidating and counting the types of conservation laws occurring in the sorts of generalised Maxwell theories that arise naturally in string/superstring theories formulated on a fixed background space-time of possibly complicated topology, we have been led to a well established area of pure mathematics. This is the theory of relative homology/cohomology associated with the space-time, assumed to have a boundary, and it seems not to be so familiar to physicists despite its evident physical relevance. Accordingly we have tried to build it up systematically, as guided by physics, and in particular, the generalised Maxwell equations, and included reasonably self-contained proofs. Given an understanding of the overall grand mathematical structure, comprising the two exact sequences of homology and cohomology and the Poincaré-Lefschetz isomorphism relating them, as depicted by (10.10), and the duality relations, (7.8), indicating a horizontal reflection symmetry of the exact sequences, it is relatively easy to explain the relevance to physics. This is what we now do because of the value of the new perspectives afforded. The first step is the recognition that there are precisely three types of conserved quantity, electric charge, (2.1), electric flux, (4.2) and magnetic flux, (6.2), and that these are associated with the three possible types of homology/cohomology, namely relative, boundary and absolute bulk, respectively. In fact these conserved quantities are invariant under the appropriate homologies/cohomologies and, indeed, constitute nonsingular bilinear forms on the free parts of these groups, thereby being responsible for the duality relations, (7.8), of the exact sequences (10.10). However this argument makes only partial use of the generalised Maxwell’s equations, (1.4) and (1.5), and the associated boundary conditions (3.1) and (3.2). What is used for each conserved quantity in turn is: Electric charge, (2.1): d ∗ j = 0 and ∗ j B = 0, Electric flux, (4.2): d{(∗F)B } = 0, Magnetic Flux, (6.2): d F = 0. With this limited information these three conserved quantities appear unrelated to each other and counted by the relevant Betti numbers, bm− p−1 (M, B), bm− p−2 (B) and b p+2 (M) as explained above. The content in Maxwell’s equations that has not so far been exploited is the inhomogeneous Maxwell in the bulk, (1.5), and it has many extra consequences, as we have seen in the text. From the point of view of de Rham cohomology the most immediate is that the dual current, ∗ j, is not just coclosed (current conservation, (1.6) but coexact, and hence an element of the subgroup K m− p−1 (M, B), (10.5), of the relative de Rham cohomology group. This is the subgroup that plays the role of kernel/image at this stage of the exact sequence of cohomology. Thus the exact sequence is now brought into play by means of the bulk Maxwell’s equations. When this current, j, is determined by the geometrical picture in terms of the p-brane world-volume, w, by (8.2), the fact that µ realises the Poincaré-Lefschetz isomorphism as explained in the Appendix, means that the world-volume w must belong to a class of K p+1 (M), (4.3), and hence be homologous to a cycle on the boundary, B, of spacetime. This was one of our main results, obtained by a more roundabout, though more self-contained, method, when we were not taking the complete mathematical structure for granted. This conclusion is contrary to what would have seemed intuitively likely, that any configuration of brane world-volumes in space-time is possible. The reason unsuitable


301

configurations are forbidden is that they provide topological obstructions to the integration of the generalised Maxwell equations for which they provide sources, as argued in the text. Notice that in obtaining this selection rule it was not necessary to take into account any equations of motion for the brane degrees of freedom. We saw that another, related, consequence of Maxwell’s inhomogeneous equations in the bulk was the reduction of the count of linearly independent electric charges from the Betti number bm− p−1 (M, B) to s p+1 (M) = bm− p−1 (M, B) − sm− p−1 (M, B), corresponding to the number of linearly independent homology classes permitted for the p-brane world-volume. In Sect. 4 we saw that p-brane electric fluxes are classified by the boundary homology group, Hm− p−2 (B; ZZ ), or, more precisely by the free part of this abelian group obtained by dividing out the torsion subgroup, namely a lattice of dimension given by the Betti number bm− p−2 (B). The effect of the inhomogeneous bulk Maxwell equations is to equate to electric charges those electric fluxes on the sublattice of dimension sm− p−2 (B), corresponding to K m− p−2 (B; ZZ ). As a result these electric fluxes are quantised, as integer multiples of q, but this result does not apply to the remaining bm− p−2 (B) − sm− p−2 (B) electric fluxes. There is no reason for them to be quantised. Through these boundary cycles there may also be magnetic fluxes, this time associated with p-branes, ˜ dual to the p-branes (so p + p˜ + 4 = m). As seen in Sect. 6, these vanish on the afore-mentioned sublattice of dimension sm− p−2 (B) whilst the remaining fluxes are quantised as integer multiples of 2π /q. Thus there is evidence of states carrying just quantised electric charge and no magnetic charge, and these must be the input p-brane states. But the quantised magnetic flux and non-quantised electric flux through the (H/K )m− p−2 (B; ZZ ) cycles is rather reminiscent of known solutions [W1] to the Zwanziger-Schwinger quantisation condition [Z2, S2] applying to particles in four dimensional space-time and so provides evidence of mysterious and intriguing dyonic objects that are not situated on the space-time, M, according to (1.4). Maybe a better understanding of this phenomenon is important in connection with electromagnetic duality. At this stage, we should explain that one motivation for the present work was to gain a better understanding of electromagnetic duality [MO]. It has been understood that in a closed space-time of four dimensions, certain partition functions exhibit a beautiful covariance under the action of the modular group implementing electro-magnetic duality transformations [V, W2] and this is further enhanced when spin is taken into account [AO2]. It is also possible to include Wilson loops [Z1]. But, in closed space-times there are neither non-vanishing electric charges nor electric fluxes, only magnetic fluxes. Yet in supersymmetric gauge theories on flat space-time it is familiar that electromagnetic duality transformations permute electric and magnetic charges [S3]. So it might be important to consider space-times with boundary, as we have. As just explained there is a beautiful topological classification of conservation laws involving electric charge, electric flux and magnetic flux but leaving no room for the classification of magnetic charge. As a result we are left with a dilemma to be resolved by future work. There are many other questions left open for future work and many of them concern undesirable simplifications that have been made relative to the full complication of superstring theory. We shall conclude by listing some of these. Some of these oversimplifications are routine practice in the subject but should none-the-less be removed when possible. 1) Branes have been treated as geometrical objects, cycles in space-time, and not assigned any sort of generalised quantum mechanical wave function as ideally they should.

302

2) 3) 4) 5)

6) 7) 8) 9)


In the absence of this there is lacking the concept of intrinsic spin which is familiar for particle ( p = 0) wave-functions on four dimensional space-time, and known to play a role in the understanding of electromagnetic duality [AO2]. No account is taken of any internal brane structure, such as gauge theories confined to the brane as sometimes required by supersymmetry. If so presumably a K -theory classification of this internal structure would be relevant, [W4]. Brane world-volumes have been treated as cycles. It might be more reasonable to allow them to have boundaries in the infinite past or future but we do not know how to do this. No equations of motion for p-brane degrees of freedom have been considered. Partly this is because these equations ought to involve the wave-functions, not yet formulated properly anyway when p > 0. No account is taken of any supersymmetry. This usually requires a spectrum of values of p and the fact that some branes may possess boundaries situated on other branes [S6]. It would be interesting to know how the charges and fluxes we have discussed could be related to the tensor charges occurring in the supersymmetry algebra. Branes have been treated as carrying only electric charge but maybe an additional magnetic charge should be allowed as an input in (1.4). No account of Chern-Simons type terms has been taken in the generalised Maxwell equations. This could only occur when p + 2 is even and divides m + 1, as for the familiar case of p = 2 and m = 11, [CJS]. No special consideration has been made of the self-dual case when m equals twice p + 2 (so p = p). ˜ Only orientable manifolds have been considered but there is a possibility of interesting phenomena when space-time (or space) is not orientable [S4, S5, DH].

12. Appendix The basic idea stems from the way the term in the action, (1.1), describing the geometrical coupling of the p-brane to the gauge potential A, defines the dual electric current, ∗ j, via (6.1) to be proportional to a distribution valued differential form, µ(w), depending on the world-volume, w. Clearly this idea is motivated by physical considerations. A mathematical version had earlier been proposed by de Rham [dR]. So far the idea applies to absolute cycles and it has to be extended to relative cycles and to chains both in the bulk and on the boundary and this is now done. If C is a q-chain containing no sub q-chain lying in the boundary, B, its dual current, µ(C), is defined by f = f ∧ µ(C), M

C

where f is an arbitrary q-form. On the other hand if γ is a q-chain lying on the boundary, B, the dual surface current, ν(γ ) is defined by g= g ∧ ν(γ ), γ

B

where g is an arbitary q-form on the boundary. Notice that even though C and γ are chains of the same dimension, q, µ(C) and ν(γ ) are forms of different degree, m − q and m − q − 1 respectively.


303

The boundary of C, ∂C, can be decomposed into two terms of the type just described, each of dimension one less: ∂C = U + α, so that µ(C), µ(U ) and ν(α) are all well-defined. The integral ∂C h can be evaluated in two ways, first as M h ∧µ(U )+ B h B ∧ν(α), and secondly as dh ∧ µ(C), using Stokes’ theorem. On integrating by parts this M d(C) equals B h B ∧ µ(C) B + (−1) M h ∧ dµ(C), where d(C) is the dimension of C. Equating the bulk and boundary terms separately yields the identities dµ(C) = (−1)d(C) µ(U )

and

µ(C)B = ν(α).

These are precisely what is needed to check the properties of the upwards PoincaréLefschetz homomorphism from homology to de Rham cohomology. This will be done by exploiting the different ways of interpreting these relations and special cases of them. C is a relative cycle if U vanishes. Then dµ(C) = 0 and so µ(U ) is an absolute bulk de Rham cocycle as µ(C)B need not vanish. C is an absolute bulk cycle if U and α both vanish. Then dµ(C) and µ(C)B both vanish, implying that µ(C) is a relative de Rham cocycle. U is a relative boundary and µ(U ) is coexact and so trivial in absolute bulk cohomology. U is an absolute boundary if α vanishes. Again µ(U ) is coexact but in addition µ(U )B = ν(α) = 0 so that now µ(U ) is trivial in relative cohomology. These four observations are sufficient to show that µ maps absolute or relative homology classes into relative or absolute cohomology classes respectively. By linearity these maps are homorphisms and it is easy to see that their kernels include the torsion subgroups so more properly µ acts on the free parts, F, of the homology groups H (obtained by dividing out the torsion). So µ:

Fq (M; ZZ ) → F m−q (M, B; ZZ )

and

Fq (M, B; ZZ ) → F m−q (M; ZZ ).

The last step is to check that µ maps the appropriate kernel subgroups into each other. U is an absolute bulk cycle homologous to a boundary cycle, −α if ∂U vanishes and so in a class of K q (M) by (5.2). But then µ(U ) = d[(−1)d(C) µ(C)] and µ(C)B = ν(α), where dν(α) = −(−1)d(C) ν(∂α) = 0. So, by (10.5), µ maps from a class of K q (M) to a class of K m−q (M, B). Finally if U vanishes and α = ∂γ , then ∂C = ∂γ meaning that C is a relative cycle in a class of K q (M, B) by (4.4). Hence dµ(C) vanishes and µ(C)B = ν(α) = ν(∂γ ) = (−1)d(γ ) dν(γ ). Thus, by (10.7), µ maps from a class of K q (M, B) to a class of K m−q (M). By the work of this Appendix, the intersection number

I (w, S) ≡

µ(w) = S

M

µ(w) ∧ µ(S).

Furthermore if w ∈ K q (M; ZZ ) and S ∈ K m−q (M, B; ZZ ) then µ(w) ∈ K m−q (M, B; ZZ ) and µ(S) ∈ K q (M; ZZ ) so that, finally, I (w, S) vanishes by (10.11), as desired.

304


Acknowledgements. D.I. Olive is belatedly grateful to G.-C. Wick for first introducing him to homology theory, long ago, and to Tobias Ekholm and Victor Pidstrigach for separately explaining important mathematical concepts to him. He thanks the Mittag-Leffler Institute (Djursholm), IFT (UNESP São Paulo), the Yukawa Institute (University of Kyoto) and NORDITA for hospitality whilst parts of this work were accomplished. M. Alvarez’s research has been supported by PPARC through the Advanced Fellowship PPA/A/S/1999/00486. Support to both of us from the European String Network HPRN-CT-2000-122 is also gratefully acknowledged.

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Communicated by G.W. Gibbons


Communications in


The Universality Classes in the Parabolic Anderson Model Remco van der Hofstad1 , Wolfgang König2 , Peter Mörters3 1 Department of Mathematics and Computer Science, Eindhoven University of Technology,

5600 MB Eindhoven, The Netherlands. E-mail: [email protected]

2 Mathematisches Institut, Universität Leipzig, 04109 Leipzig, Germany.

E-mail: [email protected]

3 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom.

E-mail: [email protected] Received: 18 March 2005 / Accepted: 20 February 2006 Published online: 3 August 2006 – © Springer-Verlag 2006

Abstract: We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on Zd . We consider general i.i.d. potentials and show that exactly four qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at ∞ that are thicker than the double-exponential tails, (2) double-exponential tails at ∞ studied by Gärtner and Molchanov, (3) a new class called almost bounded potentials, and (4) potentials bounded from above studied by Biskup and König. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities. Our analysis of class (3) relies on two large deviation results for the local times of continuous-time simple random walk. One of these results is proved by Brydges and the first two authors in [BHK05], and is also used here to correct a proof in [BK01].

1. Introduction and Main Results 1.1. The parabolic Anderson model. We consider the continuous solution v : [0, ∞) × Zd → [0, ∞) to the Cauchy problem for the heat equation with random coefficients and localised initial datum, ∂ v(t, z) = d v(t, z) + ξ(z)v(t, z), ∂t v(0, z) = 1l0 (z), for z ∈ Zd .

for (t, z) ∈ (0, ∞) × Zd ,

(1.1) (1.2)

308

R. van der Hofstad, W. König, P. Mörters

Here ξ = (ξ(z) : z ∈ Zd ) is an i.i.d. random potential with values in [−∞, ∞), and d is the discrete Laplacian, d f (z) = f (y) − f (z) , for z ∈ Zd , f : Zd → R. y∼z

The parabolic problem (1.1) is called the parabolic Anderson model. The operator d + ξ appearing on the right is called the Anderson Hamiltonian; its spectral properties are well-studied in mathematical physics. Equation (1.1) describes a random mass transport through a random field of sinks and sources, corresponding to lattice points z with ξ(z) < 0, respectively, > 0. It is a linearised model for chemical kinetics [GM90], is equivalent to Burger’s equation in hydrodynamics [CM94], and describes magnetic phenomena [MR94]. We refer the reader to [GM90, M94, CM94] for more background and to [GK05] for a survey on mathematical results. The long-time behaviour of the parabolic Anderson problem is well-studied in the mathematics and mathematical physics literature because it is the prime example of a model exhibiting an intermittency effect. This means, loosely speaking, that most of the total mass of the solution, U (t) = v(t, z), for t > 0, (1.3) z∈Zd

is concentrated on a small number of remote islands, called the intermittent islands. A manifestation of intermittency in terms of the moments of U (t) is as follows. For 0 < p < q, the main contribution to the q th moment of U (t) comes from islands that contribute only negligibly to the p th moments. Therefore, intermittency can be defined by the requirement, lim sup t→∞

U (t) p 1/ p = 0, for 0 < p < q, U (t)q 1/q

(1.4)

where · denotes expectation with respect to ξ . Whenever ξ is truly random, the parabolic Anderson model is intermittent in this sense, see [GM90, Theorem 3.2]. However, one wishes to understand the intermittent behaviour in much greater detail. The following has been heuristically argued in the literature and has been verified, at least partially, for important special examples of potentials: the intermittent islands are characterized by a particularly high exceedance of the potential and an optimal shape, which is determined by a deterministic variational formula. A universal picture is present: the location and number of the intermittent islands are random, their size and the absolute height of the potential in the islands is t-dependent, but the (rescaled) shape depends neither on randomness nor on t. Examples studied include the double-exponential distribution [GM98], potentials bounded from above [BK01] and continuous analogues on Rd instead of Zd like Poisson obstacle fields [S98] and Gaussian and other Poisson fields [GK00, GKM00]. A finer analysis of the geometry of the intermittent islands has been carried out for Poisson obstacle fields [S98] and the double-exponential distribution [GKM06]. In the present paper we initiate the study of the parabolic Anderson model for arbitrary potentials, with the aim of identifying all universality classes of intermittent behaviour that can arise for different potential distributions. Our standing assumption is that the potentials (ξ(z) : z ∈ Zd ) are independent and identically distributed and that all positive exponential moments of ξ(0) are finite, which is necessary and sufficient for

The Universality Classes in the Parabolic Anderson Model

309

the finiteness of the p th moments of U (t) at all times. The long-term behaviour of the solutions depends strongly and exclusively on the upper tail behaviour of the random variable ξ(0). It is fully described by the top of the spectrum of the Anderson Hamiltonian d + ξ in large t-dependent boxes. The outline of the remainder of this section is as follows. In Sect. 1.2, we formulate and discuss a mild regularity condition on the potential. In Sect. 1.3, we show that under this condition the potentials can be split into exactly four classes, which exhibit four different types of intermittent behaviour. Three of these classes have been studied in the literature up to now. A fourth class, the class of almost bounded potentials, is studied in the present paper for the first time. We present our results on the moment and almost-sure large-time asymptotics for U (t) in Sect. 1.4. In Sect. 1.5, we give a heuristic derivation of the moment asymptotics, and in Sect. 1.6, we explain the variational problems involved. 1.2. Regularity assumptions. We first state and discuss our regularity assumptions on the potential. Roughly speaking, the purpose of these assumptions is to ensure that the potential has the same qualitative behaviour at different scales, and therefore the system does not belong to different universality classes at different times. Our assumptions refer to the upper tail of ξ(0), and are conveniently formulated in terms of the regularity of its logarithmic moment generating function, H (t) = log etξ(0) , as t ↑ ∞. (1.5) Note that H is convex and t → H (t)/t is increasing with limt→∞ H (t)/t = esssup ξ(0). To simplify the presentation, we make the assumption that if ξ is bounded from above, then esssup ξ(0) = 0, so that limt→∞ H (t)/t ∈ {0, ∞}. This is no loss of generality, 1 as additive constants in the potential appear as additive constants both in pt logU (t) p and

1 t

log U (t). The first central assumption on H is the following:

Assumption (H). t →

H (t) t

is in the de Haan class.

is in the de Haan class if, for some regularly varyWe say that a measurable function H (λt) − H (t)) converges to a nonzero ing function g : (0, ∞) → R, the term g(t)−1 ( H ∈ g . limit as t ↑ ∞, for any λ > 1. In the notation of [BGT87] this means that H Recall that a measurable function g is called regularly varying if g(λt)/g(t) converges to a positive limit for every λ > 0. If this is the case, then the limit takes the form λ , and is called the index of regular variation. If = 0, then the function is called slowly varying. When H (t)/t is in the de Haan class, then H is regularly varying with some index γ ∈ R, see [BGT87, Theorem 3.6.6]. By convexity of H , we have γ ≥ 0. If H is regularly varying with index γ = 1, then H (t)/t is in the de Haan class, so that the statements are equivalent for γ = 1. However, if γ = 1, then this does not necessarily hold, see [BGT87, Theorem 3.7.4]. which From the theory of regular functions we derive the existence of a function H can be characterized by two parameters, γ ∈ [0, ∞) and ρ ∈ (0, ∞), and plays an important role in the sequel. : (0, ∞) Proposition 1.1. Assumption (H) is equivalent to the existence of a function H → R and a continuous auxiliary function κ : (0, ∞) → (0, ∞) such that lim

t↑∞

H (t y) − y H (t) (y) = 0, for y ∈ (0, 1) ∪ (1, ∞). =H κ(t)

(1.6)

310


The convergence holds uniformly on every interval [0, M], with M > 0. Moreover, with γ the index of variation of H , the following statements hold: (i) κ is regularly varying of index γ ≥ 0. In particular, κ(t) = t γ +o(1) as t ↑ ∞. (ii) There exists a parameter ρ > 0 such that, for every y > 0, γ (y) = ρ y − y , and lim H (t) = ρ , (a) if γ = 1, then H t↑∞ κ(t) 1−γ γ −1 |H (t)| (y) = ρy log y, and lim (b) if γ = 1, then H = ∞. t↑∞ κ(t)

Proof. See Chapter 3 in [BGT87]. More accurately, using the notation f (t) = H (t)/t and g(t) = κ(t)/t, (i) is shown in [BGT87, Sect. 3.0], see also [BGT87, Theorem 1.4.1]. The uniformity of the convergence follows since the left hand side of (1.6) is convex in y, negative on the interval (0, 1), and continuous in zero. (ii) follows from [BGT87, Lemma 3.2.1]. The implication stated in (ii)(a) follows from [BGT87, Theorems 3.2.6, 3.2.7], and the implication stated in (ii)(b) is shown in [BGT87, Theorem 3.7.4].

an asymptotic shape function for Note that κ is an asymptotic scale function, and H H . While γ ∈ [0, ∞) is unambiguously determined by the potential distribution, the . The latter option makes it possible to parameter ρ could be absorbed in either κ or H keep track of ρ in the sequel. If ξ is unbounded from above, then ξ and ξ + C have the and κ for any C ∈ R. If ξ is replaced by Cξ for some C > 0, then the pair same pair of H , κ). In the case γ = 1 one may choose κ(t) = H (t) ( H , κ) may be replaced by (C γ H t in (1.6), if γ = 1 one may take κ(t) = H (t) − 1 H (s)/s ds, see [BGT87, Theorem 3.7.3]. The three regimes 0 ≤ γ < 1, γ = 1 and γ > 1 obviously distinguish three qualitatively different classes of (upper tail behaviour of) potentials. However, in order to appropriately describe the asymptotics of the parabolic Anderson model in the case γ = 1, a finer distinction is necessary. For this we need an additional mild assumption on the auxiliary function κ: κ(t) exists as an element of [0, ∞]. Assumption (K). The limit κ ∗ = lim t→∞ t Assumption (K) is obviously satisfied in the cases γ = 1 and for potentials bounded from above in the case γ = 1. Indeed, when γ < 1, then κ ∗ = 0, while when γ > 1, then κ ∗ = ∞ by Proposition 1.1(ii)(a). When γ = 1 and H (t)/t → 0, then, by Proposition 1.1(ii)(b), H (t)/κ(t) → ∞, so that κ(t)/t → 0. Hence, Assumption (K) can be a restriction only for potentials unbounded from above in the case γ = 1. 1.3. The universality classes. In this section, we define and discuss the four universality classes of the parabolic Anderson model under the Assumptions (H) and (K). In particular, we explain the relation between the asymptotics of the parabolic Anderson model and the parameters γ and κ ∗ introduced in Assumptions (H) and (K). For the moment, we focus on the large time behaviour of the p th moment U (t) p for any p > 0. In this paper we show that there is a scale function α : (0, ∞) → (0, ∞) and a number χ ∈ R such that

H pt α( pt)−d 1 1 p χ + o(1) , as t ↑ ∞ . (1.7) − logU (t) = −d 2 pt pt α( pt) α( pt)


311

The scale function α describes how fast the expected total mass, which at time t = 0 is localised at the origin, spreads, in the sense that α(t)2 z∈Zd v(t, z) 1l{|z| ≤ R α(t)} log lim lim inf = 0. (1.8) R↑∞ t↑∞ t z∈Zd v(t, z) Moreover, in the three classes where the mass does not concentrate asymptotically in a single point, there exists R > 0 such that α(t)2 z∈Zd v(t, z) 1l{|z| ≤ R α(t)} log < 0. (1.9) lim inf t↑∞ t z∈Zd v(t, z) In three of the four classes the results (1.7), (1.8) and (1.9) are already contained in the literature, and we only give references; a further class will be the subject of the remainder of the paper. Heuristically, α(t) also determines the size of the intermittent islands for the almost sure behaviour of U (t). The order of their diameter is given as (α ◦ β)(t), where β(t) is the asymptotic inverse of t → t/α(t)2 evaluated at d log t, cf. Sect. 1.4.2 below. The numbers χ are naturally given in terms of minimisation problems, where the minimisers correspond to the typical shape of the solution on an intermittent island. A rigorous proof of these heuristic statements, however, is beyond the means of this paper. One expects that α(t) is asymptotically the larger, the thinner the upper tails of ξ(0) are. It will turn out that when κ ∗ = ∞, then (1.7) is satisfied with α(t) = 1. Therefore, we only need to analyse α(t) in the case when κ ∗ < ∞. Analytically, if κ ∗ < ∞, then α(t) may be defined by a fixed point equation as follows: Proposition 1.2 (The scale function α). Suppose that Assumptions (H) and (K) are satisfied and κ ∗ < ∞. There exists a regularly varying scale function α : (0, ∞) → (0, ∞), which is unique up to asymptotic equivalence, such that for all sufficiently large t > 0,

κ tα(t)−d 1 = . (1.10) tα(t)−d α(t)2 t = ∞. Moreover, α(t)d √ (i) If γ = 1 and 0 < κ ∗ < ∞, then limt↑∞ α(t) = 1/ κ ∗ ∈ (0, ∞). (ii) If γ = 1 and κ ∗ = 0, or if γ < 1, then limt↑∞ α(t) = ∞.

The index of regular variation is

1−γ d+2−dγ

and hence lim

t↑∞

Proof. To see that α is regularly varying and unique up to asymptotic equivalence we note that f (t) = t (κ(t)/t)−d/2 is regularly varying with index at least one. By [BGT87, Theorem 1.5.12], there exists an asymptotically unique inverse g such that f (g(t)) ∼ t for t ↑ ∞. This inverse is regularly varying. By definition, t → tα(t)−d satisfies f (tα(t)−d ) = t and hence α(t) ∼ (t/g(t))1/d is regularly varying. The index of regular variation of α is immediate from the defining equation and the fact that κ(t) is regularly varying with index γ . Under the assumptions of (i), for large t, the mapping x → κ(t x d/2 )/t x d/2 maps a compact interval centred in κ ∗ to itself, and hence the existence of a solution to (1.10) follows from a fixed-point argument. The stated properties of α( · ) follow immediately from the definition.

312


Under the assumptions of (ii), we look at the problem of finding s > 0 such that κ(s)/s = (s/t)2/d . For any fixed t, as we increase s the left hand side goes to zero and the right hand side to infinity. Hence for sufficiently large t, there exists a solution s = s(t), which is going to infinity as t ↑ ∞. Then α(t) = (t/s(t))1/d solves (1.10) and converges to infinity.

Now we introduce the four universality classes, ordered from thick to thin upper tails of ξ(0). Recall the general formula for the asymptotics of the moments U (t) p from (1.7). Uniqueness for the variational problems below is to be understood up to spatial translation. (1) γ > 1, or γ = 1 and κ ∗ = ∞. This case is included in [GM98], see also [GM90], as the upper boundary case ρ = ∞ in their notation. Examples include the Weibull-type distributions, for which Prob{ξ(0) > x} ≈ exp(−βx a ) with a > 1. Here χ = 2d, the scale function α(t) = 1 is constant, and the first term on the right-hand side in (1.7) dominates the sum, which diverges to infinity. The asymptotics in (1.8) can be strengthened to v(t, 0) 1 = 0, log t↑∞ t z∈Zd v(t, z) lim

i.e. the expected total mass remains essentially in the origin and the intermittent islands are single sites, a phenomenon of complete localisation. We call this the single-peak case. 3 (2) γ = 1 and κ ∗ ∈ (0, ∞). This case, the double-exponential case, is the main objective of [GM98]. The prime example is the double exponential distribution with parameter ρ ∈ (0, ∞),

Prob ξ(0) > r = exp{−er/ρ }, √ ∗ which implies H (t) = ρt log(ρt) − ρt + o(t). Here α(t) → 1/ κ ∈ (0, ∞), so that the size of the intermittent islands is constant in time. The first term on the right hand side in (1.7) dominates the sum, which goes to infinity. Moreover, 1

2 χ = min g 2 (x) log g 2 (x) , (1.11) g(x) − g(y) − ρ 2d g : Zd →R d d g 2 =1

x,y∈Z x∼y

x∈Z

where we write x ∼ y if x and y are neighbours. This variational problem is difficult to analyse. It has a solution, which is unique for sufficiently large values of ρ, and heuristically this minimizer represents the shape of the solution. As noted in [GH99], for any family of minimizers gρ , as ρ ↑ ∞, gρ converges to δ0 , which links to the single-peak case. Furthermore, as ρ ↓ 0, the minimisers gρ are asymptotically given by 2 √ gρ2 (x/ ρ) = (1 + o(1)) e−|x| π −d/2 , uniformly on compacts and in L 1 (Rd ). Consequently, ρ 1 χ = ρ d 1 − log + o(1) as ρ ↓ 0. 2 π 3


313

(3) γ = 1 and κ ∗ = 0. Potentials in this class are called almost bounded in [GM98] and may be seen as the degenerate case for ρ = 0 in their notation. This class contains both bounded and unbounded potentials, and is analysed for the first time in the present paper. The scale function α(t) and hence the diameter of the intermittent islands goes to infinity and is slowly varying, in particular it is slower than any power of t. The first term on the right-hand side in (1.7) dominates the sum, which may go to infinity or zero. Moreover, χ = min |∇g(x)|2 d x − ρ g 2 (x) log g 2 (x) d x , (1.12) g∈H 1 (Rd ) g2 =1

Rd

see Theorem 1.4. This variational formula is obviously the continuous variant of (1.11), and it is much easier to solve. There is a unique minimiser, given by ρ d/4 ρ gρ (x) = exp − |x|2 , π 2 representing the rescaled shape of the solution on an intermittent island. In particular, χ = ρd 1 − 21 log πρ , which is the asymptotics of (1.11) as ρ ↓ 0. Hence, on the level of variational problems, (3) is the boundary case of (2) for ρ ↓ 0. 3 (4) γ < 1. This is the case of potentials bounded from above, which is treated in [BK01]. Indeed, in [BK01], it is assumed that there exists a non-decreasing function α(t) : (0, ∞) → (−∞, 0] such that and a nonpositive function H t

d+2 lim α(t)t H α(t) d y = H (y), t↑∞

uniformly on compact sets in (0, ∞). It is easy to infer from the results of Sect. 1.2 that this assumption is equivalent to Assumption (H) with index γ < 1 (recall that in this case Assumption (K) is redundant), for α defined by (1.10) and (y) = ρ y γ . H γ −1 1−γ Here α(t) → ∞ as t → α(t) is regularly varying with index d+2−dγ . The potential ξ is necessarily bounded from above. In this case, the two terms on the right hand side in (1.7) are of the same order, and (1.7) converges to zero. Moreover, g 2γ (x) − g 2 (x) 2 dx . (1.13) |∇g(x)| d x − ρ χ = inf γ −1 g∈H 1 (Rd ) Rd Rd g2 =1

In the lower boundary case where γ = 0, the functional g 2γ must be replaced by the Lebesgue measure of supp (g). In this case the formula is well-known and well-understood. In particular, the minimizer exists, is unique up to spatial shifts, and has compact support. To the best of our knowledge, for γ ∈ (0, 1), the formula in (1.13) has not been analysed explicitly, unless in d = 1. In Proposition 1.16 below, we show that (1.13) converges to (1.12), as would follow from interchanging the limit γ ↑ 1 with the infimum on g. This means that, on the level of variational formulas, (3) is the boundary case of (4) for γ ↑ 1. 3

314


Remark 1.3. The variational problems in (1.11), (1.12), and (1.13) encode the asymptotic shape of the rescaled and normalised solution v(t, · ) in the centred ball with radius of order α(t). Informally, the main contribution to U (t) comes from the events that

v t, · α(t)

≈ g, v t, · α(t) 2 where g is a minimiser in the definition of χ . To the best of our knowledge this heuristics has not been made rigorous in any nontrivial case so far. Note that in case (1), formally, (1.11) holds with ρ = ∞ and hence the optimal g is 1l0 . 3 Since the cases (1), (2) and (4) have been studied in the literature [BK01, GM98], the possible scaling picture of the parabolic Anderson model under the Assumptions (H) and (K) is complete once the case (3) is resolved. This is the content of the remainder of this paper.

1.4. Long time tails in the almost bounded case. In this section we present our results on the almost bounded case (3). In other words, we assume that κ(t)/t is slowly varying and converges to zero. 1.4.1. Moment asymptotics. Our main result on the annealed asymptotics of U (t) gives the first two terms in the asymptotics of U (t) p for any p > 0, as t ↑ ∞. This is a substantial improvement over the result for the almost bounded case contained in [GM98, Theorem 1.2], which just states that logU (t) p = o(t) for p ∈ N. Theorem 1.4 (Moment asymptotics). Suppose Assumptions (H) and (K) hold, and assume that we are in case (3), i.e., γ = 1 and κ ∗ = 0. Let ρ > 0 be as in Proposition 1.1(ii)(b). Then, for any p ∈ (0, ∞),

H pt α( pt)−d 1 1 p logU (t) = ρd(1 − 21 log πρ ) + o(1) , as t ↑ ∞. − −d 2 pt pt α( pt) α( pt) (1.14) Remark 1.5 (The constant). Recall from (1.7) and (1.12) that the constant ρd(1 − ρ 1 2 log π ) arises as a variational problem; see Sect. 1.6. The variational problem plays an essential role in the proof. 3 Remark 1.6 (Intermittency). Note from (1.10) that the first term in (1.14) is of higher order than the second term. Formula (1.14), together with the results of Proposition 1.1 and the fact that α( · ) is slowly varying, imply that

H pt α( pt)−d H qt α(qt)−d U (t) p 1/ p log = − + o t/α(t)2 q 1/q −d −d U (t) p α( pt) q α(qt) t q ˆ p

= (1.15) H q + o(1) for p, q ∈ (0, ∞). p 2 α(t) In particular, we have intermittency in the sense of (1.4), and the convergence is exponential on the scale t/α(t)2 . 3


315

Remark 1.7 (Interpretation). The fact that the minimisers of the variational problem (1.12) are given by Gaussian densities can be interpreted in the sense that the solution u(t, x) is asymptotically a heat flow running in the ‘slow motion’ scale α(t). Observe that this heat flow is the solution of (1.1) if the potential ξ is replaced by a certain parabola in the same scale. This parabola is the optimal potential in the sense of Remark 1.13 below. 3 In spite of the simplicity of the variational formula (1.12), the derivation of (1.14) is technically rather involved and requires a number of demanding tools. We use both representations of U (t) available to us: an approximative representation in terms of an eigenfunction expansion, and the Feynman-Kac formula involving the simple random walk. The heart of the proof is an application of a large deviation principle for the rescaled local times of simple random walk. However, there are three major obstacles to be removed, which require a variety of novel techniques. The first one is a compactification argument for the space, which is based on an estimate for Dirichlet eigenvalues in large boxes against maximal Dirichlet eigenvalues in small subboxes. This is an adaptation of a method from [BK01]. The second technique is a cutting argument for the large potential values, which we trace back to a large deviations estimate for the self-intersection number of the simple random walk. This is of independent interest and is carried out in Sect. 2. Finally, the third obstacle, which appearsin the proof of the upper bound, is the lack of upper semi-continuity of the map f → f (x) log f (x) d x in the topology of the large deviation principle, even after compactification and removal of large values. Therefore, in the proof of the upper bound we replace the classical large deviation principle by a new approach, taken from [BHK05], which identifies and estimates the joint density of the family of the random walk local times. See Proposition 3.3 below. An alternative heuristic derivation of formula (1.14) is given in Sect. 1.5. The proof of Theorem 1.4 is given in Sect. 2 and 3. 1.4.2. Almost-sure asymptotics. We define another scale function β such that β(t)

2 ∼ d log t . α β(t)

(1.16)

In other words, β(t) is the asymptotic inverse of t → t/α(t)2 evaluated at d log t, which by [BGT87, Theorem 1.5.12] exists and is slowly varying. In order to avoid technical inconveniences, we assume that the field ξ is bounded from below. See Remark 1.11 for comments on this issue. Theorem 1.8 (Almost sure asymptotics). Suppose Assumptions (H) and (K) hold, and assume that we are in case (3), i.e., γ = 1 and κ ∗ = 0. Furthermore, suppose that β is defined by (1.16) and that essinf ξ(0) > −∞. Let ρ > 0 be as in Proposition 1.1. Then, almost surely,

H β(t)α(β(t))−d 1 log U (t) = t β(t) α(β(t))−d

1 ρ(d − d2 log πρ + log ρe ) + o(1) , as t ↑ ∞. (1.17) − 2 α(β(t)) Remark 1.9 (The constant). In Sect. 1.6, we will see that also the constant ρ(d − d2 log πρ + log ρe ) arises as a variational problem. A remarkable fact is that the first two leading contributions to U (t) are deterministic. 3

316


Remark 1.10 (Interpretation). Heuristically, α(β(t)) is the order of the diameter of the intermittent islands, which almost surely carry most of the mass of U (t). Note that β(t) = (log t)1+o(1) and α(β(t)) = (log t)o(1) , i.e., the size of the intermittent islands increases extremely slowly. The crucial point in the proof of Theorem 1.8 is to show the existence of an island with radius of order α(β(t)) within the box [−t, t]d on which the shape of the vertically shifted and rescaled potential is optimal, i.e., resembles a certain parabola. To prove this, we use the first moment asymptotics at time β(t) locally on that island. The exponential rate, which is β(t)/α(β(t))2 has to be balanced against the number of possible islands, which has exponential rate d log t, cf. (1.16). 3 Remark 1.11 (Lower tails of the potential). The assertion of Theorem 1.8 remains true mutatis mutandis if the assumption essinf ξ(0) > −∞ is replaced, in d ≥ 2, by the assumption that Prob{ξ(0) > −∞} exceeds the critical nearest-neighbour site percolation threshold. This ensures the existence of an infinite component in the set C = {z ∈ Zd : ξ(z) > −∞}, and thus (1.17) holds conditional on the event that the origin belongs to the infinite cluster in C. In d = 1, an infinite cluster exists if and only if Prob{ξ(0) > −∞} = 1. If we assume that ξ(0) > −∞ almost surely and log(−ξ(0) ∨ 1) < ∞, (1.17) is true verbatim, while otherwise the rate of the almost sure asymptotics depends on the lower tails of ξ(0); see [BK01a] for details. The effect of the assumption is to ensure sufficient connectivity in the sense that the mass flow from the origin to regions where the random potential assumes high values and an approximately optimal shape is not hampered by deep valleys on the way. We decided to detail the proof of the almost sure asymptotics under the stronger assertion that essinf ξ(0) > −∞. See [BK01, Sect. 5.2] for the proof of the analogous assertion in the bounded-potential case under the weaker assumptions. The arguments given there can be extended with some effort to the situation of the present paper. 3 The proof of Theorem 1.8 is given in Sect. 4. It essentially follows the strategy of [BK01]. 1.4.3. Examples. We now explain what kind of upper tail behaviour is covered by the almost bounded case, arguing separately for the bounded and unbounded case, denoted by (B) and (U), respectively. Suppose the distribution of the field ξ(0) satisfies

r ↑∞ in case (U), f (r ) log Prob ξ(0) > r ∼ −e , as (1.18) r ↑ 0 = esssup ξ(0), in case (B). Here f is a positive, strictly increasing smooth function satisfying f (r ) ↑ ∞ as r ↑ ∞ in case (U) and f (r )r ↑ ∞ as r ↑ 0 in case (B). Note that typical representatives of case (2) of the four universality classes are f (r ) ≈ cr as r ↑ ∞, violating the condition in case (U); and typical representatives of case (4) of the four universality classes γ are f (r ) ≈ − 1−γ log |r | as r ↑ 0, violating the condition in case (B). The cumulant generating function behaves like

H (t) ≈ log etr exp −e f (r ) dr ≈ sup tr − e f (r ) = tr (t) − e f (r (t)) , (1.19) r

where r (t) is asymptotically, as t ↑ ∞, defined via t = f (r (t))e f (r (t)) . Note that r (t) ↑ ∞ in case (U), while r (t) ↑ 0 in case (B), as t ↑ ∞. Hence, f (r (t)) ↑ ∞ in


317

case (U), while f (r (t))r (t) ↑ ∞ in case (B). Rewriting the definition of r (t) as e f (r (t)) =

tr (t) = o(tr (t)), f (r (t))r (t)

we thus obtain that the first term on the right hand side of (1.19) dominates the second term. Therefore, we can approximate H (t)/t ≈ r (t), as t ↑ ∞. We next assume that f (r ( · )) is slowly varying at infinity. We then see that, using the fact that r (t) = f −1 log f (rt (t)) in the last equality,

H (t y) − y H (t) ≈ t y f −1 log f (rt (ty y)) − f −1 log f (rt (t))

≈ t y f −1 log f (rt (t)) + log y − f −1 log f (rt (t)) ≈ t (y log y) ( f −1 ) log

t f (r (t))

= (y log y)

t . f (r (t))

Using Proposition 1.1, this means that the scaling relation in (1.6) is satisfied with κ(t) = t/ f (r (t)) and ρ = 1. As f (r (t)) ↑ ∞ is slowly varying, we see that we are in case (3) of the four universality classes.

1.5. Heuristic derivation of Theorem 1.4. In this section, we give a heuristic explanation of Theorem 1.4 in terms of large deviations for the scaled potential ξ . Our proof of Theorem 1.4 follows a different strategy. We use the setup and notation of Sect. 1.4.3 and handle the cases (B) respectively (U) simultaneously. Consequently, the definition (1.10) of α(t) reads α(t)2 =

tα(t)−d = f r (tα(t)−d ) . −d κ(tα(t) )

(1.20)

We introduce the shifted, scaled potential

H (tα(t)−d ) ξ t (x) := α(t)2 ξ xα(t) − tα(t)−d

d f (r (tα(t)−d )) ≈ α(t)2 ξ xα(t) − r (tα(t)−d ) + α(t) , e t

(1.21)

for x ∈ Q R = [−R, R]d . The process ξ t satisfies a large deviation principle, for every R > 0, on the cube Q R with rate tα(t)−2 and rate function ϕ → Q R eϕ(x)−1 d x. Indeed, with B R = [−R, R]d ∩ Zd ,

Prob ξ t ≈ ϕ on Q R −1 ) α(t)d f (r (tα(t)−d )) −d ≈ Prob ξ(0) ≈ ϕ(zα(t) + r (tα(t) ) − e 2 t α(t) z∈B Rα(t)

≈

z∈B Rα(t)

exp − exp f r (tα(t)−d ) +

ϕ(zα(t)−1 ) α(t)2

−

α(t)d f (r (tα(t)−d )) t e

.

318


By a Taylor expansion around r (tα(t)−d ), using that s = f (r (s))e f (r (s)) for s = tα(t)−d as well as (1.20), we can continue with

Prob ξ t ≈ ϕ on Q R ϕ(x) ≈ exp −α(t)d exp f (r (tα(t)−d )) + f (r (tα(t)−d )) − 1 d x 2 α(t) QR t eϕ(x)−1 d x = exp − f (r (tα(t)−d )) Q R t eϕ(x)−1 d x . ≈ exp − 2 α(t) Q R The asymptotics of U (t) p can now be explained as follows. Note that U (t) = u(t, 0), where u(t, · ) is the solution of the parabolic Anderson model (1.1) with initial condition u(0, · ) = 1. We can approximate u(t, 0) by wt (t, 0), where (s, z) → wt (s, z) is the solution to the initial boundary value problem (1.1) with zero boundary condition outside the box Bt and initial condition wt (0, · ) = 1l Bt . Let λdt (ξ ) denote the principal eigenvalue of d + ξ in 2 (Bt ) with zero boundary condition. Then an eigenfunction expansion shows that U (t) p = u(t, 0) p ≈ wt (t, 0) p ≈ e ptλt (ξ ) . d

This already explains why the asymptotics of the p th moments of U (t) are the same as the asymptotics of the moments of U ( pt). We proceed by taking p = 1. Now the shift invariance and the asymptotic scaling properties of the discrete Laplace operator yield that

(tα(t)−d ) (tα(t)−d ) λdt (ξ ) = Htα(t) + λdt α(t)−2 ξ t (· α(t)−1 ) ≈ Htα(t) + α(t)−2 λ(ξ t ), −d −d where λ(ψ) denotes the principal eigenvalue of +ψ in L 2 (Q tα(t)−d ), with zero boundary condition. Hence, t −d d U (t) ≈ e H (tα(t) )α(t) exp λ(ξ ) . (1.22) t α(t)2 Using the large deviation principle for ξ t with R = tα(t)−d , and anticipating that ψ → λ(ψ) has the appropriate continuity and boundedness properties, we may use Varadhan’s lemma to deduce that 1 1 (tα(t)−d ) logU (t) ≈ Htα(t) − χ, −d t α(t)2 where χ is given by χ = inf ψ

Rd

eψ(x)−1 d x − λ(ψ) .

(1.23)

We show in Sect. 1.6 that χ is equal to ρd(1 − 21 log πρ ). This completes the heuristic derivation of Theorem 1.4. The interpretation of the above heuristics is that the moments of the total mass U (t) are mainly governed by potentials ξ whose shape is approximately given as

(tα(t)−d ) ξ(·) ≈ Htα(t) + α(t)−2 ψ · α(t)−1 , −d where ψ is a minimiser of the formula in (1.23).


319

1.6. Variational representations of the constants in Theorem 1.4 and 1.8. 1.6.1. The constant in Theorem 1.4. Fix ρ > 0 and define χ (ρ) ∈ R by χ (ρ) =

inf

g∈H 1 (Rd ) g2 =1

∇g22 − H(g 2 ) ,

(1.24)

where H 1 (Rd ) is the usual Sobolev space, ∇ the usual (distributional) gradient, and H(g 2 ) = ρ

Rd

g 2 (x) log g 2 (x) d x.

(1.25)

By the logarithmic Sobolev inequality in (1.30) below, H(g 2 ) ∈ [−∞, ∞) is welldefined for g ∈ H 1 (Rd ). Furthermore, we introduce the Legendre transform of H on L 2 (Rd ) and the top of the spectrum of the operator + ψ in H 1 (Rd ), L(ψ) =

sup

g∈L 2 (Rd )

g 2 , ψ−H(g 2 )

and λ(ψ) =

sup g∈H 1 (Rd ) g2 =1

ψ, g 2 −∇g22 . (1.26)

Introduce the functions gρ (x) =

ρ d

4

π

ρ

e− 2 |x|

2

and ψρ (x) = ρ + ρ

ρ d log − ρ 2 |x|2 , for x ∈ Rd . 2 π (1.27)

Note that the Gaussian density gρ is the unique L 2 -normalized positive eigenfunction of the operator + ψρ in H 1 (Rd ) with eigenvalue λ(ψρ ) = ρ − ρd + ρ d2 log πρ . It satisfies L(ψρ ) = ρ. Proposition 1.12 (Solution of the variational formula in (1.24)). For any ρ ∈ (0, ∞), the infimum in (1.24) is, up

to horizontal shift, uniquely attained at gρ . In particular, χ (ρ) = ρd 1 − 21 log πρ is the constant appearing in Theorem 1.4. Moreover, L is identified as 1 ρ e ρ ψ(x) d x, L(ψ) = e Rd

(1.28)

and the ‘dual’ representation is χ (ρ) =

inf

ψ∈C(Rd ) L(ψ) 0. The right side of (1.31) is 1 maximal precisely for g22 = 1e e ρ ψ . Substituting this value, we arrive at (1.28). To see the last two statements, we use (1.28) and the formula in (1.26) for λ(ψ) to obtain, for any ψ ∈ C(Rd ), ψ 1 2 2 2 − log g 2 − e ρ ψ−log g −1 . ∇g2 − H(g ) − ρ g 2 L(ψ) − λ(ψ) = inf ρ g∈H 1 (Rd ) g2 =1

(1.32) The term in square brackets is equal to θ − eθ−1 for θ = ψρ − log g 2 . Since this is nonpositive and is zero only for θ = 1, we have that ‘≤’ holds in (1.29). Furthermore, by restricting the infimum over g to strictly positive continuous functions and interchanging the order of the infima, we see that inf

L(ψ) − λ(ψ)

≤

inf

ψ 1 2 − log g 2 − e ρ ψ−log g −1 ∇g22 − H(g 2 ) − ρ g 2 ρ ψ∈C (Rd )

inf

∇g22 − H(g 2 ) = χ (ρ),

ψ∈C (Rd )

≤

g∈H 1 (Rd ) g2 =1,g>0 g∈H 1 (Rd ) g2 =1,g>0

inf

by substituting ψ = ρ + ρ log g 2 , and we use that the maximizer g of the right hand side is strictly positive. Therefore, equality holds in (1.29). We also know that, by uniqueness of the solution in (1.24), the unique minimizer in (1.29) is ψ = ρ + ρ log gρ2 = ψρ .

Remark 1.13 (Interpretation). Both representations (1.24) and (1.29) may be interpreted in terms of optimal rescaled profiles for the moment asymptotics of the total mass U (t). While the minimizer ψρ in (1.29) describes the shape of the potential ξ (see Sect. 1.5), the minimizer gρ in (1.24) describes the solution u(t, ·), cf. Remark 1.3. 3


321

1.6.2. The constant in Theorem 1.8. We now turn to the variational representation of the constant appearing in Theorem 1.8. We define χ (ρ) by χ (ρ) = inf{−λ(ψ) : ψ ∈ C(Rd ), L(ψ) ≤ 1},

(1.33)

where we recall that C(Rd ) is the set of continuous functions Rd → R. Proposition 1.14 (Solution of the variational formula in (1.33)). For any ρ ∈ (0, ∞), the function ψρ −ρ log ρe , with ψρ as defined in (1.27), is the unique minimizer in (1.33), and χ (ρ) = χ (ρ) + ρ log ρe . Proof. Obviously, the condition L(ψ) ≤ 1 in (1.33) may be replaced by L(ψ) = 1. In the representation χ (ρ) = inf ρ log L(ψ) − λ(ψ) : ψ ∈ C(Rd ), L(ψ) = 1 we may omit the condition L(ψ) = 1 completely since ρ log L(ψ) − λ(ψ) is invariant under adding constants to ψ. We use the definition of λ(ψ) in (1.26), and (1.28), and obtain, after interchanging the infima, 1 ρ 2 2 χ (ρ) = inf ∇g2 − sup ψ, g − ρ log e ρ ψ(x) d x + ρ log . 1 d e g∈H (R ) ψ∈C (Rd ) g2 =1

(1.34) The supremum over ψ is uniquely (up to additive constants) attained at ψ = ρ log g 2 with value H(g 2 ), as an application of Jensen’s inequality shows: 1 1 2 ρ log e ρ ψ(x) d x = ρ log d x g 2 (x) e ρ ψ(x)−log g (x) 1 ψ(x) − log g 2 (x) ≥ ρ d x g 2 (x) ρ = ψ, g 2 − H(g 2 ). Hence, χ (ρ) = χ (ρ) + ρ log ρe . Since gρ is, up to horizontal shifts, the unique minimiser ρ = ρ log gρ2 + C is the unique minimizer in (1.34). By the above reasoning, in (1.24), ψ ρ is the unique minimizer of (1.33), where C = −ρ log ρ is determined by requiring ψ e ρ ) = 1. that L(ψ

Remark 1.15 (Interpretation). There is an interpretation of the minimiser of (1.33) in terms of the optimal rescaled profile of the potential ξ for the almost-sure asymptotics of the total mass U (t). Indeed, the condition L(ψ) ≤ 1 guarantees that, almost surely for all large t, the profile ψ appears in some ‘microbox’ in the rescaled landscape ξ within the ‘macrobox’ Bt = [−t, t]d ∩ Zd , which is one of the intermittent islands. The logarithmic rate of the total mass, 1t log U (t) ≈ λ Bt (ξ ), can be bounded from below against the eigenvalue of ξ in the microbox, which is described by λ(ψ). Optimising over all admissible ψ explains the lower bound in (1.17). Our proof of the lower bound in Sect. 4 makes this heuristics precise. The Gaussian density gρ in (1.27) is the unique positive L 2 -normalized eigenfunction of + ψρ − ρ log ρe corresponding to the eigenvalue − χ (ρ) = λ(ψρ − ρ log ρe ).

322


It describes the rescaled shape of the solution u(t, ·) in the intermittent island. An interesting consequence is that the appropriately rescaled potential and solution shapes are identical for the moment asymptotics and for the almost sure asymptotics. This phenomenon also occurs in the cases of the double-exponential distribution and the potentials bounded from above. 3 1.6.3. Convergence of the variational problem in (1.13). We close this section by showing that the variational problem in (1.13) converges to the variational problem in (1.12) as γ ↑ 1. We define g 2γ (x) − g 2 (x) 2 χ (ρ, γ ) = inf |∇g(x)| d x + ρ d x , (1.35) 1−γ g∈H 1 (Rd ) Rd Rd g2 =1

which is equal to the variational problem in (1.13). Proposition 1.16 (Convergence of the variational problem in (1.35)). For any ρ ∈ (0, ∞), lim χ (ρ, γ ) = χ (ρ). γ ↑1

(1.36)

Proof. The upper bound in (1.36) follows by substituting the Gaussian density g = gρ in (1.27) into the infimum in (1.35), and by noting that lim

γ ↑1 Rd

2γ

gρ (x) − gρ2 (x) γ −1

dx =

Rd

gρ2 (x) log gρ2 (x) d x,

by an explicit computation of the integrals involved. For the lower bound in (1.36), we bound, for any γ ∈ [0, 1) and g ∈ H 1 (Rd ), 2 g 2γ (x) − g 2 (x) e(γ −1) log g (x) − 1 dx = dx g 2 (x) 1−γ 1−γ Rd Rd ≥− g 2 (x) log g 2 (x) d x, Rd

since eθ − 1 ≥ θ for every θ ∈ R. Therefore, χ (ρ, γ ) ≤ χ (ρ) for every γ ∈ [0, 1). The remainder of the paper is as follows. In Sect. 2 we present an important auxiliary result on self-intersections of random walks, which will be used in the proof of Theorem 1.4 in Sect. 3. The proof of Theorem 1.8 is given in Sect. 4. Finally, in Sect. 5 we use the opportunity to correct an error in the proof of the moment asymptotics in case (4) from [BK01]. 2. An Auxiliary Result on Self-Intersections of Random Walks In this section we provide a result on q-fold self-intersections of random walks, for small q > 1, which is an important tool in the proofof the upper bound in Theorem 1.4. This t result is of independent interest. Let t (z) = 0 δz (X (s)) ds denote the local time at z of the simple random walk (X (s) : s ∈ [0, t]) on Zd with generator d , starting at the origin.


323

Proposition 2.1. Fix q > 1 such that q(d − 2) < d and R > 0. Let α(t) → ∞ such that α(t) = O(t 2/(2d+2)−ε ) for some ε > 0. Then α(t)2 t θ↓0 t↑∞ 1 q − q1 [d+(2−d)q] q 1l{supp (t ) ⊆ B Rα(t) } = 0. × log E exp θ α(t) t (x)

lim sup lim sup

x∈Zd

(2.1) Remark 2.2. The result is better understood when rephrasing it in terms of the normalised and rescaled local times, L t (·) = 1t α(t)d t ( · α(t)). Then the exponent may be rewritten as  1 q 1 t − q [d+(2−d)q]  q α(t) t (x) = L t q , α(t)2 d x∈Z

where · q is the norm on L q (Rd ). Hence, (2.1) is a large deviations result for the qnorm of L t on the scale t/α(t)2 . It is known that (L t : t > 0) satisfies a large deviation principle on this scale in the weak topology generated by bounded continuous functions, see for example [GKS06]. However, (2.1) does not follow from a routine application of Varadhan’s lemma, since the q-norm is neither bounded nor continuous in this topology. See [Ch04] for an analogous result for a smoothed version of L t . 3 Remark 2.3. Our proof yields (2.1) also without indicator on {supp (t ) ⊆ B Rα(t) } if the sum is restricted to a finite subset of Zd . It can easily be extended to a large class of random walks, also in discrete time. The proof is based on a combinatorial analysis of the high integer moments of the random variable x t (x)q . This method is of crucial importance in the analysis of intersections and self-intersections of random paths [KM02], and of random walk in random scenery [GKS06]. 3 Proof of Proposition 2.1. By B we denote the box B = B Rα(t) = [−Rα(t), Rα(t)]d ∩ Zd . In the exponent on the left side of (2.1), we restrict the sum to x ∈ B and forget about the indicator on {supp (t ) ⊆ B Rα(t) }. In the following we write · q for the norm in q (B). In a first step we reduce the problem to a problem on asymptotics of high integer moments. Suppose first that there are constants T, C > 0 such that kq E t q ≤ k kq C kq α(t)k[d+(2−d)q] , for any t ≥ T, k ≥

t . α(t)2

(2.2)

We now show that this assumption implies (2.1). Expanding the exponential series, we rewrite − 1 [d+(2−d)q] − q1 [d+(2−d)q] k 1 θ α(t) E exp θ α(t) q t q = ∞ E t qk . k=0 k! Abbreviate kt = qt/α(t)2 . Under our assumption, k k k k [d+(2−d)q] C α(t) q , for t ≥ T, k ≥ kt , E t qk ≤ q

(2.3)

324


and hence we obtain − 1 [d+(2−d)q] E exp θ α(t) q t q ≤

k t −1 k=0

∞ 1 k 1 θCk k − k [d+(2−d)q] E t qk + . θ α(t) q k! k! q

(2.4)

k=kt

For all sufficiently small θ > 0, the second term is estimated as follows: k t θC ∞ ∞ k k eq θC 1 θCk ≤ = , θC k! q eq 1 − eq k=k k=k t

t

and the exponential rate (in tα(t)−2 ) of the right hand side tends to −∞ as θ ↓ 0. For the first term, we bound, using Hölder’s inequality and (2.3), for k ≤ kt , k k kt kt k t [d+(2−d)q] kt C kt α(t) q E t qk ≤ E t qkt kt ≤ q k k k t [d+(2−d)q] = C k α(t) q . q Therefore, the first term in (2.4) is bounded by kt θC 1 θCkt k k ≤ e q t. k! q k=0

This proves that (2.2) implies the statement (2.1). Therefore, it suffices to prove (2.2) with some constants C, T > 0. We use C to denote a generic constant which depends on R, d and q, but not on k and t, and C may change its value from appearance to appearance. To prove (2.2), we write Ak for the set of maps β : B → N0 satisfying x∈B βx = k. First we write out kq E t q = E t (x)q#{i : zi =x} z 1 ,...,z k ∈B

=

β∈Ak

= k!

x∈B

E t (x)qβx # z ∈ B k : βx = #{z i = x}∀x

β∈Ak

x∈B

1 . E t (x)qβx βx ! x∈B

(2.5)

x∈B

Note that, for β ∈ Ak , the numbers qβx are not necessarily integers. We resolve this problem, in an upper bound, by introducing a further sum over the set Ak (β) of all : B → N0 satisfying |β x − qβx | < 1 for every x ∈ B. Then, clearly, β $ $ % % x β qβx t (x) E t (x) E ≤ . (2.6) x∈B

∈Ak (β) β

x∈B


325

x . Writing out the local times, ∈ Ak (β) and denote We fix β ∈ Ak and β k = x∈B β we have  %  $ x β t

x . t (x)βx =  dsix  P X (six ) = x ∀x ∈ B ∀i = 1, . . . , β E x∈B i=1 0

x∈B

The next step is to give new names to the integration variables six such that we can order x for the time variables. Fix some function : {1, . . . , k} → B such that |−1 ({x})| = β any x ∈ B. We continue with, denoting the set of permutations of 1, . . . , k by Sk , E t (x)βx x∈B

=

k [0,t]

dt1 . . . dtk P{X (ti ) = (i)∀i = 1, . . . , k}

=

=

0 0.

(3.1)

(3.2)

0

By [GM90, Theorem 2.1] the random potential ξ is almost surely non-percolating from below. Hence, u ξ is the solution of the parabolic Anderson problem in (1.1) with initial


329

condition u(0, z) = 1 for all z ∈ Rd , and the main object of our study is U (t) = u ξ (t, 0). Introduce the vertically shifted random potential ξt (z) = ξ(z) − H

t α(t)d . α(t)d t

(3.3)

Note that t is a parameter here, and ξt should not be seen as a time-dependent random potential. Fix p ∈ (0, ∞). Then Theorem 1.4 is equivalent to the statement α( pt)2 log u ξ pt (t, 0) p = −χ (ρ), t↑∞ pt lim

(3.4)

where χ (ρ) is defined in (1.24). We approximate u ξ pt by finite-space versions. Let R > 0 and let B R = [−R, R]d ∩ Zd be the centred box in Zd with radius R. Introduce u VR : [0, ∞) × Zd → [0, ∞) by u R (t, z) = Ez exp V

t

V X (s) ds 1l supp (t ) ⊆ B R ,

(3.5)

0

t where t (z) = 0 δz (X (s)) ds are the local times of the random walk. Note that u rV ≤ u VR ≤ u V for 0 < r < R < ∞. In the finite space setting we can work easily with eigenfunction expansions: We look at the function

fory, z ∈ Zd , (3.6) p VR (t, y, z) = E y eV,t 1l supp (t ) ⊆ B R 1l X (t) = z and the eigenvalues, λ1 > λ2 ≥ λ3 ≥ · · · ≥ λn , of the operator d + V in 2 (B R ) with zero boundary condition, where we abbreviate n = |B R |. We may pick an orthonormal basis of corresponding eigenfunctions ek . By convention, ek vanishes outside B R . Note that z∈B R p VR (t, y, z) = u VR (t, y). Furthermore, we have the eigenfunction expansion p VR (t, y, z) =

etλk ek (y)ek (z).

(3.7)

k

In particular, u VR (t, z) =

etλk ek , 1lek (z).

(3.8)

k

The following proposition carries out the necessary large deviations arguments for the case p = 1, and is the key result for the proof of (3.4). Proposition 3.1. α(t)2 ξt (i) Let R > 0. Then lim sup (t, 0) ≤ −χ (ρ). log u Rα(t) t t↑∞ 2 α(t) ξt (ii) lim inf lim inf log u Rα(t) (t, 0) ≥ −χ (ρ). t↑∞ R↑∞ t The proofs of Proposition 3.1(i) and (ii) are deferred to Sect. 3.2 and 3.3, respectively.

330


3.1. Proof of (3.4) subject to Proposition 3.1. Proof of the lower bound in (3.4).. All we have to do is to show that, as t ↑ ∞, ξ −2 ξ pt u pt (t, 0) p ≥ eo(tα( pt) ) u Rα( pt) ( pt, 0) .

(3.9)

To prove this, we repeat the proof of [BK01, Lemmas 4.1 and 4.3] for the reader’s convenience. We abbreviate r = Rα( pt), V = ξ pt , u = u V , u r = u rV and pr = prV . Note −2 that |Br | = eo(tα( pt) ) . Now we prove (3.9). First we assume that p ∈ (0, 1). Use the

p shift invariance of the p distribution of the field V and the inequality i xi ≥ x for nonnegative xi to i i estimate 1 −2 u(t, z) p ≥ eo(tα( pt) ) u(t, z) p u(t, 0) p = |Br | z∈Br z∈Br p −2 (3.10) u(t, z) . ≥ eo(tα( pt) ) z∈Br

By · we denote the norm on 2 (Br ). According to Parseval’s identity, the numbers ek , 1l2 /1l2 sum up to one. Using u ≥ u r , the Fourier expansion in (3.8) and Jensen’s inequality, we obtain p 2 ek , 1l2 p o(tα( pt)−2 ) ptλk ek , 1l ≥ 1l2 p ≥ e u(t, z) etλk e 1l2 1l2 z∈Br k k −2 −2 u r ( pt, z) ≥ eo(tα( pt) ) u r ( pt, 0) . (3.11) ≥ eo(tα( pt) ) z∈Br

Substituting (3.11) in (3.10) completes the proof of (3.9) in the case p ∈ (0, 1). Now we turn to the case p ∈ [1, ∞). We use the first equation in (3.10), Jensen’s p inequality, the eigenfunction expansion in (3.8) and the inequality ( i xi ) p ≥ i xi for nonnegative xi to obtain p p 1 −2 ≥ eo(tα( pt) ) u(t, z) etλk ek , 1l2 u(t, 0) p ≥ |Br | z∈Br k o(tα( pt)−2 ) ptλk 2p (3.12) e ek , 1l . ≥e k

Next we use Jensen’s inequality as follows ptλ k e , 1l2 p k ke ptλk 2p ptλk = e ek , 1l e ptλk ke k k 2 3p ptλ 2 k ek , 1l ke ptλk ≥ e ptλk ke k 31− p 2 ptλ k ke = u r ( pt, z) ptλ ≥ u r ( pt, 0) . 2 k ek , 1l ke z∈B r

(3.13)


331

In the last step, we have used the eigenfunction expansions in (3.7) and (3.8) to see that e ptλk ek , 1l2 = p RV (t, y, z) ≥ p RV (t, z, z) = e ptλk . z

k

y

z

k

Combining (3.12) and (3.13) completes the proof of (3.9) also in the case p ∈ [1, ∞).

Proof of the upper bound in (3.4). A main ingredient in our proof is the following preparatory lemma, which provides, for any potential V , an estimate of u V (t, 0) in terms of the maximal principal eigenvalue of d + V in small subboxes (‘microboxes’) of a ‘macrobox’. For z ∈ Zd and R > 0, we denote by λdz;R (V ) the principal eigenvalue of the operator d + V with Dirichlet boundary conditions in the shifted box z + B R . Lemma 3.2. Let r : (0, ∞) → (0, ∞) such that r (t)/t ↑ ∞. For R, t > 0 let B R (t) = Br (t)+2R . Then there is a constant C > 0 such that, for any sufficiently large R, t and any potential V : Zd → [−∞, ∞), 1/2 −r (t) t u V (t, 0) ≤ E e2 0 V (X s ) ds e

2 d + eCt/R 3r (t) exp t max λdz;2R (V ) . (3.14) z∈B R (t)

Proof. This is a modification of the proof of [BK01, Proposition 4.4], which refers to nonpositive potentials V only. The proof of [BK01, Proposition 4.4] consists of [BK01, Lemma 4.5] and [BK01, Lemma 4.6]. The latter states that

d V Ct/R 2 d u r (t) (t, 0) ≤ e 3r (t) exp t max λz;2R (V ) . (3.15) z∈B R (t)

A careful inspection of the proof shows that no use is made of nonpositivity of V and hence (3.15) applies in the present setting. In order to estimate u V (t, 0) − u rV(t) (t, 0), we introduce the exit time τ R = inf{t > 0 : X (t) ∈ / B R } from the box B R and use the Cauchy-Schwarz inequality to obtain   t u V (t, 0) − u V (t, 0) = E exp V (X (s)) ds 1l{τr (t) ≤ t} r (t)

0

t 1/2 ≤ E e2 0 V (X s ) ds P{τr (t) ≤ t}1/2 . According to [GM98, Lemma 2.5(a)], for any r > 0, r −1 . P{τr ≤ t} ≤ 2d+1 exp − r log dt Hence, we may estimate P{τr (t) ≤ t}1/2 ≤ e−r (t) , for sufficiently large t, completing the proof.

We now complete the proof of the upper bound in (3.4), subject to Proposition 3.1. Let p ∈ (0, ∞) and fix R > 0. First, notice that the second term in (3.14) can be estimated in terms of a sum, ptλd (V ) d exp pt max λz;2R (V ) ≤ e z;2R . (3.16) z∈B R (t)

z∈B R (t)

332


Thus, applying (3.14) to u ξ pt (t, 0) with R replaced by Rα( pt), raising both sides to the pth power, and using (x + y) p ≤ 2 p (x p + y p ) for x, y ≥ 0, together with (3.16), we get t p/2 − pr (t) u ξ pt (t, 0) p ≤ 2 p E e2 0 ξ pt (X (s)) ds e

pd 2 2 ptλd (ξ ) + eC pt/(R α( pt) ) 3r (t) e z;2Rα( pt) pt . z∈B Rα( pt) (t)

Next we take the expectation with respect to ξ and note that, by the shift-invariance of ξ , the distribution of λdz;2Rα( pt) (ξ ) does not depend on z ∈ Zd . This gives t p/2 − pr (t) e u ξ pt (t, 0) p ≤ 2 p E e2 0 ξt p (X (s)) ds

2 2 pd+d ptλd0;2Rα( pt) (ξ pt ) + eC pt/(R α( pt) ) 3r (t) e .

(3.17)

In order to show that the first term on the right is negligible, estimate, in the case p ≥ 2, with the help of Jensen’s inequality and Fubini’s theorem, t 1 t p/2 E e2 0 ξt p (X (s)) ds ≤ E et 0

H ( ptα( pt)−d ) exp − α( pt)−d 1 t H ( ptα( pt)−d ) ≤E e ptξ(X (s)) ds exp − t 0 α( pt)−d H ( ptα( pt)−d ) . = e H ( pt) exp − α( pt)−d ptξ(X (s)) ds

In the case p < 2, a similar calculation shows that t H ( ptα( pt)−d ) p/2 p E e2 0 ξt p (X (s)) ds ≤ e 2 H (2t) exp − . α( pt)−d Hence, for the choice r (t) = t 2 , the first term on the right hand side of (3.17) satisfies lim sup t↑∞

t p/2 − pr (t) α( pt)2 log E e2 0 ξt p (X (s)) ds e = −∞, pt

(3.18)

where we use that H (t)/t and α(t) are slowly varying. In (3.17), take the logarithm, multiply by α( pt)2 /( pt) and let t ↑ ∞. Then we have that lim sup t↑∞

≤

α( pt)2 log u ξ pt (t, 0) p pt

α( pt)2 C log exp{ ptλd0;2Rα( pt) (ξ pt )} , + lim sup R2 pt t↑∞ −2

(3.19)

where we also used that r (t) pd+d = eo(tα( pt) ) as t ↑ ∞. Now we estimate the right th d hand side of (3.19). We denote by λd,k 0;Rα( pt) (ξ pt ) the k eigenvalue of + ξ pt in the box


333

B Rα( pt) with zero boundary condition. Using an eigenfunction expansion as in (3.7), we get ξ pt

p Rα( pt) ( pt, x, x) exp ptλd0;Rα( pt) (ξ pt ) ≤ exp ptλd,k (ξ ) = 0;Rα( pt) pt k

≤

x∈B Rα( pt)

ξ pt u Rα( pt) ( pt, x)

(3.20)

x∈B Rα( pt)

≤

pt Ex e 0 ξ pt (X (s)) ds 1l{supp ( pt ) ⊆ x + B2Rα( pt) }

x∈B Rα( pt)

ξ pt ≤ |B Rα( pt) | u 2Rα( pt) ( pt, 0) , −2

where we also used the shift-invariance. Recall that |B Rα( pt) | ≤ eo(tα( pt) ) . We finally use Proposition 3.1(i) for pt instead of t to complete the proof of the upper bound in (3.4).

t 3.2. Proof of Proposition 3.1(i). Recall the local times of the walk, t (z) = 0 1l{X (s) = t z} ds. Note that 0 V (X (s)) ds = V, t , where · , · stands for the inner product on 2 (Zd ). From (3.5) with V = ξt , we have ξt (3.21) u Rα(t) (t, 0) = E0 eξt ,t 1l{supp (t ) ⊆ B Rα(t) } . Recall from (1.5) that elξ(x) = e H (l) for any l ∈ R and x ∈ Zd . We carry out the expectation with respect to the potential, and obtain, using Fubini’s theorem and the independence of the potential variables, ξt u Rα(t) (t, 0) d d = e−α(t) H (t/α(t) ) E0 exp t (x)ξ(x) 1l{supp (t ) ⊆ B Rα(t) } (3.22) = E0 exp

x∈B Rα(t) d H (t (x)) − t (x) α(t) t

H (t/α(t) ) d

1l{supp (t ) ⊆ B Rα(t) } ,

x∈B Rα(t)

where we also use that x∈Zd t (x) = t. We now split the sum in the exponent into a part where we have some control over the size of the local times, and a part with very large local times. Introducing d α(t)2 H(t) H (t (x)) − t (x) α(t) H (t/α(t)d ) M (t ) = t t x∈B Rα(t)

Mt 1l{t (x) ≤ α(t) d }, d d R(t) H (t (x)) − t (x) α(t) H (t/α(t) ) 1l{t (x) > M (t ) = t x∈B Rα(t)

we have ξt u Rα(t) (t, 0) = E0 exp

(3.23) Mt }, α(t)d

t (t) (t) H ( ) + R ( ) 1 l{supp ( ) ⊆ B } t t t Rα(t) . M α(t)2 M

(3.24)

334


(t) We will see that H(t) M gives the main term and R M a small remainder in the limit t → ∞, followed by M → ∞. To separate the two factors coming from this split, we use Hölder’s inequality. For any small η > 0, we have

ξt u Rα(t) (t, 0) ≤ E0 exp (1 + η)

1 t 1+η (t) H ( ) 1 l{supp ( ) ⊆ B } t t Rα(t) α(t)2 M η 1+η 1+η (t) ×E0 exp η R M (t ) 1l{supp (t ) ⊆ B Rα(t) } . (3.25)

We show later that the second factor is asymptotically negligible, more precisely, we show that α(t)2 lim sup lim sup t M→∞ t→∞ × log E0 exp CR(t) M (t ) 1l{supp (t ) ⊆ B Rα(t) } ≤ 0, for C > 0.

(3.26)

Let us first focus on the first term. Recall the definition of α(t) in (1.10) and the uniform convergence claimed in Proposition 1.1. For every ε > 0 and all sufficiently large times t, we obtain the upper bound H(t) M (t ) ≤ ≤

α(t)2 t t κ( α(t)d ) ρ

t (x) t/α(t)d

log

t (x)

1l{t (x) t/α(t)d

x∈B Rα(t)

2 Mt t } + ε (2R)d α(t)d α(t) t κ( α(t)d ) α(t)d

≤ρ

d 1 1 t t (x) log t t (x) α(t)

+ ε (2R)d

x∈B Rα(t)

= G t ( 1t t ) + ε (2R)d ,

(3.27)

where we dropped the indicator, which we can do for M ≥ 1 since y log y ≥ 0 for y > M, and let G t (µ) = ρ

µ(x) log α(t)d µ(x) , for µ ∈ M(Zd ).

(3.28)

x∈B Rα(t)

The further analysis makes crucial use of an inequality derived in [BHK05]. In [BHK05], the law of the local times are investigated, and an explicit formula is derived for the density of the local times on the range of the random walk. This explicit formula makes it possible to give strong upper bounds on exponential functionals: Proposition 3.3. For any finite set B ⊆ Zd and any measurable functional F : M1 (B) → R, 1 E0 et F( t t ) 1l{supp (t ) ⊆ B} 4 4

2 F(µ) − 21 (2dt)|B| |B|. (3.29) ≤ exp t sup µ(x) − µ(y) µ∈M1 (B)

x∼y


335

We substitute (3.27) into (3.25) and apply (3.29) for F = (1+η)G t /α(t)2 and B = B Rα(t) 2 and note that (2dt)|B Rα(t) | ≤ eo(t/α(t) ) . Hence, we obtain that the first term on the right hand side of (3.25) can be estimated by t (t) H ( ) 1 l{supp ( ) ⊆ B } E0 exp (1 + η) t t Rα(t) M α(t)2

2 ρ

1 ρ log α(t) + ε (2R)d , (3.30) ≤ eo(t/α(t) ) exp − t χ d α(t) 2 − α(t)2 d where we abbreviated ρ = (1 + η)ρ and introduced 4 4

2 1 χ d (δ) = inf µ(x) − µ(y) − δ µ(x) log µ(x) , 2 µ∈M1 (Zd )

for δ > 0,

x∈Zd

x∼y

(3.31) the discrete variant of χ (ρ) in (1.24), which was studied in Gärtner and den Hollander [GH99]. In Proposition 3 and the subsequent remark they show that χ d (δ) =

π e2 dδ log + o(1) , as δ ↓ 0. 2 δ

Substituting this into (3.30), we obtain t α(t)2 H(t) log E0 exp (1 + η) M (t ) 1l{supp (t ) ⊆ B Rα(t) } 2 t α(t) t→∞ 2 πe ρ d log + ε(2R)d = −χ ( ρ ) + ε(2R)d , (3.32) ≤− 2 ρ

lim sup

as can be seen from Proposition 1.12. Using (3.32) together with (3.26) in (3.25) and letting M → ∞, ε ↓ 0 and η ↓ 0, gives the desired upper bound and finishes the proof of Proposition 3.1(i) subject to the proof of (3.26). It remains to investigate the second term in (3.25), i.e., to prove (3.26). We first estimate R(t) M (t ) (recall (3.24)) from above in terms of a nice functional of t . Since we have to work uniformly for arbitrarily large local times, it is not possible to estimate against a functional of the form x t (x) log t (x), but we succeed in finding an upper 1/q q (x) for some q > 1 close to 1. Then Proposition 2.1 can bound of the form x t be applied and yields (3.26). We fix δ ∈ (0, 21 ] and note that there exist A > 1, t0 > 0 such that H (t y) − y H (t) 2 ≤ Ay 1+δ /3 for any y ≥ 1 and t > t0 . κ(t)

(3.33)

Indeed, this follows from [BGT87, Theorem 3.8.6(a)]. Therefore, we obtain that −d t H (t (x)) − t (x) α(t) t H ( α(t)d ) ≤ Aκ(tα(t) ) d

(x) 1+δ 2 /3 t . tα(t)−d

(3.34)

We pick now ε > 0 such that 1 + δ 2 /3 − ε = 1/(1 + δ).

(3.35)

336


which implies that ε < δ. For any µ ∈ M1 (Zd ), we use Jensen’s inequality together with (3.35) as follows: µ(x)ε 2 1+δ 2 /3 ε µ(x) = µ(x) µ(x)1+δ /3−ε ε x : µ(x)>M µ(x) x : µ(x)>M

x : µ(x)>M

≤

x : µ(x)>M

µ(x)

x : µ(x)>M

≤

≤M

µ(x)

M

x : µ(x)>M

x : µ(x)>M δ 1+δ (ε−1)

ε

ε−δ 1+δ

1 1− 1+δ

ε

µ(x)1+ε ε x : µ(x)>M µ(x) µ(x)

1+ε

x : µ(x)>M

µ(x)

1+δ

1 1+δ

=M

1 δ−ε 1+δ µ(x) (3.36) M

2δ ε− 1+δ

x

1 1+δ

µ(x)

1 1+δ

1+δ

,

x

where we used in the last step that in the first integral on the right, µε ≤ M ε−1 µ on {µ > M}, and hence the first term on the right is not bigger than one, as the exponent is positive and µ ∈ M1 (Zd ). We write q = 1 + δ. We apply the above to µ = 1t t and M M replaced by α(t) d , to obtain that t (x) 1+δ 2 /3 1l{t (x) > tα(t)−d x

Mt } α(t)d

≤ α(t)d(1+δ

2 /3)

2δ M ε− 1+δ −1 t t q d α(t)

2δ

δ

= M ε− 1+δ α(t)d(1+ 1+δ ) t −1 t q .

(3.37)

We recall (3.24), use (3.34) and the definition of α(t) in (1.10). With the help of (3.37) we arrive at t (x) 1+δ 2 /3 t Mt R(t) ( ) ≤ A κ 1l t (x) > α(t) t M d −d α(t)d tα(t) x 2δ

δ

2δ

− q1 [d+(2−d)q]

≤ AM ε− 1+δ α(t)−(2+d)+d(1+ 1+δ ) t q = AM ε− 1+δ α(t)

t q , δ = − q1 [d + (2 − d)q]. where we recall that q = 1 + δ and therefore −(2 + d) + d 1 + 1+δ 2δ

Put θ = AM ε− 1+δ , and observe that θ ↓ 0 as M ↑ ∞ for δ > 0 small enough, since 2 2δ δ ε − 1+δ = δ3 − 1+δ < 0 for δ > 0 small enough. Hence, (3.26) follows immediately from Proposition 2.1. This completes the proof of Proposition 3.1(i). 3.3. Proof of Proposition 3.1(ii). Recall from (1.21) the rescaled version, ξ t , of the vertically shifted potential, ξt , defined in (3.3). Furthermore, introduce the normalised, scaled version of the random walk local times, L t (x) :=

α(t)d t xα(t) , t

for x ∈ Rd ,


337

and note that L t is an L 1 -normalised random step function. Note that supp (L t ) ⊆ Q R if supp (t ) ⊆ B Rα(t) , where we abbreviated Q R = [−R, R]d . We start from (3.22). Let t

t

H y α(t) d − y H α(t)d t (y) = t

, for t, y > 0, H κ α(t) d , uniformly on all compact sets. Now the exponent on t converges to H and recall that H the right hand side of (3.22) can be rewritten as follows. t

H (t (z)) −α(t)d H α(t) d +

z∈B Rα(t)

t

d x + α(t) L t (x) H H α(t) d L t (x) d x QR QR

t t L t (x) d x = t 2 H(t) = α(t)d κ α(t) H R (L t ), d α(t)

= −α(t)

d

t α(t)d

d

QR

where we use the definition of α(t) in (1.10) and introduce the functional

t f (x) d x. H(t) ( f ) = H R QR

Hence, ξt (t, 0) u Rα(t)

= E0

t (t) H (L t ) 1l{supp (L t ) ⊆ Q R } . exp α(t)2 R

(3.38)

A key ingredient in the proof of Proposition 3.1(ii) is the large deviation principle for (L t : t > 0) as formulated in the following proposition: Proposition 3.4. Fix R > 0. Under P0 { · , supp (L t ) ⊆ Q R }, the rescaled local times process (L t : t > 0) satisfies a large deviation principle as t ↑ ∞ on the set of L 1 -normalized functions Q R → R, equipped with the weak topology induced by test integrals against all continuous functions, where the speed of the large deviation principle is tα(t)−2 , and the rate function is g 2 → ∇g22 , on the set of all g ∈ H 1 (Rd ) with supp (g) ⊆ Q R , and is equal to ∞ outside this set. Proof. This large deviation principle is stated in [GKS06, Lemma 3.2] in the discretetime case, and is proved in [GKS06, Sect. 6]. The proof in the continuous-time case is very similar, we briefly sketch the argument. Let f : Q R → R be continuous. The core of the argument is to show that

α(t)2 log E0 exp tα(t)−2 f, L t 1l{supp (L t ) ⊆ Q R } = λ R ( f ) , (3.39) t↑∞ t lim

where λ R ( f ) is the principal eigenvalue of + f in H01 (Q R ), see also (4.12). The rest of the argument is an application of the Gärtner-Ellis theorem, see [GKS06, Sect. 6] for details. To show (3.39), consider the discrete approximation

d f t (z) = α(t) for z ∈ Zd . f x + zα(t)−1 d x , [0,α(t)−1 )d

338


Then t 1 f, L t = α(t)2 α(t)2

t

t/α(t)2

f t (X (s)) ds =

f t X (sα(t)2 ) ds .

0

0

Denoting by µt the normalised occupation measure of a Brownian motion {B(s) : s ≥ 0}, an application of the local functional central limit theorem yields that

E0 exp tα(t)−2 f, L t 1l{supp (L t ) ⊆ Q R } t/α(t)2

2 f t α(t) B(s) ds 1l{supp (µt/α(t)2 ) ⊆ Q R } eo(t/α(t) ) . = E0 exp 0

Since f t (α(t) · ) → f ( · ) uniformly on Q R , (3.39) follows from E exp

T

f B(s) ds 1l{supp (µT ) ⊆ Q R } = exp T λ R ( f ) + o(T ) , for T ↑ ∞,

0

see, e.g. [S98, Theorem 3.1.2], with T = tα(t)−2 .

In order to apply the large deviation principle in Proposition 3.4 to obtain a lower bound for the right hand side of (3.38), we need the lower-bound half of Varadhan’s lemma, and we have to replace H(t) R by its limiting version HR ( f ) = ρ

f (x) log f (x) d x.

(3.40)

QR

However, the latter is technically not so easy. Inserting the indicator on the event {L t ∞ < M} for any M > 1 would make it possible to use the locally uniform cont (y) towards ρy log y, but this event is not open in the topology of the vergence of H large deviation principle. Therefore, similarly to the proof of the upper bound, we have to split H(t) R (L t ) into the sum of H R (L t ) and a remainder term, separate these two from each other by the use of Hölder’s inequality and apply Proposition 2.1 to the remainder term. Let us turn to the details. Since H is convex with H (0) = 0, we have H (yt) ≥ y H (t) for all t > 0 and all t ( f (x)) ≥ 0 on {x : f (x) > M} for any M > 1. Hence, we may y ≥ 1. Therefore, H estimate H(t) R (f) ≥

t f (x) d x 1l{ f (x) ≤ M} H QR

=ρ

1l{ f (x) ≤ M} f (x) log f (x) d x + o(1) = HR ( f ) − ρ 1l{ f (x) > M} f (x) log f (x) d x + o(1). QR

QR


339

The remainder can be estimated, for any δ > 0, as follows. For any f : Q R → [0, ∞) satisfying f = 1, 1+δ/2 f 2 2 f >M f δ/2 log f ≤ f log f = f f log δ f >M δ f >M f >M f >M f >M f f >M f M −δ/2 f >M f 1+δ 2 f log ≤ δ f >M f >M f 1 1+δ 1+δ M −δ/2 2 f >M f ≤ f δ f >M f >M f δ δ δ 2 2 1+δ = M − 2+2δ f f q ≤ M − 2+2δ f q , δ δ f >M δ

where we put q = 1 + δ. Altogether, we have, abbreviating θ = 2 ρδ M − 2+2δ , t ξt

2 u Rα(t) (t, 0) ≥ E0 exp H R (L t ) − θ L t q 1l{supp (L t ) ⊆ Q R } eo(t/α(t) ) . 2 α(t) (3.41) Similarly to the proof of the upper bound, the main contribution will turn out to come from H R , and the q-norm is a small remainder. In order to separate the two from each other, we use Hölder’s inequality to estimate, for some small η > 0, t E0 exp (1 − η)H (L ) 1 l{supp (L ) ⊆ Q } R t t R α(t)2 1−η t

H R (L t ) − θ L t q 1l{supp (L t ) ⊆ Q R } ≤ E0 exp α(t)2 η t 1−η θ L t q 1l{supp (L t ) ⊆ Q R } . ×E0 exp (3.42) α(t)2 η This effectively yields a lower bound on the expected value in (3.41) of the form

ξt u Rα(t) (t, 0) ≥ E0 exp

1 1−η t (1 − η)H (L ) 1 l{supp (L ) ⊆ Q } R t t R 2 α(t) − η t 1−η 1−η 2 θ L t q 1l{supp (L t ) ⊆ Q R } ×E0 exp eo(t/α(t) ) . 2 α(t) η (3.43)

From Proposition 2.1 it follows that the second expectation on the right is negligible in the limit t → ∞, followed by M → ∞, i.e., θ ↓ 0. Hence, we can concentrate on the first term. To apply the lower-bound half of Varadhan’s lemma, see [DZ98, Lemma 4.3.4], we need the following lower semi-continuity property of the function H R : Lemma 3.5. Let f : Q R → [0, ∞) be continuous. Then H R is lower semi-continuous in f in the topology induced by pairing with all continuous functions Q R → [0, ∞).

340


Proof. Let ( f n : n ∈ N) be a family in L 1 (Q R ) such that f n , ψ → f, ψ as n → ∞ for any continuous function ψ : Q R → R. We have to show that lim inf n→∞ H R ( f n ) ≥ H R ( f ). For any s ∈ (0, ∞) we denote by gs the tangent to y → φ(y) := ρy log y in s, i.e., gs (y) = ρ(1 + log s)y − ρs, for all y ∈ R. By convexity we have gs ≤ φ for any s ∈ (0, ∞). Therefore, for any 0 < ε < 1/e,

φ f n (x) d x ≥ g f (x)∨ε f n (x) d x HR ( fn ) = QR QR = ρ 1 + log( f ∨ ε), f n − ρ f ∨ ε, f n . Letting n → ∞, we obtain, using the boundedness and continuity of log( f ∨ ε), lim inf H R ( f n ) ≥ ρ 1 + log( f ∨ ε), f − ρ f ∨ ε, f n→∞ f (x) log f (x)1l{ f (x)>ε} d x + gε ( f (x))1l{ f (x)≤ε} d x ≥ρ Q QR R

≥ρ f (x) log f (x) d x + f (x)(1 + log ε) − ε 1l{ f (x)≤ε} d x. QR

QR

The second summand is bounded from below by Leb(Q R )ε log ε, which converges to zero as ε ↓ 0. This completes the proof.

Now we can apply [DZ98, Lemma 4.3.4] and obtain t α(t)2 lim inf log E0 exp (1 − η)H R (L t ) 1l{supp (L t ) ⊆ Q R } t→∞ t α(t)2 ≥ − inf ∇g22 −(1−η)H R (g 2 ) : g ∈ H 1 (Rd ) ∩ C(Rd ), g2 = 1, supp (g) ⊆ Q R . Letting η ↓ 0 and R ↑ ∞, it is easy to see that the right hand side tends to −χ (ρ) defined (R) in (1.24). Indeed, use appropriate continuous cut-off versions g(1−η)ρ of the minimiser g(1−η)ρ in (1.27) to verify this claim. Using this on the right hand side of (3.43) and recalling Proposition 2.1, we see that the proof of the lower bound in Proposition 3.1(ii) is finished. 4. The Almost-Sure Asymptotics: Proof of Theorem 1.8 We again derive upper and lower bounds, following the strategy in [BK01, Sect. 5]. Recall the scale function β(t) defined in (1.16) and let β(t) α(β(t))d ξβ(t) (z) = ξ(z) − H (4.1) α(β(t))d β(t) denote the appropriately vertically shifted potential (compare to (3.3)). Then Theorem 1.8 is equivalent to the assertion α(β(t))2 log u ξβ(t) (t, 0) = − χ (ρ), almost surely, (4.2) t↑∞ t where χ (ρ) = ρ d − d2 πρ + log ρe = − sup{λ(ψ) : ψ ∈ C(Rd ), L(ψ) ≤ 1}, see Sect. 1.6.2. lim


341

4.1. Proof of the upper bound in (4.2). Let r (t) = t log t and apply Lemma 3.2 with V = ξβ(t) and with R replaced by Rα(β(t)). Furthermore, take logarithms, multiply with α(β(t))2 /t and let t ↑ ∞. As in (3.18), one shows that the first term is negligible. Hence, we obtain that lim sup t↑∞

α(β(t))2 C log u ξβ(t) (t, 0) ≤ 2 + lim sup α(β(t))2 max λz;2Rα(β(t)) (ξβ(t) ) , z∈B(t) t R t↑∞

where B(t) = B Rα(β(t)) (t) (recall the definition B R (t) = Br (t)+2R from Lemma 3.2). Let (λi (t) : i = 1, . . . , N (t)), with N (t) = |B R (t)|, be a deterministic enumeration of the random variables λz;2Rα(β(t)) (ξβ(t) ) with z ∈ B(t). Note that these random variables are identically distributed (but not independent) and that, by (3.20) and Proposition 3.1(i), their exponential moments are estimated by lim sup t↑∞

α(β(t))2 log eβ(t) λ1 (t) ≤ −χ (ρ). β(t)

(4.3)

We next show that, for any ε > 0, almost surely, N (t)

lim sup α(β(t))2 max λi (t) ≤ − χ (ρ) + ε, i=1

t↑∞

(4.4)

which completes the proof of the upper bound in (4.2). To prove (4.4), one first realizes that it suffices to show (4.4) only for t ∈ {en : n ∈ N}, since the functions t → α(t), t → β(t), and t → H (t)/t are slowly varying, and t → N (t), R → λ R (ξβ(t) ) are increasing. Let

− χ (ρ) + ε pn = Prob max λi (e ) ≥ . i=1 α(β(en ))2 N (en )

n

We recall that β(en )α(β(en ))−2 ∼ dn. Using Chebyshev’s inequality and (4.3), we estimate, for any k > 0, n n n n −2 χ (ρ)−ε) pn ≤ N (en )Prob ekβ(e )λ1 (e ) ≥ e−kβ(e )α(β(e )) ( χ (ρ)−ε) kβ(en )λ1 (en ) ≤ en(d+o(1)) enkd( e .

(4.5)

In order to evaluate the last expectation, we intend to apply (4.3) with β(t) replaced by kβ(t). For this purpose, we note that we can replace α(β(t)) by α(kβ(t)) in (4.3), since α is slowly varying. Also, kβ(t)λ Rα(kβ(t)) (ξβ(t) ) = kβ(t)λ Rα(kβ(t)) (ξkβ(t) ) − kβ(t)

H (β(t)α(β(t))−d ) H (kβ(t)α(kβ(t))−d ) , − β(t)α(β(t))−d kβ(t)α(kβ(t))−d

342


where we use that by (4.1), the field ξβ(t) − ξkβ(t) is constant and deterministic. Now we use (1.6) and (1.10), to see that the deterministic term is equal to H (β(t)α(β(t))−d ) H (kβ(t)α(kβ(t))−d ) kβ(t) − β(t)α(β(t))−d kβ(t)α(kβ(t))−d = α(β(t))d k H (β(t)α(β(t))−d ) − H (kβ(t)α(β(t))−d ) + o(n)

(k) + o(1) κ β(t)α(β(t))−d + o(n) = −α(β(t))d H

β(t) ρk log k + o(1) (1 + o(1)) + o(n) =− 2 α(β(t)) = −nd (ρk log k)(1 + o(1)). Hence,

ekβ(e

n )λ (en ) 1

≤ exp − nd kχ (ρ) − ρk log k + o(1) .

Using this in (4.5), we arrive at

pn ≤ exp nd 1 + k( χ (ρ) − ε) − kχ (ρ) + ρk log k + o(1) . Choosing k = ρ1 , we see that pn ≤ e−nd(kε+o(1)) . This is summable over n ∈ N, and the Borel-Cantelli lemma yields that (4.4) holds almost surely. This completes the proof of the upper bound in (4.2). 4.2. Proof of the lower bound in (4.2). Our proof of the lower bound in (4.2) follows the strategy of [BK01, Sect. 5.2]. First we establish that, with probability one, for any sufficiently large t, there is, inside a ‘macrobox’ of radius roughly t, centred at the origin, some ‘microbox’ of radius Rα(β(t)) in which the random field ξβ(t) has some shape with optimal spectral properties. Then we obtain a lower bound for the Feynman-Kac formula in (3.2) by requiring that the random walk moves quickly to that box and stays there for approximately t time units. As a result, the contribution from that strategy is basically given by the largest eigenvalue of d + ξ in that microbox. Rescaling and letting R ↑ ∞, the lower bound is derived from this. Let us go to the details. We pick an increasing auxiliary scale function t → γt satisfying γt = t 1−o(1) , t − γt = t (1 + o(1)), H (β(t)α(β(t))−d ) t t γt = o , γ . = o t α(β(t))2 β(t)α(β(t))−d α(β(t))2

(4.6)

(Note that the second requirement follows from the third.) For example, γt = tα(β(t))−2 εt with some suitable εt ↓ 0 as a small inverse power of log t satisfies (4.6). This is obvious in the case where lims↑∞ H (s)/s = 0, and in the case where lims↑∞ H (s)/s = ∞, it is also clear since H (s)/s diverges only subpolynomially in s, while β(t) = (log t)1+o(1) and α is slowly varying. The crucial step is to show that, in the ‘macrobox’ Bγt , we find an appropriate ‘microbox’. To fix some notation, let Q R = [−R, R]d and let C(Q R ) denote the set of


343

continuous functions Q R → R. We need finite-space versions of the functionals H, L and λ defined in (1.25) and (1.26). Recall the definition of H R from (3.40) and define its Legendre transform L R : C(Q R ) → (−∞, ∞] by

L R (ψ) = sup f, ψ − H R ( f ) : f ∈ C(Q R ), f ≥ 0, supp f ⊆ supp ψ . (4.7) As in the proof of Proposition 1.12 one can see that f = eψ/ρ−1 is the unique maximizer in (4.7) with ρ eψ(x)/ρ d x. L R (ψ) = e QR Proposition 4.1 (Existence of an optimal microbox). Fix R > 0 and let ψ ∈ C(Q R ) satisfy L R (ψ) < 1. Let ε > 0. Then, with probability one, there exists t0 > 0, depending also on ξ , such that, for all t > t0 , there is yt ∈ Bγt , depending on ξ , such that z ε 1 − ψ , for z ∈ B Rα(β(t)) . (4.8) ξβ(t) (yt + z) ≥ 2 α(β(t)) α(β(t)) α(β(t))2 The proof of Proposition 4.1 is deferred to the end of this section. Now we finish the proof of the lower bound in (4.2) subject to Proposition 4.1. Let R, ε > 0, and let ψ ∈ C(Q R ) be twice continuously differentiable with L R (ψ) < 1. Fix ξ not belonging to the exceptional set of Proposition 4.1, i.e., let t0 and (yt : t > t0 ) in Bγt be chosen such that (4.8) holds for every t > t0 . Fix t > t0 . In the Feynman-Kac formula t u ξβ(t) (t, 0) = E0 exp ξβ(t) (X (s)) ds , 0

we obtain a lower bound by requiring that the random walk is at yt at time γt and remains within the microbox B yt ,t = yt + B Rα(β(t)) during the time interval [γt , t]. Using the Markov property at time γt , we obtain by this the lower bound γt ξβ(t) (X (s)) ds δ yt (X (γt )) u ξβ(t) (t, 0) ≥ E0 exp 0 t−γt × E yt exp ξβ(t) (X (s)) ds 1l{τ yt ,t > t − γt } , (4.9) 0

/ B yt ,t } denotes the exit time from the microbox B yt ,t . where τ yt ,t = inf{s > 0 : X (s) ∈ In the first expectation on the right side of (4.9), we estimate ξ from below by its minimum K = essinf ξ(0) > −∞, and in the second expectation we use (4.8) and shift spatially by yt to obtain H (β(t)α(β(t))−d ) ξβ(t) P0 {X (γt ) = yt } u (t, 0) ≥ exp γt K − β(t)α(β(t))−d t−γt −2 ×e−ε(t−γt )α(β(t)) E0 exp ψt (X (s)) ds 1l{τ0,t > t − γt } , 0

(4.10)

344


where we have denoted ψt (·) = α(β(t))−2 ψ(· α(β(t))−1 ). By our choice in (4.6), the −2 first term on the right side of (4.10) is eo(tα(β(t)) ) . Now, by choosing a path from the origin to yt consisting of k steps for k = γt or k = γt + 1, k

1 P{σ (1) + · · · + σ (k) ≤ γt < σ (1) + · · · + σ (k + 1) , P0 X (γt ) = yt } ≥ 2d where σ (1), σ (2), . . . are independent exponential random variables with mean 1/2d. Using that

P σ (1) + · · · + σ (k) ≤ γt < σ (1) + · · · + σ (k + 1)

≥ P σ (1) + · · · + σ (k) ∈ [ γ2t , γt ) P σ (0) ≥ γ2t , and Cramér’s theorem, we obtain the lower bound P0 {X (γt ) = yt } ≥ e−O(γt ) = e−o(tα(β(t))

−2 )

.

By an eigenfunction expansion we have that t−γt E0 exp ψt (X (s)) ds 1l{τ0,t > t − γt } 0

t−γt ψt (X (s)) ds 1l{τ0,t > t − γt , X (t − γt ) = 0} ≥ E0 exp 0 ≥ exp (t − γt )λd (t) et (0)2 ,

where λd (t) is the principal eigenvalue of d + ψt in the box B Rα(β(t)) with zero boundary condition, and et is the corresponding positive 2 -normalized eigenvector. Putting together these estimates and recalling from (4.6) that t − γt = t (1 + o(1)), we obtain, almost surely, lim inf t↑∞

α(β(t))2 log u ξβ(t) (t, 0) ≥ −ε + lim inf α(β(t))2 λd (t) t↑∞ t α(β(t))2 log et (0)2 . + lim inf t↑∞ t

(4.11)

We now define the continuous counterpart λ R of λd (t), which is the finite-space version of the spectral radius defined in (1.26):

λ R (ψ) = sup ψ, g 2 − ∇g22 : g ∈ H 1 (Rd ), g2 = 1, supp g ⊆ Q R . (4.12) According to [BK01, Lemma 5.3], lim inf α(β(t))2 λd (t) ≥ λ R (ψ) t↑∞

and

lim inf t↑∞

α(β(t))2 log et (0)2 ≥ 0. t

Using this in (4.11), we obtain lim inf t↑∞

α(β(t))2 log u ξβ(t) (t, 0) ≥ −ε + λ R (ψ), t

(4.13)


345

for any ε > 0 and for any twice continuously differentiable function ψ ∈ C 2 (Q R ) satisfying L R (ψ) < 1. Hence, lim inf t↑∞

where

α(β(t))2 χR , log u ξβ(t) (t, 0) ≥ − t

χ R = inf −λ R (ψ) : ψ ∈ C 2 (Q R ) and L R (ψ) < 1 .

(4.14)

R ≤ χ (ρ). This can be It remains to show that, for any ρ > 0, we have lim sup R↑∞ χ seen as follows: By Proposition 1.14 the variational problem in (1.33) has a minimizer ψ ∗ , a parabola with L(ψ ∗ ) = 1. Pick ψ R = ε R + ψ ∗ | Q R , where ε R > 0 is chosen such that L R (ψ R ) = 1 − R1 . Obviously ε R ↓ 0. It is easy to show, using the explicit principal eigenfunction of + ψ ∗ that lim R→∞ λ R (ψ R ) = λ(ψ ∗ ). This completes the proof of the lower bound in (4.2) subject to Proposition 4.1. We finally prove Proposition 4.1: Proof of Proposition 4.1. This is very similar to the proof of [BK01, Prop. 5.1]. Recall that ψt (·) = α(β(t))−2 ψ(· α(β(t))−1 ). Consider the event 5 ε ξβ(t) (y + z) ≥ ψt (z) − , for y ∈ Zd . A(t) y = 2α(β(t))2 z∈B Rα(β(t))

Note that the distribution of A(t) y does not depend on y. Our first goal is to show that, for every ε > 0,

−d L R (ψ)−Cε+o(1) Prob A(t) , as t ↑ ∞, (4.15) 0 ≥t where C > 0 depends only on R and ψ, but not on ε. It is convenient to abbreviate st = β(t)α(β(t))−d .

(4.16)

Let f ∈ C(Q R ) be some positive auxiliary function (to be determined later), and consider the tilted probability measure Probt,z ( · ) = e ft (z)ξβ(t) (z) 1l{ξ(z) ∈ · } e−H ( f t (z))+ ft (z)H (st )/st , for z ∈ Zd , where f t (z) = st f (zα(β(t))−1 ) is the scaled version of f . The purpose of this tilting is to make the event A(t) 0 typical. We denote the expectation with respect to Probt,z by · t,z . Consider the event ε ε . ≥ ξβ(t) (z) − ψt (z) ≥ − Dt (z) = 2α(β(t))2 2α(β(t))2 6 Using that z∈B Rα(β(t)) Dt (z) ⊆ A(t) 0 and the left inequality in the definition of Dt (z), we obtain     H (s ) (t)

ε t H ( f t (z)) − f t (z) + ψt (z) + Prob A0 ≥ exp  st 2α(β(t))2  z∈B Rα(β(t))

Probt,z Dt (z) . (4.17) × z∈B Rα(β(t))

346


Since β(t)α(β(t))−2 = d log t, it is clear from a Riemann sum approximation that     ε − f t (z) ψt (z) + exp  2α(β(t))2  z∈B Rα(β(t))

= exp −

z ε 1 β(t) z ψ α(β(t))) + 2 f α(β(t))) 2 d α(β(t)) α(β(t)) z∈B Rα(β(t))

=t

−d f,ψ−d 2ε f,1l+o(1)

,

as t ↑ ∞.

(4.18)

We use the uniformity of the convergence in (1.6), the definitions (1.10) of α( · ) and (1.16) of β(t), and a Riemann sum approximation to obtain H ( f t (z)) − f t (z) Hs(st t ) z∈B Rα(β(t))

=

z

z H st f α(β(t)) − f α(β(t)) H (st )

z∈B Rα(β(t))

= κ(st )

ρf

z α(β(t))

log f

z α(β(t))

+ o(α ◦ β(t)d )

z∈B Rα(β(t))

= H R ( f ) + o(1) (1 + o(1))

β(t) = H R ( f ) d(log t) + o(1) . (4.19) 2 α(β(t))

Using (4.18) and (4.19) in (4.17), we arrive at

d H R ( f )− f,ψ− 2ε f,1l +o(1) ≥ t Prob A(t) 0

Probt,z Dt (z) ,

as t ↑ ∞.

z∈B Rα(β(t))

Recall from (4.7) that L R is the Legendre transform of H R . We choose f as the minimizer on the right of (4.7), i.e., such that H R ( f ) − f, ψ = −L R (ψ). Hence, to show that (4.15) holds, it is sufficient to show that

Probt,z Dt (z) ≥ t o(1) , as t ↑ ∞. (4.20) z∈B Rα(β(t))

To show this, note that

ε Probt,z Dt (z) = 1 − Probt,z ξβ(t) (z) > ψt (z) + 2α(β(t))2 ε . −Probt,z ξβ(t) (z) < ψt (z) − 2α(β(t))2

(4.21)

Since both terms are handled in the same way, we treat only the second term. For any a > 0 we use the exponential Chebyshev inequality to bound ε Probt,z ξβ(t) (z) < ψt (z) − 2α(β(t))2 ε ≤ e−H ( ft (z))+ ft (z)H (st )/st exp f t (z)ξβ(t) (z) + a ψt (z) − ξβ(t) (z) − 2α(β(t))2 = e H ( ft (z)−a)−H ( f t (z))+a H (st )/st ea[ψt (z)−ε/2α(β(t)) ] . 2


347

We pick a = δt f t (z) with some δt ↓ 0. Then the terms involving H can be treated similarly to (4.19). Indeed, abbreviating f = f (zα(β(t))−1 ), we obtain H (st ) st

= H st (1 − δt ) f − (1 − δt ) f H st − H st f − f H st

ρ + o(1) (1 − δt ) = κ(st )(1 + o(1)) H f −H f =− f , f δt d log t 1 + log d α(β(t))

H ( f t (z) − a) − H ( f t (z)) + a

where we also used the approximation log(1 − δt ) = −δt (1 + o(1)). Hence, we obtain Probt,z ξβ(t) (z) < ψt (z) − ≤ t δt α(β(t))

−d

ε 2α(β(t))2

−ε/2−ρ(1+log f [ψ f )](d+o(1))

,

as t ↑ ∞,

= ψ(zα(β(t))−1 ). Recall that we chose f optiwhere we recall (4.1) and abbreviate ψ mally in (4.7), which in particular means that log f (x) = ψ(x)/ρ − 1. Hence, for some C > 0, not depending on t nor on z, we have, for t > 1 large enough, Probt,z ξβ(t) (z) < ψt (z) −

ε 1 −d ≤ t −Cdεδt α(β(t)) ≤ . 2 2α(β(t)) 4

Going back to (4.21) and assuming that the first probability term satisfies the same bound, we have

|B

d | Probt,z Dt (z) ≥ 1 − 21 Rα(β(t)) = eC(Rα(β(t))) = eo(log t) = t o(1) ,

(4.22)

z∈B Rα(β(t))

where we use that α is slowly varying and β(t) = (log t)1−o(1) , so that α(β(t))d ≤ β(t)dη = o(log t) for t → ∞. This proves (4.20), and therefore (4.15). We finally complete the proof of Proposition 4.1. As in the proof of [BK01, Prop. 5.1] it suffices to prove the almost sure existence of a (random) n 0 ∈ N such that, for any n+1 n ≥ n 0 , there is a yn ∈ Bγen such that the event A(eyn ) occurs. In the following, we abbreviate t = en . Let Mt = Bγt ∩ 3Rα(β(et))Zd . Note that |Mt | ≥ t d−o(1) as t ↑ ∞ and that the events A(et) y with y ∈ Mt are independent. It suffices to show the summability of (et) (et) 1 pt = Prob 1l{A y } ≤ 2 |Mt |Prob(A0 ) y∈Mt

on t ∈ eN . Indeed, since, by (4.15), d−d L R (ψ)−Cε−o(1) |Mt |Prob(A(et) 0 )≥t

(4.23)

tends to infinity if ε > 0 is small enough (recall that L R (ψ) < 1), the summability ensures, via the Borel-Cantelli lemma, that, for all sufficiently large t, even a growing

348


number of the events A(et) y with y ∈ Mt occurs. To show the summability of pt for t ∈ eN , we use the Chebyshev inequality to estimate pt ≤ Prob

(et)

1l{A y } −

y∈Mt

≤4

(et)

1l{A y }

y∈Mt

1 − Prob(A(et) 0 )

|Mt |Prob(A(et) 0 )

2

2 1 > |Mt |Prob(A(et) 0 ) 4

.

The summability over all t ∈ eN is clear from (4.23).

5. Appendix: Corrected Proof of Lemma 4.2 in [BK01] We use the opportunity to correct an error in the proof of one of the main results of [BK01], the analogue of Theorem 1.4 for case (4) in Sect. 1.3. In the original proof the large deviation principle of Proposition 3.4 and Varadhan’s lemma are applied to the functional f → − f γ d x, which fails to be continuous in the topology of the large deviation principle. Here we adapt the techniques of the present paper to derive this result. We use the notation of Sect. 3. Recall case (4) from Sect. 1.3. That is, we are in the case where esssup ξ(0) = 0, γ ∈ (0, 1) and κ ∗ = 0. The case γ = 0 is easier and can be treated analogously. The t (x) = −Dx γ , uniformly in x on compact subsets main assumption is that limt→∞ H of [0, ∞), where d+2 t (x) = α(t) H x t , H t α(t)d

(5.1)

and D > 0 is a parameter. We have α(t) = t ν+o(1) as t → ∞, where ν = (0,

1 d+2 ).

1−γ d+2−dγ

∈

The step which needs amendment in [BK01] is the following analogue of Proposition 3.1: Proposition 5.1. ξ α(t)2 log u Rα(t) (t, 0) ≤ −χ (M) . t t↑∞ ξ α(t)2 (ii) For any R > 0, lim inf log u Rα(t) (t, 0) ≥ −χ R , t↑∞ t (i) For any R > 0 and M > 0, lim sup

where χ (M) = χR =

inf

g∈H 1 (Rd ) g2 =1

inf

∇g22 + D

g∈H 1 (Rd ) g2 =1,supp (g)⊆Q R

g 2 (x) ∧ M

γ

dx ,

∇g22 + D g 2γ (x) d x .


Proof. Introduce H(t) R (f) =

t f (x) d x, H

349

for f ∈ L 1 (Q R ), f ≥ 0.

QR

As in (3.38), we have ξ u Rα(t) (t, 0) = E0 exp

t (t) H (L ) 1 l{supp (L ) ⊆ Q } , t t R R α(t)2

(5.2)

where we recall the rescaled and normalized local times L t . We start with the proof of (i). Fix M > 0. With H R ( f ) = −D Q R f (x)γ d x, we have, uniformly in f ∈ L 1 (Q R ), f ≥ 0, (t) lim sup H(t) R ( f ) ≤ lim sup H R ( f ∧ M) = H R ( f ∧ M).

t↑∞

t↑∞

Note that H R (L t ∧ M) = α(t)2 G t ( 1t t ), where we introduce γ D −d d G t (µ) = − (α(t) α(t) µ(z)) ∧ M , for µ ∈ M1 (B Rα(t) ). α(t)2 z∈B Rα(t)

We now use Proposition 3.3 for B = B Rα(t) and F = G t to obtain from (5.2) that, for any large t, ξ −2 u Rα(t) (t, 0) ≤ eo(tα(t) ) E0 exp t G t ( 1t t ) 1l{supp (t ) ⊆ B Rα(t) } −2 ≤ eo(tα(t) ) exp − tχt(M) , where (M)

χt

4 2 4 1 = inf µ(x) − µ(y) − G t (µ) . µ∈M1 (B Rα(t) ) 2 x∼y

The proof of the upper bound is finished as soon as we have shown that lim inf α(t)2 χt(M) ≥ χ (M) .

(5.3)

t↑∞

This is shown as follows. Let (tn : n ∈ N) be a sequence of positive numbers tn → ∞ along which lim inf t↑∞ α(t)2 χt(M) is realized. We may assume that its value is finite. Let (µn : n ∈ N) be a sequence of approximative minimizers, i.e., probability measures on Zd having support in B Rα(t) such that 2 γ 4 4

21 −d d α(tn ) µn (z) ∧ M µn (z) − µn (y) + Dα(tn ) lim inf α(tn ) 2 n→∞

z∼y

z

is equal to the left-hand side of (5.3). For any i ∈ {1, . . . , d} consider gn(i) : Rd → R given by 7

α(tn )xi gn(i) (x) = α(tn )d µn α(tn )x + xi − α(tn ) 7

7

×α(tn ) α(tn )d µn α(tn )x + ei − α(tn )d µn α(tn )x ,

350


where ei ∈ Zd is the i th unit vector. For x = (x j : j = 1, . . . , d) ∈ Rd , we abbreviate (i) (i) xi = (x j : j = i) ∈ Rd−1 and denote gn, xi ∈ Rd−1 , xi (x i ) = gn (x). For almost every (i) 1 the map gn, xi is continuous and piecewise affine, and hence lies in H (R) with support in [−R, R]. Furthermore, 7

7

∂gn(i) (i) d µ α(t )x + e − α(t )d µ α(t )x . (gn, ) (x ) = (x) = α(t ) α(t ) i n n n n i n n n xi ∂ xi Hence, using Fubini’s theorem and Fatou’s lemma, we see that 7 2 7 21 d ∞ > lim inf α(tn ) α(tn ) µn (z) − α(tn )d µn (y) n→∞ 2 z∼y = lim inf n→∞

≥

d i=1

d i=1

Rd−1

Rd−1

d xi

R

d xi lim inf n→∞

R

82 8 (i) 8 d xi 8(gn, xi ) (x i ) 82 8 (i) 8 d xi 8(gn, xi ) (x i ) .

Since |xi − α(tn )xi /α(tn )| ≤ α(tn )−1 , this also shows that lim gn(i) −

n→∞

7 α(tn )d µn (α(tn ) · )2 = 0.

(5.4)

(5.5)

In particular, gn(i) is asymptotically L 2 -normalized. Furthermore, it follows that, along a (i) (i) 1 suitable subsequence, for almost all xi ∈ Rd−1 , gn, xi converges to some g xi ∈ H (R). 2 The convergence is (i) strong in L , (ii) pointwise almost everywhere, and (iii) weak in L 2 for the gradients. The limit satisfies 7 7 2 1 lim inf α(tn )2 α(tn )d µn (z) − α(tn )d µn (y) n→∞ 2 z∼y ≥

d i=1

Rd−1

d xi

R

82 8 d xi 8(gx(i)i ) (xi )8 .

(5.6)

4 (i) (i) (i) d Since gn, xi (x i ) = gn (x) and lim n→∞ gn − α(tn ) µn (α(tn ) · )2 = 0, there is g ∈ (i) L 2 (Rd ) such that g(x) = gxi (xi ) for almost all x ∈ Rd . In particular, (a) g ∈ H 1 (Rd ) with (b) g2 = 1, (c) supp (g) ⊂ Q R and (d) 7 2 7 2 21 d ∇g2 ≤ lim inf α(tn ) α(tn ) µn (z) − α(tn )d µn (y) . n→∞ 2 z∼y Indeed, (a) follows from (b) and (d). Item (b) follows from (5.5), while item (c) is trivially satisfied. We are left to prove item (d). Since gx(i)i (xi ) = g(x) for almost every x, we get

∂ (i) ∂ gx(i)i (xi ) = g (xi ) = g(x), ∂ xi xi ∂ xi

(5.7)


351

and hence d

d−1 i=1 R

d xi

R

82 8 d xi 8(gx(i)i ) (xi )8 =

Rd

dx

d 82 8 ∂ 8 g(x)8 = ∇g22 . ∂ xi

(5.8)

i=1

Therefore, item (d) follows from (5.6). It remains to show that (g(x)2 ∧M)γ d x ≤ lim inf n→∞ α(tn )−d z ((α(tn )d µn (z))∧ M)γ . Note that γ

α(tn )d µn (z) ∧ M α(tn )−d z

γ

= α(tn )d µn (α(tn )x) ∧ M d x 8 82γ α(tn )xi (i) 8 (i) 8 γ (gn,xi ) (xi )8 ∧ M d x. = 8gn (x) − xi − α(tn )

(5.9)

We next use the inequality |a − b|2γ ≥ (|a|γ − |b|γ )2 ≥ |a|2γ − 2|ab|γ and for the subtracted term use Jensen’s inequality and the Cauchy-Schwarz inequality, as well as (5.4), to see that 8 8γ α(tn )xi (i) 8 8 (i) (gn,xi ) (xi )8 d x 8gn (x) xi − α(tn ) 8 γ 8 (i) 888 8g (x)88 xi − α(tn )xi (g (i) ) (xi )88 d x ≤ (2R)d (2R)−d n n, xi α(tn ) QR γ γ ≤ α(tn )−γ (2R)(1−γ )d g (i) ∂ g (i) , n

2

∂ xi n

2

which is negligible. Next we use the fact that gn(i ) → g pointwise and Fatou’s lemma to see that the limit inferior of the right hand side of (5.9) is not smaller than (g(x)2γ ∧M γ ) d x. This completes the proof of (5.3) and therefore the proof of (i). We next turn to the proof of (ii). First we show that, for any f ∈ C(Q R ) and any family of L 1 (Q R )-normalized functions f t ∈ L 1 (Q R ) satisfying f t → f in the weak topology induced by test integrals against all continuous functions, lim inf H(t) R ( f t ) ≥ H R ( f ).

(5.10)

t↑∞

(t) (t) We fix a large M > 0 and estimate H(t) R ( f t ) ≥ H R ( f t ∧ M) + H R ( f t 1l{ f t > M}). We γ first handle the first term. Introduce φ(x) = x and let g y (x) = (1 − γ )y γ + γ y γ −1 x denote the tangent of φ at y ∈ (0, ∞). By concavity, we have φ ≤ g y on (0, ∞) for any y > 0. This implies that, as t ↑ ∞, for any ε > 0,

H(t) φ f t (x) ∧ M d x ≥ o(1) − D g f (x)∨ε f t (x) d x R ( f t ∧ M) = o(1) − D QR QR

γ

γ −1 ≥ o(1)− D(1 − γ ) f (x) ∨ ε d x − Dγ f t (x) f (x) ∨ ε dx QR QR = o(1) − D(1 − γ ) ( f ∨ ε)γ − Dγ f ( f ∨ ε)γ −1 ,

QR

QR

352


where in the last step we used that ( f ∨ ε)γ −1 is continuous and f t → f . Letting ε ↓ 0, we see that lim inf t↑∞ H(t) R ( f t ∧ M) ≥ H R ( f ) for any M > 0. It remains to show that lim inf M↑∞ lim inf t↑∞ H(t) R ( f t 1l{ f t > M}) ≥ 0. Fix δ > 0 such that γ + δ < 1. Recall (5.1). Since H is regularly varying with exponent γ , by [BGT87, Proposition 1.3.6], there is an M > 0 such that, for any sufficiently large t, t (x) ≥ −x γ +δ , H Hence,

for any x > M.

f t (x)γ +δ 1l{ f t (x) > M} d x ≥ −M γ +δ−1 f t (x) d x = −M γ +δ−1 ,

H(t) R ( f t 1l{ f t > M}) ≥ −

QR

QR

since f t is L 1 -normalized. This completes the proof of (5.10). We complete the proof of Proposition 5.1(ii) by using (5.10) in (5.2) and use the lower bound of Varadhan’s lemma in [DZ98, Lemma 4.3.4] to conclude that the assertion in (ii) holds.

Acknowledgements. We would like to thank Laurens de Haan for helpful discussions on regularly varying functions, and the organisers of the Workshop on Interacting stochastic systems in Cologne, 2003, where this work was initiated. The work of the first author was supported in part by Netherlands Organisation for Scientific Research (NWO). The second author would like to thank the German Science Foundation for awarding a Heisenberg grant (realized in 2003/04), and the third author would like to acknowledge the support of the Nuffield Foundation through grant NAL/00631/G and the EPSRC through grant EP/C500229/1.

References [BGT87] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge: Cambridge University Press 1987 [BK01] Biskup, M., König, W.: Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29(2), 636–682 (2001) [BK01a] Biskup, M., König, W.: Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model. J. Stat. Phys. 102(5/6), 1253–1270 (2001) [BHK05] Brydges, D., van der Hofstad, R., König, W.: Joint density for the local times of continuous-time Markov chains (2005) available at http://www.math.uni-leipzig.de/∼koenig/www/localtimes.pdf [CM94] Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108(518), (1994) [Ch04] Chen, X.: Exponential asymptotics and law of the iterated logarithm for self-intersection local times of random walks. Ann. Probab. 32(4), 3248–3300 (2004) [DZ98] Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd Edition. New York: Springer, 1998 [DV79] Donsker, M., Varadhan, S.R.S.: On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32, 721–747 (1979) [GKS06] Gantert, N.,König, W., Shi, Z.: Annealed deviations for random walk in random scenery. Annales Inst. H. Poincaré: Probab. et Stat., to appear (2006). [GH99] Gärtner, J., den Hollander, F.: Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Relat. Fields 114, 1–54 (1999) [GK00] Gärtner, J., König, W.: Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10(3), 192–217 (2000) [GK05] Gärtner, J., König, W.: The parabolic Anderson model. In: J.-D. Deuschel, A. Greven (eds.), Interacting Stochastic Systems, Berlin Heidelberg Newyork: Springer 2005, pp.153–179 [GKM00] Gärtner, J., König, W., Molchanov, S.: Almost sure asymptotics for the continuous parabolic Anderson model. Probab. Theory Relat. Fields 118(4), 547–573 (2000)


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[GKM06] Gärtner, J., König, W., Molchanov, S.: Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab., to appear (2006) [GM90] Gärtner, J., Molchanov, S.: Parabolic problems for the Anderson model I. Intermittency and related topics. Commun. Math. Phys. 132, 613–655 (1990) [GM98] Gärtner, J., Molchanov, S.: Parabolic problems for the Anderson model II. Second-order asymptotics and structure of high peaks. Probab. Theory Relat. Fields 111, 17–55 (1998) [dH00] den Hollander, F.: Large Deviations. Fields Institute Monographs. Providence, RI: Amer. Math. Soc. 2000 [KM02] König, W., Mörters, P.: Brownian intersection local times: upper tail asymptotics and thick points. Ann. Probab. 30, 1605–1656 (2002) [LL01] Lieb, E.H., Loss, M.: Analysis, 2nd Edition. AMS Graduate Studies, Vol. 14, Providence, RI: Amer. Math. Soc., 2001 [M94] Molchanov, S.: Lectures on random media. In: D. Bakry, R.D. Gill, S. Molchanov, Lectures on Probability Theory, Ecole d’Eté de Probabilités de Saint-Flour XXII-1992, LNM 1581, Berlin: Springer, 1994, pp. 242–411 [MR94] Molchanov, S., Ruzmaikin, A.: Lyapunov exponents and distributions of magnetic fields in dynamo models. In The Dynkin Festschrift: Markov Processes and their Applications. Mark Freidlin, (ed.) Basel: Birkhäuser, (1994) [S98] Sznitman, A.-S.: Brownian motion, Obstacles and Random Media. Berlin: Springer 1998 Communicated by H. Spohn


Communications in


Convergence in Higher Mean of a Random Schrödinger to a Linear Boltzmann Evolution Thomas Chen Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ 08544-1000, USA. E-mail: [email protected] Received: 20 April 2005 / Accepted: 16 May 2006 Published online: 1 August 2006 – © Springer-Verlag 2006

Abstract: We study the macroscopic scaling and weak coupling limit for a random Schrödinger equation on Z3 . We prove that the Wigner transforms of a large class of “macroscopic” solutions converge in r th mean to solutions of a linear Boltzmann equation, for any 1 ≤ r < ∞. This extends previous results where convergence in expectation was established. 1. Introduction We study the macroscopic scaling and weak coupling limit of the quantum dynamics in the three dimensional Anderson model, generated by the Hamiltonian 1 Hω = − + λVω (x) 2

(1)

on 2 (Z3 ). Here, is the nearest neighbor discrete Laplacian, the coupling constant λ > 0 defines the disorder strength, and the random potential is given by Vω (x) = ωx , where {ωx }x∈Z3 , are independent, identically distributed Gaussian random variables. While the phenomenon of impurity-induced insulation is, for strong disorders λ 1 or extreme energies, mathematically well understood (Anderson localization, [1, 6]), establishing the existence of electric conduction in the weak coupling regime λ 1 is a central open problem in this research field. A particular strategy to elucidate aspects of the latter, which has led to important recent successes (especially [5]), is to analyze the macroscopic transport properties derived from the microscopic quantum dynamics generated by (1), [2–5, 9]. Let φt ∈ 2 (Z3 ) be the solution of the random Schrödinger equation i∂t φt = Hω φt (2) φ0 ∈ 2 (Z3 ),

356

T. Chen

with a deterministic initial condition φ0 which is supported on a region of diameter O(λ−2 ). Let Wφt (x, v) denote its Wigner transform, where x ∈ 21 Z3 ≡ (Z/2)3 , and v ∈ T3 = [0, 1]3 . We consider a scaling for small λ defined by the macroscopic time, position, and velocity variables (T, X ) := λ2 (t, x), V := v, while (t, x, v) are the microscopic variables. Likewise, we introduce an appropriately rescaled, macroscopic counterpart Wλr esc (T, X, V ) of Wφt (x, v). It was proved by Erdös and Yau for the continuum, [4, 3], and by the author for the lattice model, [2], that globally in macroscopic time T , and for any test function J (X, V ), lim E d X d V J (X, V )Wλr esc (T, X, V ) λ→0 = d X d V J (X, V )F(T, X, V ), where F(T, X, V ) is the solution of a linear Boltzmann equation. For the random wave equation, a similar result is proved by Lukkarinen and Spohn, [7]. The corresponding local in T result was established much earlier by Spohn, [9]. The main goal of this paper is to strengthen the mode of convergence. We establish convergence in r th mean, lim E d X d V J (X, V )Wλr esc (T, X, V ) λ→0 r (3) − d X d V J (X, V )F(T, X, V ) = 0, for any J and any r ∈ 2N, and hence for any 1 ≤ r < ∞. Thus, in particular, we observe that the variance of d X d V J (X, V )Wλr esc (T, X, V ) vanishes in this macroscopic, hydrodynamic limit. Our proof comprises generalizations and extensions of the graph expansion methods introduced by Erdös and Yau in [4, 3], and further elaborated on in [2]. The structure of the graphs entering the problem is significantly more complicated than in [4, 3, 2], and the number of graphs in the expansion grows much faster than in [4, 3, 2] (superfactorial versus factorial). As a main technical result in this paper, it is established that the associated Feynman amplitudes are sufficiently small to compensate for the large number of graphs, which is shown to imply (3). This is similar to the approach in [4, 3, 2]. The present work addresses a time scale of order O(λ−2 ) (as in [4, 3, 2]), in which the average number of collisions experienced by the electron is finite, so that ballistic behavior is observed. Accordingly, the macroscopic dynamics is governed by a linear Boltzmann equation. Beyond this time scale, the average number of collisions is infinite, and the level of difficulty of the problem increases drastically. In their recent breakthrough result, Erdös, Salmhofer and Yau have established that over a time scale of order O(λ−2−κ ) for an explicit numerical value of κ > 0, the macroscopic dynamics in d = 3 derived from the quantum dynamics is determined by a diffusion equation, [5]. We note that control of the macroscopic dynamics up to a time scale O(λ−2 ) produces lower bounds of the same order (up to logarithmic corrections) on the localization lengths of eigenvectors of Hω , see [2] for d = 3 (the same arguments are valid for d ≥ 3). This extends recent results of Schlag, Shubin and Wolff, [8], who derived similar lower bounds for the weakly disordered Anderson model in dimensions d = 1, 2 using harmonic analysis techniques.

Convergence in Higher Mean of a Random Schrödinger to a Linear Boltzmann Evolution

357

This work comprises a partial joint result with Laszlo Erdös (Lemma 5.2), to whom the author is deeply grateful for his support and generosity. 2. Definition of the Model and Statement of Main Results To give a mathematically well-defined meaning to all quantities occurring in our analysis, we first introduce our model on a finite box L = {−L , −L + 1, . . . , −1, 0, 1, . . . , L − 1, L}3 ⊂ Z3 ,

(4)

for L ∈ N much larger than any relevant scale of the problem, and take the limit L → ∞ later. All estimates derived in the sequel will be uniform in L. We consider the discrete Schrödinger operator 1 Hω = − + λVω 2

(5)

on 2 ( L ) with periodic boundary conditions. Here, is the nearest neighbor Laplacian, f (y), (6) ( f )(x) = 6 f (x) − |y−x|=1

and Vω (x) = ωx

(7)

is a random potential with {ω y } y∈ L i.i.d. Gaussian random variables satisfying E[ωx ] = 0, E[ω2x ] = 1, for all x ∈ L . Expectations of higher powers of ωx satisfy Wick’s theorem, cf. [4], and our discussion below. Clearly, Vω ∞ ( L ) < ∞ almost surely (a.s.), and Hω is a.s. self-adjoint on 2 ( L ), for every L < ∞. 1 1 L−1 3 3 Let ∗L = L1 L = {−1, − L−1 L , . . . , − L , 0, L , . . . , L , 1} ⊂ T denote the lat3 3 tice dual to L , where T = [−1, 1] the 3-dimensional unit torus. For 0 < ρ ≤ 1 with 1 is given by ∗L ,ρ = ρ ∈ N, we define L ,ρ := ρρ −1 L , and note that its dual lattice 1 −1 3 k∈ L ,ρ , L ρ−1 L ⊂ ρ T . For notational convenience, we shall write L ,ρ dk ≡ and ρ −1 T3 dk for the Lebesgue integral. For the Fourier transform and its inverse, we use the convention 3 −2πik·x ∨ e f (x), g (x) = dkg(k)e2πik·x , (8) f (k) = ρ ∗L ,ρ

x∈ L ,ρ

for L ≤ ∞ (where ∞,ρ = ρZ3 and ∗∞,ρ = T3 /ρ). We will mostly use ρ = 1, and sometimes ρ = 21 . On ∗L ,ρ , we define δ(k) = 1(k) with δ(0) = | L ,ρ | if k = 0 and d d δ(k) = 0 if k = 0. On T or R , δ will denote the usual d-dimensional delta distribution. The nearest neighbor lattice Laplacian defines the Fourier multiplier f (k), (− f ) (k) = 2e (k)

(9)

where e (k) =

3

(1 − cos(2π ki )) = 2

i=1

determines the kinetic energy of the electron.

3 i=1

sin2 (π ki )

(10)

358

T. Chen

Let φt ∈ 2 ( L ) denote the solution of the random Schrödinger equation i∂t φt = Hω φt φ0 ∈ 2 ( L ),

(11)

for a fixed realization of the random potential. We define its (real, but not necessarily positive) Wigner transform Wφt : L , 1 × ∗L → R by 2

Wφt (x, v) := 8

φt (y)φt (z)e2πi(y−z)v .

(12)

y,z∈ L y+z=2x

Fourier transformation with respect to the variable x ∈ L , 1 (i.e. (8) with ρ = 21 , see 2 [5] for more details) yields ξ ξ φt (ξ, v) = φ t (v + ), t (v − )φ W 2 2 for v ∈ ∗L and ξ ∈ ∗

L , 12

(13)

⊂ 2T3 .

The Wigner transform is the key tool in our derivation of the macroscopic limit for the quantum dynamics described by (19). For η > 0 small, we introduce macroscopic variables T := ηt, X := ηx, V := v, and consider the rescaled Wigner transform η

Wφt (X, V ) := η−3 Wφt (η−1 X, V )

(14)

for T ≥ 0, X ∈ η L , 1 , and V ∈ ∗L . 2

For a Schwartz class function J ∈ S(R3 × T3 ), we write η η d V J (X, V )Wφt (X, V ). J, Wφt := X ∈η L , 1

(15)

∗L

2

φt as in (13), we have With W η

φt = J, Wφt = J η , W

∗

×∗L L , 21

φt (ξ, v), dξ dv J η (ξ, v)W

(16)

where Jη (x, v) := η−3 J (ηx, v), and J η (ξ, v) = η−3

J (ηx, v)e−2πi xξ = η−3

x∈ L , 1 2

J (X, v)e

− 2πiηX ξ

.

(17)

X ∈η L , 1 2

We note that in the limit L → ∞, J η (ξ, v) tends to a smooth delta function with respect to the ξ -variable, of width O(η) and amplitude O(η−1 ), but remains uniformly bounded with respect to η in the v-variable. The macroscopic scaling limit obtained from letting η → 0, with η = λ2 , is determined by a linear Boltzmann equation. This was proven in [2] for Z3 , and nonGaussian distributed random potentials (the Gaussian case follows also from [2]). The corresponding result for the continuum model in dimensions 2, 3 was proven in [4].


359

Theorem 2.1. For η > 0, let η φ0 (x)

2πi S(ηx)

η h(ηx)e := η ,

h 2 (ηZ3 ) 3 2

(18)

with h, S ∈ S(R3 , R) of Schwartz class, and h L 2 (R3 ) = 1. Assume L sufficiently large η η η (see (73)) that φ0 L = φ0 . Let φt be the solution of the random Schrödinger equation η

η

i∂t φt = Hω φt

(19)

η

on 2 ( L ) with initial condition φ0 , and let (η)

η

WT (X, V ) := Wφ η

η−1 T

(X, V )

(20)

denote the corresponding rescaled Wigner transform. Choosing η = λ2 ,

(21)

where λ is the coupling constant in (5), it follows that

2 lim lim E J, WT(λ ) = J, FT ,

λ→0 L→∞

(22)

where FT (X, V ) solves the linear Boltzmann equation ∂T FT (X, V ) + =

T3

3 (sin 2π V j )∂ X j FT (X, V ) j=1

dU σ (U, V ) [FT (X, U ) − FT (X, V )]

(23)

with initial condition η

F0 (X, V ) = w − lim Wφ η = |h(X )|2 δ(V − ∇ S(X )), η→0

(24)

0

and where σ (U, V ) := 2π δ(e (U ) − e (V )) denotes the collision kernel. The purpose of the present work is to obtain a significant improvement of the mode of convergence. Our main result is the following theorem.

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T. Chen

η , satisfies the concentration Theorem 2.2. Assume that the Fourier transform of (18), φ 0 of singularity property (29)–(31). Then, for any fixed, finite r ∈ 2N, any T > 0, and for any Schwartz class function J , the estimate lim

L→∞

r1 1 (λ2 ) (λ2 ) r E J, WT − E J, WT ≤ c(r, T )λ 300r

(25)

holds for λ sufficiently small, and a finite constant c(r, T ) that does not depend on λ. Consequently,

r 2 (26) lim lim E J, WT(λ ) − J, FT = 0 λ→0 L→∞

for any 1 ≤ r < ∞ (i.e. convergence in r th mean), and any T ∈ R+ .

(λ2 ) We observe that, in particular, the variance of J, WT vanishes in the limit λ → 0. Moreover, the following result is an immediate consequence. Corollary 2.1. Under the assumptions of Theorem 2.2, the rescaled Wigner transform 2 WT(λ ) converges weakly, and in probability, to a solution of the linear Boltzmann equations, globally in T > 0, as λ → 0. That is, for any finite T > 0, any ν > 0, and any J of Schwartz class,

(λ2 ) (27) P lim J, WT − J, FT > ν = 0, λ→0

where FT solves (23) with initial condition (24). η

. We have obtained a well-defined semiclassical initial condition 2.1. Singularities of φ 0 (24) for the linear Boltzmann evolution (23) from initial data in (2) of WKB type (18). This can in general not be expected if the initial data in (2) are only required to be in 2 (Z3 ), but 2 (Z3 ) initial data in (2) suffice for the expectation value of the quantum fluctuations in (22) to converge to zero as λ → 0, see [4, 2]. As we will see, a key point in proving that as λ → 0, the quantum fluctuations vanish η with in higher mean, (26), consists of controlling the overlap of the singularities of φ 0 −1 those of the resolvent multipliers (e (k) − α ± iε) , where α ∈ R and ε = O(η) 1. As opposed to the case in (22), it cannot be expected that the quantum fluctuations vanish in higher mean for general 2 inital data (for (22), the overlap of the singularities of η and of those of the resolvent multipliers plays no rôle). Moreover, we note that the φ 0 singularities of the WKB initial condition η (k) = η 2 φ 0

3

h(ηx)e

2πi( S(ηx) η −kx)

x∈Z3 3

= η2

h(X )e

X 2πi( S(X )−k ) η

(28)

X ∈ηZ3

(which are determined by the zeros of detHessS(X ), the determinant of the Hessian of S) will possess a rather arbitrary structure for generic choices of S ∈ S(R3 , R). At present, we do not know if for WKB initial data of the form (18), the quantum fluctuations would


361

converge to zero in higher mean without any further restrictions on the phase function S ∈ S(R3 , R). A more detailed analysis of these questions is left for future work. In this paper, we shall assume that the Fourier transform of the WKB initial condition (18) satisfies a concentration of singularity condition: η η η (k) = f ∞ φ (k) + f sing (k), 0

(29)

where η

f ∞ L ∞ (T3 ) < c,

(30)

and η

η

η

4

| f sing | ∗ | f sing | L 2 (T3 ) = | f sing |∨ 24 (Z3 ) ≤ c η 5

(31)

for finite, positive constants c, c independent of η. This condition imposes a restriction on the phase function S. η satisfy (29)–(31). The following simple, but physically important examples of φ 0 2.1.1. Example Let S(X ) = p X for X ∈ supp{h}, and p ∈ T3 . Then, η− 2 h(η−1 (k − p)) =: δη (k − p).

h 2 (ηZ3 ) 3

η (k) = φ 0

(32)

Since h is of Schwartz class, δη is a smooth bump function concentrated on a ball of radius O(η), with δη L 2 (T3 ) = 1. Accordingly, we find (|δη | ∗ |δη |)(k) ≈ χ (|k| < cη),

(33)

and 3

|δη | ∗ |δη | L 2 (T3 ) = |δη |∨ 24 (Z3 ) ≤ cη 2 .

(34)

η

Hence, (29)–(31) is satisfied, with f ∞ = 0. In this example, p ∈ T3 corresponds to the velocity of the macroscopic initial condition F0 (X, V ) in (24) for the linear Boltzmann evolution. 2.1.2. Example As a small generalization of the previous case, we may likewise assume for S that for every k ∈ T3 , there are finitely many solutions X j (k) of ∇ X S(X j (k)) = k, and that X j (·) ∈ C 1 (supp{h}) for each j. Moreover, we assume that |detHessS(X )| > c uniformly on supp{h}. Then, by stationary phase arguments, [10], one finds that η η η η (k) = f ∞ φ (k) + f sing (k), f ∞ L ∞ (T3 ) < c 0

with η

f sing (k) =

c j δη( j) (k − ∇ X S(X j (k))),

(35)

(36)

j ( j)

for constants c j independent of η, and smooth bump functions δη similar to (32). One 3 η again obtains | f sing |∨ 24 (Z3 ) ≤ cη 2 , which verifies that (29)–(31) holds. ∇ S determines the velocity distribution of the macroscopic initial condition F0 (X, V ) in (24).

362

T. Chen

3. Proof of Theorem 2.2 We expand φt into a truncated Duhamel series N −1

φt =

φn,t + R N ,t ,

(37)

n=0

where

φn,t := (−iλ)n

Rn+1 +

ds0 · · · dsn δ

n

s j − t eis0 2 Vω eis1 2 · · · Vω eisn 2 φ0

(38)

j=0

denotes the n th Duhamel term, and where t dse−i(t−s)Hω Vω φ N −1,s R N ,t = −iλ

(39)

0

η

is the remainder term. Here and in the sequel, we write φ0 ≡ φ0 for brevity. The number ω is well-defined and bounded N remains to be optimized. Since Vω 1 ( L ) < ∞ a.s., V on ∗L , with probability one, for every L < ∞. Then, n n,t (k0 ) = (−iλ)n ds0 · · · dsn δ φ sj − t j=0

×

(∗L )n

ω (k1 − k0 )e−is1 e (k1 ) · · · dk1 · · · dkn e−is0 e (k0 ) V

ω (kn − kn−1 )e−isn e (kn ) φ 0 (kn ). ×···V

(40)

Expressed as a resolvent expansion in momentum space, we find (−λ)n εt n,t (k0 ) = φ e dαe−itα 2πi R 1 ω (k1 − k0 ) × dk1 · · · dkn V e (k0 ) − α − iε (∗L )n ω (kn − kn−1 ) ×···V

1 0 (kn ). φ e (kn ) − α − iε

(41)

1 as a particle propagator. Likewise, we We refer to the Fourier multiplier e (k)−α−iε th note that (41) is equivalent to the n term in the resolvent expansion of 1 1 φ0 . dze−it z (42) φt = 2πi −iε+R Hω − z

By the analyticity of the integrand in (41) with respect to the variable α, the path of the α-integration can, for any fixed n ∈ N, be deformed into the closed contour I = I0 ∪ I1 , away from R, with

(43)


363

I0 := [−1, 13], I1 := ([−1, 13] − i) ∪ (−1 − i(0, 1]) ∪ (13 − i(0, 1]), which encloses spec (− − iε) = [0, 12] − iε. Next, we apply the time partitioning method introduced in [4]. To this end, we choose κ ∈ N with 1 κ ε−1 , and subdivide [0, t] into κ subintervals bounded by θ j = jtκ , j = 1, . . . , κ. Then, R N ,t = −iλ

κ−1

e

−i(t−θ j+1 )Hω

θ j+1

θj

j=0

dse−i(θ j+1 −s)Hω Vω φ N −1,s .

(44)

Let φn,N ,θ (s) denote the n th Duhamel term, conditioned on the requirement that the first N collisions occur in the time interval [0, θ ], and all remaining n − N collisions take place in the time interval (θ, s]. That is, n−N n−N φn,N ,θ (s) := (−iλ) ds0 · · · dsn−N δ s j − (s − θ ) +1 Rn−N +

×e

is0 2

Vω · · · Vω e

j=0 2

Vω φ N −1,θ .

(45)

n,N ,θ (s) := −iλVω φn−1,N ,θ (s) φ

(46)

isn−N

Moreover, let

denote its “truncated” counterpart. Further expanding e−is Hω in (44) into a truncated Duhamel series with 3N terms, we find ( C0 . By Part (1), f − Ptop ( f ) > C on the partition set which contains x. Denote this partition set by [x 0 ], write x 0 = [b, ξ , a], and consider the point z := (x 0 , wab , x 0 , wab , x 0 , wab , . . .). This is a periodic point of order 1 + n ab , and ( f − Ptop ( f ))1+n ab (z) > C − C = 0. N −1 [ f (T k z) − Ptop ( f )] > 0. Write z = π(z). Then for some N , T N (z) = z and k=0 N −1 1 The measure μ:= N k=0 δT k z is T –invariant, has zero entropy, and satisfies h μ (T ) +

+ A

[ f − Ptop ( f )]dμ =

N −1 1 [ f (T k z) − Ptop ( f )] > 0. N k=0

It follows that h μ (T ) + f dμ > Ptop ( f ), in contradiction to the definition of Ptop ( f ). Part (2) is proved. Before proving Part (3), we recall from [BS] that μ[a] = 0 for any state a ∈ S and every equilibrium measure μ of a potential with summable variations on a topologically mixing shift. Therefore, μ is well defined. Next we note that μ is shift invariant, because μ| A is T A –invariant. The formulæ of Kac and Abramov and the conjugacy between T A and the induced shift give f − Ptop ( f )dμ Ptop ( f − Ptop ( f )) ≥ h μ◦π −1 (T A ) + 1 = h μ (T ) + f − Ptop ( f )dμ = 0. μ(A) The other inequality is more delicate, because it is not true that every T A –invariant probability measure is induced by a T –invariant probability measure: We can only guarantee this for T A –invariant measures for which ϕ is integrable.

Continuous Phase Transitions for Dynamical Systems

657

To deal with this difficulty, we note that since f − Ptop ( f ) has summable variations (Part 1) and is bounded from above (Part 2), then Ptop ( f − Ptop ( f )) is equal to the Gurevich pressure of f − Ptop ( f ). Therefore, by Theorem 2 of [S1], Ptop ( f − Ptop ( f )) = sup h m (T ) + f − Ptop ( f )dm : m has compact support . For such measures ϕ ◦ π is essentially bounded, whence integrable. Therefore Ptop ( f − Ptop ( f )) is achieved as a supremum over invariant measures which are in+ . Such measures ν satisfy duced by shift invariant measures on A 1 h ν (T ) + f − Ptop ( f )dν = f − Ptop ( f )dν ≤ 0. h ν◦π −1 (T A ) + + ν(A) + A A

Passing to the supremum, we get Ptop ( f − Ptop ( f )) ≤ 0. In the first part of the proof we saw that h μ◦π −1 (T A ) + f − Ptop ( f )dμ = 0 for μ induced by the equilibrium measure of f . Consequently, this is an equilibrium measure for f − Ptop ( f ) (by [BS] the only one), and the pressure is zero. Proof of Theorem 8. The convexity of Ptop (φ +tψ) and the assumption that Ptop (φ) = 0 imply that either Ptop (φ +tψ) = 0 on some right neighborhood of 0, or Ptop (φ +tψ) = 0 for all t > 0 small. In the first case the theorem holds trivially by Lemma 1, Part (3). We may therefore assume without loss of generality that Ptop (φ + tψ) = 0 for all t > 0 small. Recall the definitions of ϕ, + , and of the functions φ, ψ induced by φ, ψ. By A assumption, A is a finite union of states such that ψ ≤ 0 = Eμφ [ψ] outside A. Thus: ϕ−1

sup ψ < ∞.

To see this write ψ = k=0 ψ ◦ T k , and observe that the first summand is dominated by sup ψ, while the other summands are non-positive (they correspond to the part of the orbit which lies outside A). Note also that by Lemma 1 Part (2) sup φ < ∞. Step 1. Ptop (φ + tψ) > 0 for all t > 0. Proof. Kac’ formula and the assumption Eμφ [ψ] = 0 imply that Eμφ [ψ] = 0, where μ ◦π μφ = μφφ(A) . By Lemma 1, Ptop (φ) = 0, and μφ is the equilibrium measure of φ: μφ = μφ . Consequently Eμφ [ψ] = 0.

By Theorem 4 for BIP systems, Ptop (φ +tψ) = o(t) as t → 0+ (note that the assumptions listed at the beginning of Sect. 3 are satisfied). We see that the right–derivative of t → Ptop (φ +tψ) at t = 0 vanishes. But t → Ptop (φ +tψ) is convex, so Ptop (φ +tψ) ≥ 0 for t ≥ 0. Lemma 1 tells us that Ptop (φ + tψ − Ptop (φ + tψ)) = 0. If Ptop (φ + tψ) were negative, then by the properties of the topological pressure and since ϕ ≥ 1, 0 = Ptop (φ + tψ − Ptop (φ + tψ)ϕ) ≥ Ptop (φ + tψ) + |Ptop (φ + tψ)| > 0. Therefore Ptop (φ + tψ) ≥ 0 for t > 0. The inequality is strict, otherwise by convexity Ptop (φ + tψ) vanishes on some right-neighbourhood of 0, contrary to our assumptions.

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Step 2. Set f t := φ + tψ and φt := f t − Ptop ( f t ). The induced potentials φt , f t have Gibbs measures μ ft , μφt , and Eμφ [ϕ] ≤ t

Ptop ( f t ) Ptop ( f t )

≤ Eμ f [ϕ] for all 0 ≤ t ≤ 0 . t

Proof. Any locally Hölder continuous potential with finite pressure on a shift with the BIP property has an invariant Gibbs measure [S3]. Therefore, since + has the BIP propA

erty, it is enough to check that f t , φt have finite pressure. They do, because sup ψ < ∞, Ptop ( f t ) ≥ 0 (Step 1), and Ptop (φ) = 0 < ∞ (Lemma 1 Part 3). Fix 0 > 0 such that f t := φ + tψ has an equilibrium measure for 0 ≤ t ≤ 0 (regularity). Fix 0 ≤ t ≤ 0 , and consider the function p(s):= Ptop ( f t − sϕ) for s ≥ 0. This is a convex function, and therefore p+ (0) ≤

p(Ptop ( f t )) − p(0) ≤ p+ (Ptop ( f t )), Ptop ( f t )

where p+ denotes one-sided derivative from the right (which can be infinite). The term P

(f )

t in the middle is − Ptop (Lemma 1, Part (3)). Theorem 4 for BIP systems gives the top ( f t ) one-sided derivatives (see the remark after Theorem 4): $ d $$ p+ (0) = Ptop ( f t + s(−ϕ)) = −Eμ f [ϕ], t ds $s=0+

p+ (Ptop ( f t ))

$ d $$ = Ptop (φt + s(−ϕ)) = −Eμφ [ϕ]. t ds $s=0+

This completes the proof. Step 3. Eμ f [ϕ] −−−→ + t

t→0

1 μφ (A) .

Proof. We work on the BIP shift ( + , T ). Define as in Sect. 3 the space L and the A

operators R0 , Rt corresponding to φ and ψ: R0 ( f )(x):= eφ(y) f (y) , Rt ( f ):= R0 [etψ f ]. T y=x

Here and throughout T denotes the shift on + . A As in the beginning of Sect. 3, we may assume without loss of generality that φ(y) = 1 (otherwise pass to φ + h − h ◦ T with some bounded locally Hölder T y=x e continuous function h : +A → R, and note that this does not affect μ ft or μφ ). This reduction allows us to assume that R0 1 = 1. By Proposition 2 Part (3), Rt − R0 −−−→ 0. It follows that the eigenprojections + t→0

Pt := P(Rt ) are well–defined for t small, and converge in norm to P0:= P(R0 ). The operator Rt is the Ruelle operator of f t . The theory of Ruelle operators for shifts with BIP says that λ(Rt ) = exp Ptop ( f t ) and that Pt F = ht Fdνt where νt is an eigenmeasure of Rt , h t is a positive eigenfunction of Rt , and h t dνt = 1. The Gibbs measure of f t is h t dνt . Consequently, Pt [F Pt 1] Fdμ ft = for all F ∈ L. (12) Pt 1 (The RHS is a scalar, because dim Im(Pt ) = 1.)


659

Since Pt → P0 in norm and P0 1 = 1 (because R0 1 = 1), Eμ f [F] −−−→ Eμφ [F] t t→0+

for all F ∈ L. In particular, Eμ f ϕ1[ϕ 0). 't − R0 −−−→ 0 we can proceed exactly as in the preNow that we know that R + t→0

't := P( R 't ) replacing Pt , to deduce that vious step, but with the eigenprojections P Eμφ [ϕ] −−−→ E [ϕ]. The theorem follows from Step 2. μ φ + t

t→0

5. Proofs for Shifts not Satisfying the BIP Property Reduction of the General Case to the BIP Case. Let φ and ψ be two locally Hölder continuous functions bounded from above and assume ( ), ( ), and that φ + tψ has an equilibrium measure for 0 ≤ t ≤ 0 . We also assume without loss of generality that Ptop (φ) = 0 and Eμφ [ψ] = 0 (otherwise subtract suitable constants). Let A ⊂ S be a finite union of states such that ψ ≤ Eμφ [ψ] = 0 outside A. Let a ∈ S be some state such that Eμφ [ra ] < ∞, where ra (x):= min{k: xk = a}. Without loss of generality, [a] ⊆ A (otherwise add a to A). Set ϕ(x):= 1 A (x) min{k ≥ 1: T k x ∈ A}, and let T A: A → A, T A (x):= T ϕ(x) (x) be the induced map. We have seen that this map can be coded by a topological Markov shift with the BIP property. Let φ and ψ be as before. These are locally Hölder continuous functions, and as in the proof of Theorem 8, sup φ = sup φ − Ptop (φ) < ∞; sup ψ = sup ψ − Eμφ [ψ] < ∞. We conclude that + , φ, ψ satisfy the standing assumptions listed at the beginning of A Sect. 3 – the assumptions needed to prove theorems 2, 3, 4 for BIP systems. In order to pass from the induced system to the original system, we need to apply + , φ, ψ); we check Theorems 7 and 8. The conditions of Theorem 8 are satisfied (by A the conditions of Theorem 7. The only thing to check is that the tightness assumption holds in all relevant cases. If α ∈ (1, 2) one must show that B1n (ϕ n − n/μφ (A)) is tight for any sequence Bn √ regularly varying of index α1 ; If α = 2 one must check tightness for Bn = n or for Bn √ s.t. n = o(Bn ). (The case α = 1 does not require Theorem 7). We show √ 1 n [ϕ − n/μφ (A)] is tight for all {Bn } positive s.t. lim sup < ∞. Bn n B n→∞ n This covers all possibilities.

(14)


661

Observe that Eμφ [ϕ 2 ] < ∞. To see this recall from Lemma 1 that μφ = μφ , note that ra (x) ≥ min{k ≥ 1: T k (x) ∈ A} =: r A (x), and observe that n ∞ ∞ 2 2 n μφ [ϕ = n] ≤ k μφ [ϕ = n] 2 Eμφ [ϕ ] = n=1

=2

∞

⎛ ⎝

n=1

n=1 ϕ−1 [ϕ=n] k=0

k=1

⎞

r A ◦ T k ⎠ dμφ = 2

r A dμφ ≤ 2

ra dμφ < ∞.

It follows that −ϕ satisfies Case (2)(c) of Theorem 5. By Theorem 3 for BIP systems, ϕ n satisfies the central limit theorem, and (14) follows. Proof of Theorem 4 for Systems without the BIP Property. It is enough to treat the case Eμφ [ψ], Ptop (φ) = 0. By Lemma 1, Ptop (φ) = 0 and μφ = μφ . By Kac’ formula, ψ ∈ L 1 , and Eμφ [ψ] = 0. We deduce from Theorem 4 in the BIP case that Ptop (φ + tψ) = o(t). By Theorem 8, Ptop (φ + tψ) = o(t). The remaining part of the Theorem is because of the ergodicity of μφ , see [BS]. Proof of Theorem 2 for Systems without the BIP Property. It is enough to treat the case Eμφ [ψ], Ptop (φ) = 0. Suppose Ptop (φ + tψ) = ct + t α L(1/t) with c ∈ R, 1 < α < 2 and L(x) slowly varying at infinity. The previous section shows that c = 0. By Theorem 8 Ptop (φ + tψ) = μφ1(A) t α [1+o(1)]L(1/t). L(x):= 1+o(1) μφ (A) L(x) is slowly varying at infinity, therefore by Theorem 2 for BIP systems, ∃Bn regularly varying of dist. 1 −−→ G α . Bn ψ n − √ n→∞ Since 1 < α < 2, Bnn → 0, so B1n [ϕ n − n/μφ (A)] is tight. Theorem 7 now implies dist. that B1n ψn −−−→ G ∗α , where G ∗α is equal to G α up to change of scale. Renormalizing n→∞ Bn , we obtain (2) in Theorem 2, and we proved (1)⇒(2). The other direction is handled

index α such that

in the same way.

Proof of Theorem 2 for Systems without the BIP Property. It is enough to treat the case Ptop (φ), Eμφ [ψ] = 0. We saw above that c = 0. Part 1 (Taylor expansion). By Theorem 8, Ptop (φ+tψ) = 21 σ 2 t 2 +o(t 2 ) iff Ptop (φ + tψ) = 2 dist. 1 √ σ t 2 +o(t 2 ). Our results for BIP maps say that this is equivalent to √1n ψ n −−−→ 2 μφ (A)

σ2 μφ (A) )

n→∞

dist. √1 ψn −−−→ n n→∞

N (0, σ 2 )

N (0, w.r.t. μφ . By Theorem 7, this happens iff (see the remark at the end of the proof of Theorem 7). We explain why in this case ψ ∈ L 2 (μφ ). By Theorem 5, ψ ∈ L 2 (μφ ). When we proved (14), we saw that ( ) ⇒ ϕ ∈ L 2 (μφ ). Therefore, ψ − sup ψ ∈ L 2 . It follows ϕ−1 2 k 2 2 that k=0 (ψ − sup ψ) ◦ T dμφ + positive terms < ∞, whence (ψ − sup ψ) ∈ L . By Kac’ formula, ψ − sup ψ ∈ L 2 (μφ ), and so ψ ∈ L 2 . Part 2 (Critical expansion). By Theorem 8, Ptop (φ + tψ) = t 2 L(1/t) with L(x) slowly 2 varying and not asymptotically constant, iff Ptop (φ + tψ) = 1+o(1) μφ (A) t L(1/t) with such L. By the BIP property, this is equivalent to the existence of Bn r.v. of index

1 2

such that

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O. Sarig

√ dist. n 1 1 ψ − − − → N (0, 1), E [ψ] = 0, and μ n Bn Bn → 0. By (14) Bn [ϕ n − n/μφ (A)] is tight, φ n→∞ dist. dist. so B1n ψ n −−−→ N (0, 1) is equivalent to B1∗ ψn −−−→ N (0, 1) for Bn∗ proportional to n n→∞ n→∞ Bn . This gives the equivalence in Theorem 3, Part 2.

To finish the proof, it is enough to observe that the BIP property, the expansion 2 Ptop (φ + tψ) = 1+o(1) μφ (A) t L(1/t), and Theorem 5 Case (2)(c) show that L(x) → ∞ whenever it is not asymptotic to a constant. Proof of Theorem 6. Without loss of generality φ has zero pressure, and ψ has zero expectation (and then Ptop (φ + tψ) = t α L(1/t)). Fix an arbitrary finite union of states A so large that ψ < Eμφ [ψ] − = − outside A, and let ϕ(x):= 1 A (x) min{n ≥ 1: T n (x) ∈ A}. Let ψ be the induced version of ψ on A. By Theorem 8, Ptop (φ + tψ) =

1 + o(1) α t L(1/t) as t → ∞. μφ (A)

Since 1 < α < 2, L(t) must be asymptotically non-negative (see Sect. 3). By Theorem t −α 5, μφ (A ∩ [|ψ| > t]) ∼ |(1−α)| L(t) as t → ∞. ϕ−1 By choice of A, ψ = ψ + k=1 ψ ◦ T k , where each summand under the sigma symbol is less than −. It follows that (ϕ − 1) − ψ∞ ≤ |ψ| ≤ ϕψ∞ . Since L(x) is slowly varying, L(λx), L(λ + x) ∼ L(x) as x → ∞ for all λ ∈ R+ , and so

−α t −α L(t) μφ [ϕ > t] ≤ μφ |ψ| > (t − 1) − ψ∞ ∼ , |(1 − α)| −α L(t)

ψ−α ∞ t . μφ [ϕ > t] ≥ μφ |ψ| > tψ∞ ] ∼ |(1 − α)| Consequently, μφ [ϕ > n] L(n) nα . We now appeal to Gouëzel [Gou1], Theorem 1.3 (see also [S5]), which says that in our context for every f, g locally Hölder continuous supported inside A with non-zero expectation, ∞ Covμφ ( f, g ◦ T n ) ∼ μφ [ϕ > k] f dμφ gdμφ . k=n+1

The theorem follows from Karamata’s Theorem.

Appendix A. Slow and Regular Variation Slow and Regular Variation. A positive function L : (c0 , ∞) → R is called slowly varying (at infinity) if it is Borel measurable and L(ts) −−−→ 1 for all s > 0. L(t) t→∞ A positive sequence {cn }n≥1 is called slowly varying (at infinity) if L(t):= c[t] is slowly varying (at infinity). A positive function f : (c0 , ∞) → R is called regularly varying at infinity with index α, if f (x) = x α L(x) with L(x) slowly varying at infinity. A positive sequence {cn }n≥1


663

is called regularly varying at infinity with index α, if f (t):= c[t] is regularly varying at infinity with index α. For example, log x, 1/ ln ln x are slowly varying at infinity, and x α ln x(ln ln x)2 , α x / ln x are regularly varying with index α. Sufficient Condition for Regular Variation. Let f (x) be a positive continuous function, and {an }, {bn } some positive numbers such that lim sup bn =∞ and lim sup bbn+1 =1. n n→∞

n→∞

If lim an f (bn x) exists, is positive, and is continuous on some open interval (a, b) ⊂ n→∞

R+ , then f (x) is regularly varying at infinity ([BGT], Theorem 1.9.2). The General Form of Regularly Varying Functions. A Borel function f (x) is regularly varying at infinity with index α iff x du α as x → ∞, f (x) = [c + o(1)]x exp (u) u 1 where c > 0 and (u) −−−→ 0 ([BGT], Theorem 1.3.1). x→∞

In particular, any regularly varying function f (x) with index α satisfies f (x) → ∞ when α > 0 and f (x) → 0 when α < 0 ([BGT], Prop. 1.5.1). Uniform Convergence Theorem. If L(t) is slowly varying at infinity, then uniformly on compact subsets of (0, ∞) ([BGT], Theorem 1.2.1).

L(ts) −−→ 1 L(t) − t→∞

Asymptotic Inversion Theorem. If f (x) is regularly varying at infinity with positive index α, then there exists g(x) regularly varying at infinity with index 1/α such that ( f ◦ g)(x) ∼ (g ◦ f )(x) ∼ x as x → ∞ ([BGT], Theorem 1.5.12). Differentiating Asymptotic Relations: The Monotone Density Theorem. Suppose x U (t) = 0 u(y)dy, and L(x) is slowly varying at infinity. (1) If u(y) is monotone at some interval (0, δ) and ρ ≥ 0, then U (t) ∼ t ρ L(1/t) as t → 0+ implies u(t) ∼ ρt ρ−1 L(1/t) as t → 0+ . (2) If u(y) is monotone at some interval (δ, ∞) and ρ ∈ R, then U (x) ∼ x ρ L(x) as x → ∞ implies u(x) ∼ ρx ρ−1 L(x) as x → ∞. Here and throughout f (x) ∼ 0 · g(x) means f (x) = o(g(x)) ([BGT], Theorems 1.7.2 and 1.7.2b). Integrating Asymptotic Relations: Karamata’s Theorem. Suppose L(x) is slowly varying at infinity and locally bounded. Then as x → ∞, x x ρ+1 L(x), f orallρ > −1, t ρ L(t)dt ∼ ρ+1 a ∞ x ρ+1 L(x), f orallρ < −1. t ρ L(t)dt ∼ − ρ+1 x

664

O. Sarig

The converse is also true: Any positive locally bounded L(x) for which one of these relations holds for some ρ = −1 must be slowly varying ([BGT], Theorems 1.5.11 and 1.6.1) After a change of variables, Karamata’s theorem implies that if L(x) is slowly varying at infinity and α > −1, then t t 1+α L(1/t) as t → 0+ . τ α L(1/τ )dτ ∼ 1 + α 0 Conversely, if L satisfies the above, then it must be slowly varying at infinity. Karamata’s Tauberian Theorem. Let U (x) be a non-decreasing function on R, which is continuous from the right, and such that U (0) = 0. Suppose L(x) is slowly varying at infinity, and c > 0, ρ ≥ 0. The following are equivalent: cx ρ L(x), as x → ∞. (1 + ρ) ∞ c e−t x dU (x) ∼ ρ L(1/t), as t → 0+ t 0 ([BGT], Theorem 1.7.1). U (x) ∼

Truncated Variance: Feller’s Theorem. Let F(x) be a right x continuous probability distribution function such that F(0) = 0, and set U (x):= 0 y 2 d F(y). Suppose L(x) is slowly varying at infinity as x → ∞, c = 0, and 0 < ρ < 2. The following are equivalent: U (x) ∼ cx ρ L(x), as x → ∞, cρ ρ−2 x L(x), as x → ∞. 1 − F(x) ∼ 2−ρ (See Feller [F] VIII.9 for generalizations). ∞ ∞

y Proof. Start with the identity 1 − F(x) = x y −2 d 0 t 2 d F(t) = x y −2 dU (y). Integration by parts gives: y=∞ ∞ 1 − F(x) = y −2 U (y) +2 y −3 U (y − )dy. y=x

If U (y) ∼

cy ρ L(y),

then

U (y − )

∼

x

cy ρ L(y).

By Karamata’s theorem: ∞ y ρ−3 dy 1 − F(x) = −cx ρ−2 L(x)[1 + o(1)] + 2cL(x)[1 + o(1)] x

cρ ρ−2 x = L(x)[1 + o(1)]. 2−ρ

x To see the other direction, integrate by parts U (y) = 0 y 2 d F(y): y=x x x U (x) = y 2 F(y) −2 y F(y − )dy = x 2 F(x) − 2 y F(y − )dy y=0

0

0

x

= −x 2 [1 − F(x)] + 2

y[1 − F(y − )]dy.

0

Now plug into this expression the asymptotic formula for 1 − F(x) and conclude as before, using Karamata’s theorem.


665

Appendix B. The Fisher–Felderhoff Droplet Model We describe a crude simplification of a model in [FF]. A ‘vapor’ close to the condensation point consists of microscopic droplets. The interaction between particles in different droplets is negligible, but the interaction between particles in the same droplet is strong, and long–range.7 When two droplets ‘touch’, they become one. ‘Condensation’ is the appearance of macroscopic droplets. Here is a lattice–gas model of this situation. Space is discretized and described by a one–sided one-dimensional string of sites, each of which can be in one of two states: empty (state ‘0’) or occupied (‘1’). The configuration space is {0, 1}N0 . A ‘droplet’ is a maximal string of occupied sites. We describe the interaction by prescribing the function φ(x0 , x1 , . . .) := −βU (x0 |x1 , x2 , . . .), where β is a constant (‘inverse temperature’) and U (x0 |x1 , x2 , . . .) is the energy due to the interaction of site zero and the other sites.8 Note that the energy due to the interaction between the first n sites and the rest is minus the nth Birkhoff sum of φ. It follows that the Helmholtz free energy(=Energy – β1 ×Entropy) per site is up to a constant ⎡

n−1

⎤

1⎣ 1 ⎦ φ ◦ T k dμφ − μφ [a] log − n μφ [a] k=0 n–cylinders = − h μφ (T ) + φdμφ = −Ptop (φ),

lim

n→∞

at least when φ has an equilibrium measure μφ . Since different droplets do not interact, φ takes the form φ(0, ∗) = 0 , φ(1, 1, . . . , 1, 0, ∗):= f (n) , -. / n

for some function f (n). If the interaction is ‘long range’, then this function is not locally Hölder, because the effect of far away sites is not exponentially small. Consider now the following re-coding of a configuration:(x0 , x1 , . . .) → (y0 , y1 , . . .), where xi = 0 ⇒ yi = 0, and xi = 1 ⇒ yi = 1+number of occupied sites to the right of i until the first unoccupied site, for example: (0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, . . .) → (0, 2, 1, 0, 0, 1, 0, 3, 2, 1, 0, . . .). In this coding, the configuration space becomes the renewal shift: the topological Markov shift with state space N ∪ {0} and transition matrix ⎧ ⎪ ⎨1 i = 0; A = (ti j ) where ti j = 1 i > 0, j = i − 1; ⎪ ⎩0 otherwise. 7 One example of long–range interactions in liquid droplets is ‘surface tension’. 8 It is useful to think of U (x |x , x , . . .) as of the energy cost of separating site zero from sites n, n > 0, 0 1 2

and moving it to infinity – that is, if site zero is occupied.

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In the new coordinates the interaction becomes locally Hölder (‘short range’): '(y0 , y1 , . . .) = f (y0 ) y0 = 0; φ 0 y0 = 0. Thus a compact shift with a long range potential is recoded as a non-compact shift with a short range potential. The critical phenomena for the Fisher–Felderhoff model for various choices of f (n) is described in [FF] and [Wa1, Wa2]. Acknowledgements. The author wishes to thank the referee for his careful reading of the manuscript and for many valuable suggestions, and Michael Fisher and David Ruelle for useful discussions

References [ADU]

Aaronson, J., Denker, M., Urbański, M.: Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. AMS 337, 495–548 (1993) [AD] Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1(2), 193–237 (2001) [BG] Bálint, P., Gouëzel, S.: Limit theorems in the stadium billiard. Commun. Math. Phys. 263, 2, 461– 512 (2006) [BGT] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Encyclopedia of Math. and its Appl. 27, Cambridge: Cambridge Univ. Press 1987. [BDFN] Binney, J.J., Dowrick, N.J., Fisher, A.J., Newman, M.E.J.: The theory of critical phenomena, an introduction to the renormalization group. Oxford Science Publications, Oxford: Oxford University Press, 1992 [Bo] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Berlin-New York: Springer-Verlag, 1975 [BS] Buzzi, J., Sarig, O.: Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Erg. Th. Dynam. Sys. 23, 1383–1400 (2003) [Ea] Eagleson, G.K.: Some simple conditions for limit theorems to be mixing. (Russian) Teor. Verojatnost. i Primenen. 21(3), 653–660 (1976) Engl. Transl. in Theor. Probab. Appl. 21(3), 637–642 (1976) [El] Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Grund. Math. Wissenschaften 271, Berlin Heidelberg-Newyork: Springer Verlag 1985 [F] Feller, W.: An introduction to probability theory and its applications. Volume II, Second edition, Newyork: John Wiley & Sons, 1971 [FF] Fisher, M.E., Felderhof, B.U.: Phase transition in one–dimensional cluster–interaction fluids: IA. Thermodynamics, IB. Critical behavior. II. Simple logarithmic model. Ann. Phy. 58, 177–280 (1970) [GK] Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Translated and annotated by K.L. Chung, with an Appendix by J.L. Doob. Readings MA: Addison–Wesley Publishing Company, 1954 [Gou1] Gouëzel, S.: Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139, 29–65 (2004) [Gou2] Gouëzel, S.: Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128(1), 82–122 (2004) [Gu1] Gureviˇc, B.M.: Topological entropy of a countable Markov chain. Dokl. Akad. Nauk SSSR 187, 715–718 (1969) [Gu2] Gurevich, B.M.: A variational characterization of one-dimensional countable state Gibbs Random field. Z. Wahrscheinlichkeitstheorie verw. Gebiete 68, 205–242 (1984) [H1] Haydn, N., Isola, S.: Parabolic rational maps. J. London Math. Soc. 63(2), 673–689 (2001). [Hi] Hilfer, R.: Classification theory for anequilibrium phase transitions. Phys. Rev. E 48(4), 2466–2475 (1993) [Ho] Hofbauer, F.: Examples for the non–uniqueness of the equilibrium states. Trans. AMS 228, 223–241 (1977) [Ka] Kato, T.: Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics.Berlin: Springer-Verlag, 1995 [Ke] Keane, M.: Strongly mixing g-measures. Invent. Math. 16, 309–324 (1972)


[Lo] [ML] [MU1] [MU2] [MT] [N] [PS] [Ru]

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Lopes, A.O.: The Zeta function, non–differentiability of pressure, and the critical exponent of transition. Adv. Math. 101(2), 133–165 (1993) Martin–Löf, A.: Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature. Commun. Math. Phys. 32, 75–92 (1973) Urbański, M., Mauldin, R.D.: Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125, 93–130 (2001) Mauldin, R.D., Urbański, M.: Graph directed Markov systems; Geometry and dynamics of limit sets. Cambridge Tracts in Mathematics, 148, Cambridge: Cambridge University Press, Cambridge, 2003. Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 194, 191– 210 (2004) Nagaev, S.V.: Some limit theorems for stationary Markov chains. (Russian) Teor. Veroyatnost. i Primenen. 2, 389–416 (1957) Prellberg, T., Slawny, J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66(1–2), 503–514 (1992) Ruelle, D.: Thermodynamic Formalism, The mathematical structures of equilibrium statistical mechanics. 2nd Ed. . Cambridge Mathematical Library. Cambridge: Cambridge University Press, 2004 Sarig, O.M.: Thermodynamic Formalism for Countable Markov shifts. Erg. Th. Dyn. Sys. 19, 1565–1593 (1999) Sarig, O.: Phase Transitions for Countable Markov Shifts. Commun. Math. Phys. 217, 555–577 (2001). Sarig, O.: Characterization of existence of Gibbs measures for Countable Markov shifts. Proc. of AMS. 131(6), 1751–1758 (2003) Sarig, O.: Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121, 285–311 (2001) Sarig, O.: Subexponential decay of correlations. Invent. Math. 150, 629–653 (2002) Sarig, O.: Thermodynamic formalism for countable Markov shifts. Tel-Aviv University Thesis (2000). Sarig, O.: On an example with a non-analytic topological pressure. C. R. Acad. Sci. Paris Sér. I Math. 330(4), 311–315 (2000) Stanley, H.E.: Introduction to phase transitions and critical phenomena. Oxford: Oxford University Press 1971 Walters, P.: Ruelle’s operator theorem and g-measures. Trans. Amer. Math. Soc. 214, 375–387 (1975) Wang, X.-J.: Abnormal fluctuations and thermodynamic phase transition in dynamical systems. Phys. Review A 39(6), 3214–3217 Wang, X.-J.: Statistical physics of temporal intermittency. Phys. Review A 40(11), 6647–6661 (1989) Yuri, M.: Thermodynamic formalism for countable to one Markov systems. Trans. Amer. Math. Soc. 355(7), 2949–2971 (2003) Yuri, M.: Phase transition, non-Gibbsianness and subexponential instability, Ergodic Thy Dynam. Syst. 25, 1325–1342 (2005) Zolotarev, V.M.: One–dimensional stable distributions . Transl. Math. Monog. 65, Providence, RI: Amer. Math. Sec., 1986

Communicated by G. Gallavotti


Communications in


Lifshitz Tails in Constant Magnetic Fields Frédéric Klopp1 , Georgi Raikov2 1 Université de Paris Nord et Institut Universitaire de France, Département de mathématiques, Avenue J.

Baptiste Clément, 93430 Villetaneuse, France. E-mail: [email protected]

2 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Santiago,

Chile. E-mail: [email protected] Received: 12 September 2005 / Accepted: 25 January 2006 Published online: 19 August 2006 – © Springer-Verlag 2006

Abstract: We consider the 2D Landau Hamiltonian H perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of H . If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained by the first author and T. Wolff in [25] for the case of a vanishing magnetic field. 1. Introduction Let H0 = H0 (b) := (−i∇ − A)2 − b

(1.1)

be the unperturbed Landau Hamiltonian, essentially self-adjoint on C0∞ (R2 ). Here A = (− bx22 , bx21 ) is the magnetic potential, and b ≥ 0 is the constant scalar magnetic field. It is well-known that if b > 0, then the spectrum σ (H0 ) of the operator H0 (b) consists of the so-called Landau levels 2bq, q ∈ Z+ , and each Landau level is an eigenvalue of infinite multiplicity. If b = 0, then H0 = −, and σ (H0 ) = [0, ∞) is absolutely continuous. Next, we introduce a random Z2 -ergodic alloy-type electric potential V (x) = Vω (x) :=

ωγ u(x − γ ), x ∈ R2 .

γ ∈Z2

Our general assumptions concerning the potential Vω are the following ones:

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• H1 : The single-site potential u satisfies the estimates 0 ≤ u(x) ≤ C0 (1 + |x|)−κ , x ∈ R2 ,

(1.2)

with some κ > 2 and C0 > 0. Moreover, there exists an open non-empty set ⊂ R2 and a constant C1 > 0 such that u(x) ≥ C1 for x ∈ . • H2 : The coupling constants ωγ γ ∈Z2 are non-trivial, almost surely bounded i. i. d. random variables. Evidently, these two assumptions entail M := ess-sup sup |Vω (x)| < ∞. ω

(1.3)

x∈R2

On the domain of H0 define the operator H = Hω := H0 (b) + Vω . The integrated density of states (IDS) for the operator H is defined as a non-decreasing left-continuous function Nb : R → [0, ∞) which almost surely satisfies ϕ(E)dNb (E) = lim R −2 Tr 1 R ϕ(H )1 R , ∀ϕ ∈ C0∞ (R). (1.4) R

R→∞

Here and in the sequel 1O denotes the characteristic function of the set O, and R := R R 2 − 2 , 2 . By the Pastur-Shubin formula (see e.g. [36, Sect. 2] or [11, Cor. 3.3]) we have ϕ(E)dNb (E) = E Tr 11 ϕ(H )11 , ∀ϕ ∈ C0∞ (R), (1.5) R

where E denotes the mathematical expectation. Moreover, there exists a set ⊂ R such that σ (Hω ) = almost surely, and supp dNb = . The aim of the present article is to study the asymptotic behavior of Nb near the edges of . It is well known that, for many random models, this behavior is characterized by a very fast decay which goes under the name of “Lifshitz tails”. It was studied extensively in the absence of magnetic field (see e.g. [31, 15]), and also in the presence of magnetic field for other types of disorder (see [2, 6, 12, 7, 13]). 2. Main Results In order to fix the picture of the almost sure spectrum σ (Hω ), we assume b > 0, and make the following two additional hypotheses: • H3 : The support of the random variables ωγ , γ ∈ Z2 , consists of the interval [ω− , ω+ ] with ω− < ω+ and ω− ω+ ≤ 0. • H4 : We have M+ − M− < 2b where ±M± := ess-supω supx∈R2 (±Vω (x)). ∞ [2bq + M , 2bq + Assumptions H1 – H4 imply M− M+ ≤ 0. Moreover, the union ∪q=0 − 2 M+ ] which contains , is disjoint. Introduce the bounded Z -periodic potential W (x) := u(x − γ ), x ∈ R2 , γ ∈Z2

Lifshitz Tails in Constant Magnetic Fields

671

and on the domain of H0 define the operators H ± := H0 + ω± W . It is easy to see that ∞ ∞ σ (H − ) ⊆ ∪q=0 [2bq + M− , 2bq], σ (H + ) ⊆ ∪q=0 [2bq, 2bq + M+ ],

and σ (H − ) ∩ [2bq + M− , 2bq] = ∅, σ (H + ) ∩ [2bq, 2bq + M+ ] = ∅, ∀q ∈ Z+ . Set E q− := inf σ (H − ) ∩ [2bq + M− , 2bq] ,

E q+ := sup σ (H + ) ∩ [2bq, 2bq + M+ ] .

Following the argument in [16] (see also [31, Theorem 5.35]), we easily find that ∞ = ∪q=0 [E q− , E q+ ],

i.e. is represented as a disjoint union of compact intervals, and each interval [E q− , E q+ ] contains exactly one Landau level 2bq, q ∈ Z+ . In the following theorems we describe the behavior of the integrated density of states Nb near E q− , q ∈ Z+ ; its behavior near E q+ could be analyzed in a completely analogous manner. Our first theorem concerns the case where E q− = 2bq, q ∈ Z+ . This is the case if and only if ω− = 0; in this case, the random variables ωγ , γ ∈ Z2 , are non-negative. Theorem 2.1. Let b > 0 and Assumptions H1 – H4 hold. Suppose that ω− = 0, and that P(ω0 ≤ E) ∼ C E κ , E ↓ 0,

(2.1)

for some C > 0 and κ > 0. Fix the Landau level 2bq = E q− , q ∈ Z+ . i) Assume that C− (1 + |x|)−κ ≤ u(x) ≤ C+ (1 + |x|)−κ , x ∈ R2 , for some κ > 2, and C+ ≥ C− > 0. Then we have lim

E↓0

ii) Assume have

e−C+ |x| C+

β

2 ln | ln (Nb (2bq + E) − Nb (2bq))| =− . ln E κ−2

≤ u(x) ≤

e−C− |x| C−

β

, x ∈ R2 , β ∈ (0, 2], C+ ≥ C− > 0. Then we

2 ln | ln (Nb (2bq + E) − Nb (2bq))| = 1+ . E↓0 ln | ln E| β

lim 1

−C− |x|2

2

(2.2)

0 | 0. Then there exists δ > 0 such that

(2.3)

for some C+ ≥ C− > 0, x0 ∈ R2 , and

ln | ln (Nb (2bq + E) − Nb (2bq)| ln | ln E| ln | ln (Nb (2bq + E) − Nb (2bq)| ≤ lim sup ≤ 2. ln | ln E| E↓0

1 + δ ≤ lim inf E↓0

(2.4)

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F. Klopp, G. Raikov

The proof of Theorem 2.1 is contained in Sects. 3–5. In Sect. 3 we construct a periodic approximation of the IDS Nb which plays a crucial role in this proof. The upper bounds of the IDS needed for the proof of Theorem 2.1 are obtained in Sect. 4, and the corresponding lower bounds are deduced in Sect. 5. Remarks. i) In the first and second part of Theorem 2.1 we consider one-site potentials u respectively of power-like or exponential sub-Gaussian decay at infinity, and obtain the values of the so-called Lifshitz exponents. Note however that in the case of power-like decay of u the double logarithm of Nb (2bq + E) − Nb (2bq) is asymptotically proportional to ln E (see (2.2)), while in the case of exponentially decaying u this double logarithm is asymptotically proportional to ln | ln E| (see (2.3)); in both cases the Lifshitz exponent is defined as the corresponding proportionality factor. In the third part of the theorem which deals with one-site potentials u of super-Gaussian decay, we obtain only upper and lower bounds of the Lifshitz exponent. It is natural to conjecture that the value of this exponent is 2, i.e. that the upper bound in (2.4) reveals the correct asymptotic behavior. ii) In the case of a vanishing magnetic field, the Lifshitz asymptotics for random Schrödinger operator with repulsive random alloy-type potentials has been known since long ago (see [17]). To the authors’ best knowledge the Lifshitz asymptotics for the Landau Hamiltonian with non-zero magnetic field, perturbed by a positive random alloy-type potential, is considered for the first time in the present article. However, it is appropriate to mention here the related results concerning the Landau Hamiltonian with repulsive random Poisson potential. In [2] the Lifshitz asymptotics in the case of a power-like decay of the one-site potential u, was investigated. The case of a compact support of u was considered in [6]. The results for the case of a compact support of u were essentially used in [12] and [7] (see also [13]), in order to study the problem in the case of an exponential decay of u. Our second theorem concerns the case where E q− < 2bq, q ∈ Z+ . This is the case if and only if ω− < 0. In order to handle this case, we need some facts from the magnetic Floquet-Bloch theory. Let := g1 Z ⊕ g2 Z with g j > 0, j = 1, 2. Introduce the tori T := R2/ , T∗ := R2/ ∗ ,

(2.5)

where ∗ := 2πg1−1 Z ⊕ 2πg2−1 Z is the lattice dual to . Denote by O and O ∗ the fundamental domains of T and T∗ respectively. Let W : R2 → R be a -periodic bounded real-valued function. On the domain of H0 define the operator HW := H0 + W. Assume that the scalar magnetic field b ≥ 0 satisfies the integer-flux condition with respect to the lattice , i.e. that bg1 g2 ∈ 2π Z+ . Fix θ ∈ T∗ . Denote by h 0 (θ ) the self-adjoint operator generated in L 2 (O ) by the closure of the non-negative quadratic form |(i∇ + A − θ ) f |2 d x O

defined originally on the set

f = g O | g ∈ C ∞ (R2 ), (τγ g)(x) = g(x), x ∈ R2 , γ ∈ ,

where τ y , y ∈ R2 , is the magnetic translation given by (τ y g)(x) := eib

y1 y2 2

eib

x∧y 2

g(x + y), x ∈ R2 ,

(2.6)


673

with x ∧ y := x1 y2 − x2 y1 . Note that the integer-flux condition implies that the operators τγ , γ ∈ , commute with each other, as well as with operators i ∂∂x j + A j , j = 1, 2 (see (1.1)), and hence with H0 and HW . In the case b = 0, the domain of the operator h 0 is isomorphic to the Sobolev space H2 (T ), but if b > 0, this is not the case even under the integer-flux assumption since h 0 acts on U (1)-sections rather than on functions over T (see e.g [30, Subsect. 2.2]). On the domain of h 0 define the operator h W (θ ) := h 0 (θ ) + W, θ ∈ T∗ . Set

H0 :=

(2.7)

O ∗

⊕ h 0 (θ )dθ, HW :=

O ∗

⊕ h W (θ )dθ.

(2.8)

It is well-known (see e.g [10, 35 or 30, Subsect. 2.4]) that the operators H0 and HW are unitarily equivalent to the operators H0 and HW respectively. More precisely, we have H0 = U ∗ H0 U and HW = U ∗ HW U , where U: L 2 (R2 ) → L 2 (O × O ∗ ) is the unitary Gelfand-type operator defined by (U f )(x; θ ) :=

1

vol T∗ γ ∈

e−iθ(x+γ ) (τγ f )(x), x ∈ O , θ ∈ T∗ .

(2.9)

Evidently for each θ ∈ T∗ the spectrum of the operator h W (θ ) is purely discrete. Denote ∞ by E j (θ ) j=1 the non-decreasing sequence of its eigenvalues. Let E ∈ R. Set J (E) := j ∈ N ; there exists θ ∈ T∗ such that E j (θ ) = E . Evidently, for each E ∈ R the set J (E) is finite. If E ∈ R is an end of an open gap in σ (H0 + W), then we will call it an edge in σ (H0 + W). We will call the edge E in σ (H0 + W) simple if # J (E) = 1. Moreover, we will call the edge E non-degenerate if for each j ∈ J (E) the number of points θ ∈ T∗ such that E j (θ ) = E is finite, and at each of these points the extremum of E j is non-degenerate. Assume at first that b = 0. Then H0 = −, and we will consider the general d-dimensional situation; the simple and non-degenerate edges in σ (−+W) are defined exactly as in the two-dimensional case. If W : Rd → R is a bounded periodic function, it is well-known that: • The spectrum of − + W is absolutely continuous (see e.g. [33, Theorems XIII.90, XIII.100]). In particular, no Floquet eigenvalue E j : T∗ → R, j ∈ N, is constant. • If d = 1, all the edges in σ (− + W) are simple and non-degenerate (see e.g. [33, Theorem XIII.89]). • For d ≥ 1 the bottom of the spectrum of − + W is a simple and non-degenerate edge (see [19]). • For d ≥ 1, the edges of σ (− + W) generically are simple (see [24]). Despite the widely spread belief that generically the higher edges in σ (− + W) should also be non-degenerate in the multi-dimensional case d > 1, there are no rigorous results in support of this conjecture. Let us go back to the investigation of the Lifshitz tails for the operator − + Vω . It follows from the general results of [16] that E − (respectively, E + ) is an upper (respectively, lower) end of an open gap in σ (−+ Vω ) if and only if it is an upper (respectively,

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F. Klopp, G. Raikov

lower) end of an open gap in the spectrum of − + ω− W (respectively, − + ω+ W ). For definiteness, let us consider the case of an upper end E − . The asymptotic behavior of the IDS N0 (E) as E ↓ E − has been investigated in [28, 29] in the case d = 1, and in [18] in the case d ≥ 1 and E − = inf σ (− + ω− W ). Note that the proofs of the results of [28, 29 and 18], essentially rely on the non-degeneracy of E − . Later, the Lifshitz tails for the operator − + Vω near the edge E − were investigated in [21] under the assumptions that d ≥ 1, E − > inf σ (− + ω− W ), and that E − is a non-degenerate edge in the spectrum of − + ω− W ; due to the last assumption these results are conditional. However, it turned out possible to lift the non-degeneracy assumption in the two-dimensional case considered in [25]. First, it was shown in [25, Theorem 0.1] that for any single-site potential u satisfying assumption H1 , we have lim sup E↓0

ln | ln (N0 (E − + E) − N0 (E − ))| 0 such that ln | ln (N0 (E − + E) − N0 (E − ))| = −α E↓0 ln E

lim

(2.10)

under the unique generic hypothesis that E − is a simple edge. Note that the absolute continuity of σ (− + ω− W ) plays a crucial role in the proofs of the results of [25]. Assume now that the scalar magnetic field b > 0 satisfies the rational flux condition b ∈ 2π Q. More precisely, we assume that b/2π is equal to the irreducible fraction p/r , p ∈ N, r ∈ N. Then b satisfies the integer-flux assumption with respect, say, to the lattice = r Z ⊕ Z, and the operator H − is unitarily equivalent to Hω− W . As in the non-magnetic case, in order to investigate the Lifshitz asymptotics as E ↓ E q− of Nb (E), we need some information about the character of E q− as an edge in the spectrum of H − . For example, if we assume that E q− is a simple edge, and the corresponding Floquet band does not shrink into a point, we can repeat almost word by word the argument of the proof of [25, Theorem 0.2], and obtain the following Theorem 2.2. Let b > 0, b ∈ 2π Q, and Assumptions H1 –H4 hold. Assume that the support of u is compact, ω− < 0, and P(ω0 − ω− ≤ E) ∼ C E κ , E ↓ 0, for some C > 0 and κ > 0. Fix q ∈ Z+ . Suppose E q− is a simple edge in the spectrum of the operator H − , and that the function E j , j ∈ J (E q− ), is not identically constant. Then there exists α > 0 such that lim

E↓0

ln | ln (Nb (E q− + E) − Nb (E q− ))| ln E

= −α.

(2.11)

Remarks. i) It is believed that under the rational-flux assumption the Floquet eigenvalues E j , j ∈ N, for the operator H − generically are not constant. Note that this property may hold only generically due to the obvious counterexample where u = 11 , H − = H0 + ω− , and for all j ∈ N the Floquet eigenvalue E j is identically equal to 2b( j − 1) + ω− . Also, in contrast to the non-magnetic case, we do not know whether the edges in the spectrum of H − generically are simple. ii) The definition of the constant α in (2.11) is completely analogous to the one in (2.10) which concerns the non-magnetic case. This definition involving the concepts of


675

Newton polygon, Newton diagram, and Newton decay exponent, is not trivial, and can be found in the original work [25], or in [22, Subsect. 4.2.8]. 3. Periodic Approximation 2

Pick a > 0 such that ba 2π ∈ N. Set L := (2n + 1)a/2, n ∈ N, and define the random 2 2LZ -periodic potential per V per (x) = Vn,ω (x) := Vω 12L (x + γ ), x ∈ R2 . γ ∈2L Z2 per

per

On the domain of H0 define the operator H per = Hn,ω := H0 + Vn,ω . For brevity set T2L := T2L Z2 , T∗2L := T∗2L Z2 (see (2.5)). Note that the square 2L is the fundamental domain of the torus T2L , while ∗2L := π L −1 is the fundamental domain of T∗2L . As in (2.7), on the domain of h 0 define the operator h(θ ) = h per (θ ) := h 0 (θ ) + V per , θ ∈ T∗2L , and by analogy with (2.8) set Hper :=

∗2L

⊕ h per (θ )dθ.

As above, the operators H0 and H per are unitarily equivalent to the operators H0 and Hper respectively. Set per N per (E) = Nn,ω (E) := (2π )−2 N (E; h per (θ ))dθ, E ∈ R. (3.1) ∗2L

Here and in the sequel, if T is a self-adjoint operator with purely discrete spectrum, then N (E; T ) denotes the number of the eigenvalues of T less than E ∈ R, and counted with the multiplicities. The function N per plays the role of IDS for the operator H per since, similarly to (1.4) and (1.5), we have ϕ(E)dN per (E) = lim R −2 Tr 1 R ϕ(H per )1 R R

R→∞

almost surely, and E

R

ϕ(E)dN

per

(E) = E Tr 11 ϕ(H per )11 ,

(3.2)

for any ϕ ∈ C0∞ (R) (see e.g. the proof of [21, Theorem 5.1] where however the case of a vanishing magnetic field is considered). Theorem 3.1. Assume that Hypotheses H1 and H2 hold. Let q ∈ Z+ , η > 0. Then there exist ν > 0 and E 0 > 0 such that for E ∈ (0, E 0 ] and n ≥ E −ν we have −η E N per (2bq + E/2) − N per (2bq − E/2) −e−E ≤ Nb (2bq + E) − Nb (2bq − E) −η (3.3) ≤ E N per (2bq + 2E) − N per (2bq − 2E) + e−E .

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The main technical steps of the proof of Theorem 3.1 which is the central result of this section, are contained in Lemmas 3.1 and 3.2 below. Lemma 3.1. Let Q = Q ∈ L ∞ (R2 ), X := H0 + Q, D(X ) = D(H0 ). Then there exists = (b) > 0 such that for each α, β ∈ Z2 , and z ∈ C\σ (X ) we have

1 b+1 −1 e−η(z)|α−β| , χα (X − z) χβ HS ≤ 2 1/2 1 + (3.4) π η(z) where χα := 11 +α , α ∈ Z2 , η(z) = η(z; b, Q) := Hilbert-Schmidt norm, and |Q|∞ := Q L ∞ (R2 ) .

dist(z,σ (X )) |z|+|Q|∞ +1 ,

· HS denotes the

Proof. We will apply the ideas of the proof of [20, Prop. 4.1]. For ξ ∈ R2 set X ξ := eξ ·x X e−ξ ·x = (i∇ + A − iξ )2 + Q = X − 2iξ · (i∇ + A) + |ξ |2 . Evidently, X ξ − z = (X − z) 1 + (X − z)−1 |ξ |2 − 2iξ · (i∇ + A) .

(3.5)

Let us estimate the norm of the operator (X − z)−1 |ξ |2 − 2iξ · (i∇ + A) appearing at the right-hand side of (3.5). We have (X − z)−1 |ξ |2 ≤ |ξ |2 dist(z, σ (X ))−1 , (X − z)−1 2iξ · (i∇ + A) ≤ 2(H0 + 1)−1 (i∇ + A) · ξ − (X − z)−1 (Q − z − 1)(H0 + 1)−1 (i∇ + A) · ξ

1 |ξ | ≤ 2C 1 + η(z) with ((2q + 1)b)1/2 . 2bq + 1 q∈Z+

C = C(b) := (H0 + 1)−1 (i∇ + A) = sup

1 Choose ∈ 0, 8(C+1) and ξ ∈ R2 such that |ξ | = η(z). Then, by the above estimates, we have

1 η(z) (X − z)−1 |ξ |2 − 2iξ · (i∇ + A) ≤ 2 η(z)2 dist(z, σ (X ))−1 + 2C 1 + η(z) ≤ 2 η(z) + 2C(1 + η(z)) < 2 + 4C < 3/4,

(3.6)

since the resolvent identity implies η(z) < 1. Therefore, the operator X ξ −z is invertible, and (3.7) χα (X − z)−1 χβ = e−ξ ·x χα χα (X ξ − z)−1 χβ eξ ·x χβ .


677

Moreover, (3.5) and (3.6) imply χα (X ξ − z)−1 χβ HS ≤ 4(X − z)−1 χβ HS ≤ 4(H0 + 1)−1 χβ − (X − z)−1 (Q − z − 1)(H0 + 1)−1 χβ HS ≤ 4(H0 + 1)−1 χβ HS (1 + (X − z)−1 (Q − z − 1)) ≤ 4(H0 + 1)−1 χβ HS

1 . (3.8) × 1+ η(z) Finally, applying the diamagnetic inequality for Hilbert-Schmidt operators (see e.g. [1]), we get (H0 + 1)−1 χβ HS ≤ (H0 + 1)−1 (H0 + b + 1)(H0 + b + 1)−1 χβ HS ≤ (H0 + 1)−1 (H0 + b + 1)(− + 1)−1 χβ HS 2bq + b + 1 b+1 (− + 1)−1 χβ HS = = sup . 2π 1/2 q∈Z+ 2bq + 1

(3.9)

The combination of (3.7), (3.8), and (3.9) yields χα (X − z)−1 χβ HS ≤

1 2(b + 1) −ξ(α−β) 1 + . e π 1/2 η(z)

α−β Choosing ξ = η(z) |α−β| , we get (3.4).

Lemma 3.2. Assume that Hypotheses H1 and H2 hold. Then there exists a constant C > 1 such that for any ϕ ∈ C0∞ (R), and any n ∈ N, l ∈ N, we have

per E ϕ(E)dN (E) − ϕ(E)dN (E) b R R j −l Cl log l l+5 d ϕ ≤ cn e (3.10) sup (|x| + C) d x j (x) . x∈R, 0≤ j≤l+5 Proof. We will follow the general lines of the proof of [23, Lemma 2.1]. Due to the fact that we consider only the two-dimensional case, and an alloy-type potential which is almost surely bounded, the argument here is somewhat simpler than the one in [23]. By (1.5) and (3.2) we have

E ϕ(E)dNb (E) − ϕ(E)dN per (E) = E Tr 11 (ϕ(H ) − ϕ(H per ))11 . R

R

Next, we introduce a representation of the operator ϕ(H ) − ϕ(H per ) by the HelfferSjöstrand formula (see e.g. [4, Chap. 8]). Let ϕ˜ be an almost analytic extension of the function ϕ ∈ C0∞ (R) appearing in (3.10). We recall that ϕ˜ possesses the following properties: 1. 2. 3. 4.

If Im z = 0, then ϕ(z) ˜ = ϕ(z). supp ϕ˜ ⊂ {x + i y ∈ C; |y| < 1}. ϕ˜ ∈ S ({x + i y ∈ C; |y| < 1}). The family of functions x → ∂∂ϕz¯˜ (x + i y)|y|−m , |y| ∈ (0, 1), is bounded in S(R) for any m ∈ Z+ .

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F. Klopp, G. Raikov

Such extensions exist for ϕ ∈ S(R) (see [27, 4, Chapt. 8]), and there exists a constant C > 1 such that for any m ≥ 0, α ≥ 0, β ≥ 0, we have

∂β ∂ ϕ˜ sup sup x α β |y|−m (x + i y) ∂x ∂ z¯ 0≤|y|≤1 x∈R d β ϕ(x) α m log m+α log α+β+1 ≤C (3.11) sup sup x . β d x β ≤m+β+2, α ≤α x∈R Then the Helffer-Sjöstrand formula yields

E Tr 11 (ϕ(H ) − ϕ(H per ))11

1 ∂ ϕ˜ −1 per −1 (z) 11 (H − z) − (H − z) = E Tr 11 d xd y π C ∂ z¯

∂ ϕ˜ 1 −1 per per −1 (z) 11 (H − z) (V − V )(H − z) 11 d xd y . (3.12) = E Tr π C ∂ z¯

Next, we will show that 11 (H − z)−1 (V per − V )(H per − z)−1 11 is a trace-class operator for z ∈ C\R, and almost surely

M + |z| + 1 2 M(b + 1)2 −1 per per −1 11 (H − z) (V − V )(H − z) 11 Tr ≤ 1+ , 2π |Im z| (3.13) where .Tr denotes the trace-class norm. Evidently, 11 (H − z)−1 (V per − V )(H per − z)−1 11 Tr ≤

11 (H0 + 1)−1 2HS (V per

− V )(H0 + 1)(H − z)−1 (H0 + 1)(H per − z)−1 . (3.14)

per − By (3.9) we have 11 (H0 + 1)−1 2HS ≤ (b+1) 4π . Moreover, almost surely V V ≤ 2M. Finally, it is easy to check that both norms (H0 + 1)(H − z)−1 and (H0 + 1)(H per − z)−1 are almost surely bounded from above by 1 + M+|z|+1 |Im z| , so that (3.13) follows from (3.14). Taking into account estimate (3.13) and Properties 2, 3, and 4 of the almost analytic continuation ϕ, ˜ we find that (3.12) implies per E Tr 11 (ϕ(H ) − ϕ(H ))11 1 ∂ ϕ˜ (z)E Tr 11 (H − z)−1 (V per − V )(H per − z)−1 11 d xd y. (3.15) = π C ∂ z¯ 2

Our next goal is to obtain a precise estimate (see (3.19) below) on the decay rate as n → ∞ of E Tr 11 (H − z)−1 (V per − V )(H per − z)−1 11 with z ∈ C \ R and |Im z| < 1. Evidently, E Tr 11 (H − z)−1 (V per − V )(H per − z)−1 11 = E Tr 11 (H − z)−1 χα (V per − V )(H per − z)−1 11 , α∈Z2 ,|α|∞ >na


679

where |α|∞ := max j=1,2 |α j |, since V per = V on 2L , and therefore χα (V per − V ) = 0 if |α|∞ ≤ na. Hence, bearing in mind estimates (1.3) and (3.4), we easily find that |E Tr 11 (H − z)−1 (V per − V )(H per − z)−1 11 | ≤ E χ0 (H − z)−1 χα (V per − V )(H per − z)−1 χ0 Tr α∈Z2 ,|α|∞ >na

≤ 2M

E χ0 (H − z)−1 χα HS χα (H per − z)−1 χ0 HS

α∈Z2 ,|α|∞ >na

≤

M(b + 1)2 2π

|x| + M + 2 2 1+ |y|

α∈Z2 ,|α|∞ >na

2|α||y| (3.16) exp − |x| + M + 2

for every z = x + i y with 0 < |y| < 1. Using the summation formula for a geometric series, and some elementary estimates, we conclude that there exists a constant C depending only on such that

|x| + M + 2 an|y| 2|α||y| ≤ 1+C exp − , exp − |x| + M + 2 |y| |x| + M + 2 2 α∈Z ,|α|∞ >na

(3.17) provided that 0 < |y| < 1. Putting together (3.16) and (3.17), we find that there exists a constant C = C(M, b, , a) such that E Tr 11 (H − z)−1 (V per − V )(H per − z)−1 11

|x| + C 3 an|y| ≤C . (3.18) exp − |y| |x| + C Writing

|x| + C 3 |x| + C 3+l an|y| l an|y| an|y| = (an)−l exp − exp − |y| |x| + C |y| |x| + C |x| + C with l ∈ N, and bearing in mind the elementary inequality t l e−t ≤ (l/e)l , t ≥ 0, l ∈ N, we find that (3.18) implies E Tr 11 (H − z)−1 (V per − V )(H per − z)−1 11

|x| + C 3+l l log l ≤ C(ae)−l n −l e , l ∈ N. (3.19) |y| Combining (3.19) and (3.15), we get |E Tr 11 (ϕ(H ) − ϕ(H per ))11 | C ≤ (|x| + C)−2 d x (ae)−l n −l el log l sup sup (|x| + C)l+5 |y|−(l+3) π R 0 0, and choosing ν > ( + C)η and n ≥ E −ν , we find that

−η per E + (t)dNb (t) − + (t)dN (t) ≤ e−E (3.23) R

R

for sufficiently small E > 0. Now the combination of (3.21) and (3.23) yields the upper bound in (3.3). The proof of the first inequality in (3.3) is quite similar, so that we will just outline it. Let ϕ− ∈ C0∞ (R) be a non-negative Gevrey-class function with Gevrey exponent > 1, such that R ϕ+ (t)dt = 1, and supp ϕ+ ⊂ − E4 , E4 . Set + := 1 3E 3E ∗ ϕ+ . Then − is Gevrey-class function with Gevrey exponent 2bq−

4

,2bq+

4

. Similarly to (3.21) we have E N per (2bq + E/2) − N per (2bq − E/2)

− E − (t)dNb (t) − − (t)dN per (t) R

R

≤ Nb (2bq + E) − Nb (2bq − E).

(3.24)

Arguing as in the proof of (3.23), we obtain

−η per E − (t)dNb (t) − − (t)dN (t) ≤ e−E R

R

which combined with (3.24) yields the lower bound in (3.3). Thus, the proof of Theorem 3.1 is now complete. Further, we introduce a reduced IDS ρq related to a fixed Landau level 2bq, q ∈ Z+ . ∞ {2bq}, and It is well-known that for every fixed θ ∈ T∗2L we have σ (h(θ )) = ∪q=0


681

dim Ker (h(θ ) − 2bq) = 2bL 2 /π for each q ∈ Z+ (see [5]). Denote by pq (θ ) : L 2 (2L ) → L 2 (2L ) the orthogonal projection onto Ker (h(θ ) − 2bq), and by rq (θ ) = per rq,n,ω (θ ) the operator pq (θ )Vn,ω pq (θ ) defined and self-adjoint on the finite-dimensional Hilbert space pq (θ )L 2 (2L ). Set −2 N (E; rq,n,ω (θ ))dθ, E ∈ R. (3.25) ρq (E) = ρq,n,ω (E) = (2π ) ∗2L

By analogy with (3.1), we call the function ρq,n,ω the IDS for the operator Rq = Rq,n,ω := ∗ ⊕rq,n,ω dθ defined and self-adjoint on Pq L 2 (2L × ∗2L ) where Pq := 2L per P . q ∗2L ⊕ pq (θ )dθ . Note that Rq = Pq V Denote by Pq , q ∈ Z+ , the orthogonal projection onto Ker(H0 − 2bq). Evidently, Pq = U Pq U ∗ . As mentioned in the Introduction, rank Pq = ∞ for every q ∈ Z+ . Moreover, the functions q! b ( j−q+1)/2 ( j−q) q e j (x) = e j,q (x) := (−i) (x1 + i x2 ) j−q L q π j! 2

b 2 − b |x|2 |x| e 4 , j ∈ Z+ , × (3.26) 2 form the so-called angular-momentum orthogonal basis of Pq L 2 (R2 ), q ∈ Z+ (see [8] or [3, Sect. 9]). Here ( j−q)

Lq

(ξ ) :=

q l=max{0,q− j}

(−ξ )l j! , ξ ∈ R, q ∈ Z+ , ( j − q + l)!(q − l)! l!

j ∈ Z+ ,

are the generalized Laguerre polynomials. For further references we give here several estimates concerning the functions e j,k . If q ∈ Z+ , j ≥ 1, and ξ ≥ 0, we have ( j−q)

Lq

( jξ )2 ≤ j 2q e2ξ

(3.27)

(see [14, Eq. (4.2)]). On the other hand, there exists j0 > q such that j ≥ j0 implies ( j−q) ( jξ )2 Lq

1 ≥ (q!)2

2+2q 1 ( j − q)2q 2

(3.28)

if ξ ∈ [0, 1/2] (see [32, Eq. (3.6)]). Moreover, for j ∈ Z+ and q ∈ Z+ we have 1 (a ∗ )q e0,q (x), x ∈ R, e j,q (x) = √ q!(2b)q

(3.29)

where a ∗ := −i

∂ ∂ ∂ 2 2 − A1 − i −i − A2 = −2ieb|z| /4 e−b|z| /4 , z := x1 + i x2 , ∂ x1 ∂ x2 ∂z (3.30)

682

F. Klopp, G. Raikov

is the creation operator (see e.g. [3, Sect. 9]). Evidently, a ∗ commutes with the magnetic translation operators τγ , γ ∈ 2LZ2 (see (2.6)). Finally, the projection Pq , q ∈ Z+ , admits the integral kernel

b b −i b x∧x e 2 |x − x |2 , x, x ∈ R2 , K q,b (x, x ) = q (3.31) 2π 2 (0)

where q (ξ ) := L q (ξ )e−ξ/2 , ξ ∈ R. Since Pq is an orthogonal projection in L 2 (R2 ) ∗ we have Pq L 2 (R2 )→L 2 (R2 ) = 1. Using the facts that Pq = U Pq U and Pq := ∗2L ⊕ pq (θ )dθ , as well as the explicit expressions (2.9) for the unitary operator U , and (3.31) for the integral kernel of Pq , q ∈ Z+ , we easily find that the projection pq (θ ), θ ∈ T∗2L , admits an explicit kernel in the form b iθ(x −x) −i b x∧x e Kq,b (x, x ; θ ) = e 2 2π

b b b 2 |x − x + α| e−iθα ei 2 (x+x )∧α ei 2 α1 α2 , x, x ∈ 2L . (3.32) × q 2 2 α∈2L Z

Lemma 3.3. Let the assumptions of Theorem 3.1 hold. Suppose, moreover, that the random variables ωγ , γ ∈ Z2 , are non-negative. M , ∞ there exists E 0 ∈ (0, 2b) such that for each E ∈ (0, E 0 ), a) For each c0 ∈ 1 + 2b θ ∈ T∗2L , almost surely (3.33) N (E; r0 (θ )) ≤ N (E; h(θ )) ≤ N (c0 E; r0 (θ )). M M b) Assume H4 , i.e. 2b > M. Then for each c1 ∈ 0, 1 − 2b , c2 ∈ 1 + 2b , ∞ , there ∗ exists E 0 ∈ (0, 2b) such that for each E ∈ (0, E 0 ), θ ∈ T2L , and q ≥ 1, almost surely N (c1 E; rq (θ )) ≤ N (2bq + E; h(θ )) − N (2bq; h(θ )) ≤ N (c2 E; rq (θ )). (3.34) Proof. In order to simplify the notations we will omit the explicit dependence of the operators h, h 0 , pq , and rq , on θ ∈ T∗2L . Moreover, we set Dq := pq D(h) = pq L 2 (2L ), and Cq := (1 − pq )D(h). At first we prove (3.33). The minimax principle implies N (E; h) ≥ N (E; p0 hp0 |D0 ) = N (E; r0 ), which coincides with the lower bound in (3.33). On the other hand, the operator inequality h ≥ p0 (h 0 + (1 − δ)V per ) p0 + (1 − p0 )(h 0 + (1 − δ −1 )V per )(1 − p0 ), δ ∈ (0, 1), (3.35) combined with the minimax principle, entails N (E; h) ≤ N (E; p0 (h 0 + (1 − δ)V per ) p0 |D0 ) +N (E; (1 − p0 )(h 0 + (1 − δ −1 )V per )(1 − p0 )|C0 ) ≤ N ((1 − δ)−1 E; r0 ) + N (E + M(δ −1 − 1); (1 − p0 )h 0 (1 − p0 )|C0 ). (3.36)


Choose M(δ −1 − 1) < 2b, and, hence, c0 := (1 − δ)−1 > 1 + M(δ −1 − 1)). Since

683 M 2b ,

and E ∈ (0, 2b −

inf σ ((1 − p0 )h 0 (1 − p0 )|C0 ) = 2b, we find that the second term on the r.h.s. of (3.36) vanishes, and N (E; h) ≤ N (c0 E; r0 ) which coincides with the upper bound in (3.33). Next we assume q ≥ 1 and M < 2b, and prove (3.34). Note for any E 1 ∈ (0, 2b− M) we have N (2bq; h) = N (2bq − E 1 ; h). Pick again δ ∈

M 2b+M , 0

M so that c2 := (1 − δ)−1 > 1 + 2b . Then the operator inequality

h ≥ pq (h 0 + (1 − δ)V per ) pq + (1 − pq )(h 0 + (1 − δ −1 )V per )(1 − pq ), δ ∈ (0, 1), analogous to (3.35), yields N (2bq + E; h) ≤ N (2bq + E; pq (h 0 + (1 − δ)V per ) pq |Dq ) +N (2bq + E; (1 − pq )(h 0 + (1 − δ −1 )V per )(1 − pq )|Cq ) ≤ N (c2 E; rq ) + N (2bq + E + M(δ −1 − 1); (1 − pq )h 0 (1 − pq )|Cq ). On the other hand, the minimax principle implies N (2bq − E 1 ; h) ≥ N (2bq − E 1 ; (1 − pq )h(1 − pq )|Cq ) ≥ N (2bq − E 1 − M; (1 − pq )h 0 (1 − pq )|Cq ). Thus we get N (2bq + E; h) − N (2bq − E 1 ; h) ≤ N (c2 E; rq ) +N (2bq + E + M(δ −1 − 1); (1 − pq )h 0 (1 − pq )|Cq ) −N (2bq − E 1 − M; (1 − pq )h 0 (1 − pq )|Cq ).

(3.37)

It is easy to check that 2bq − E 1 − M > 2b(q − 1), 2bq + E + M(δ −1 − 1) < 2(q + 1)b, provided that E ∈ (0, 2b − M(δ −1 − 1)). Since σ ((1 − pq )h 0 (1 − pq )|Cq ) ∩ (2(q − 1)b, 2(q + 1)b) = ∅, we find that the the r.h.s. of (3.37) is equal to N (c2 E; rq ), thus getting the upper bound in (3.34). M Finally, we prove the lower bound in (3.34). Pick ζ ∈ 2b−M , ∞ , and, hence M c1 := (1 + ζ )−1 ∈ 0, 2b . Bearing in mind the operator inequality h ≤ pq (h 0 + (1 + ζ )V per ) pq + (1 − pq )(h 0 + (1 + ζ −1 )V per )(1 − pq ),

684

F. Klopp, G. Raikov

and applying the minimax principle, we obtain N (2bq + E; h) ≥ N (2bq + E; pq (h 0 + (1 + ζ )V per ) pq |Dq ) +N (2bq + E; (1 − pq )(h 0 + (1 + ζ −1 )V per )(1 − pq )|Cq ) ≥ N (c1 E; rq ) + N (2bq + E − M(ζ −1 + 1); (1 − pq )h 0 (1 − pq )|Cq ). On the other hand, since V per ≥ 0, the minimax principle directly implies N (2bq − E 1 ; h) ≤ N (2bq − E 1 ; h 0 ) = N (2bq − E 1 ; (1 − pq )h 0 (1 − pq )|Cq ). Combining the above estimates, we get N (2bq + E; h) − N (2bq − E 1 ; h) ≥ N (c1 E; rq ) − N (2bq + E − M(ζ −1 + 1); (1 − pq )h 0 (1 − pq )|Cq ) −N (2bq − E 1 ; (1 − pq )h 0 (1 − pq )|Cq ) .

(3.38)

Since 2(q −1)b < 2bq + E − M(ζ −1 + 1) < 2(q +1)b, 2(q −1)b < 2bq − E 1 < 2(q +1)b, provided that E ∈ (0, 2b + M(ζ −1 + 1)), we find that the r.h.s of (3.38) is equal to N (c1 E; rq ) which entails the lower bound in (3.34). Integrating (3.33) and (3.34) with respect to θ and ω, and combining the results with (3.3), we obtain the following Corollary 3.1. Assume that the hypotheses of Theorem 3.1 hold. Let q ∈ Z+ , η > 0. If q ≥ 1, assume M < 2b. Then there exist ν = ν(η) > 0, d1 ∈ (0, 1), d2 ∈ (1, ∞), and E˜ 0 > 0, such that for each E ∈ (0, E˜ 0 ) and n ≥ E −ν , we have −η −η E ρq,n,ω (d1 E) − e−E ≤ Nb (2bq + E) − Nb (2bq) ≤ E ρq,n,ω (d2 E) + e−E . (3.39) 4. Proof of Theorem 2.1: Upper Bounds of the IDS In this section we obtain the upper bounds of Nb (2bq + E) − Nb (2bq) necessary for the proof of Theorem 2.1. Theorem 4.1. Assume that H1 – H4 hold, that almost surely ωγ ≥ 0, γ ∈ Z2 , and (2.1) is valid. Fix the Landau level 2bq, q ∈ Z+ . i) Assume that u(x) ≥ C(1 + |x|)−κ , x ∈ R2 , for some κ > 2, and C > 0. Then we have lim inf E↓0

2 ln | ln (Nb (2bq + E) − Nb (2bq))| ≥ . | ln E| κ−2

(4.1)

β

ii) Assume u(x) ≥ Ce−C|x| , x ∈ R2 , for some β > 0, C > 0. Then we have lim inf E↓0

2 ln | ln (Nb (2bq + E) − Nb (2bq))| ≥1+ . ln | ln E| β

(4.2)


685

iii) Assume u(x) ≥ C1{x∈R2 ; |x−x0 | 0, x0 ∈ R2 , and ε > 0. Then there exists δ > 0 such that we have lim inf E↓0

ln | ln (Nb (2bq + E) − Nb (2bq)| ≥ 1 + δ. ln | ln E|

(4.3)

Fix θ ∈ T∗2L . Denote by λ j (θ ), j = 1, . . . , rank rq,n,ω (θ ), the eigenvalues of the operator rq,n,ω (θ ) enumerated in non-decreasing order. Then (3.25) implies 1 E ρq,n,ω (E) = E(N (E; rq,n,ω (θ ))dθ (2π )2 ∗2L 1 = (2π )2

rank rq,n,ω (θ)

∗2L

P(λ j (θ ) < E)dθ

(4.4)

j=1

with E ∈ R. Since the potential V is almost surely bounded, we have rank rq,n,ω (θ ) ≤ rank pq (θ ) = 2bL 2/π . Therefore, (4.4) entails bL 2 E ρq,n,ω (E) ≤ 2π 3

∗2L

P(rq,n,ω (θ ) has an eigenvalue less than E)dθ. (4.5)

In order to estimate the probability in (4.5), we need the following Lemma 4.1. Assume that, for n ∼ E −ν , the operator rq,n,ω (θ ) has an eigenvalue less than E. Set L := (2n + 1)a/2. Pick E small and l large such that L l. Decompose 2L = ∪γ ∈2l Z2 ∩2L (γ + 2l ). Fix C > 1 sufficiently large and m = m(L , l) such that 1 2 bl ≤ m ≤ CbL 2 , C 2 l 2 2 E > Ce−bl /2+m ln(C bl /m) . L

(4.6) (4.7)

Then, there exists γ ∈ 2lZ2 ∩2L and a non-identically vanishing function ψ ∈ L 2 (R2 ) in the span of {e j,q }0≤ j≤m , the functions e j,q being defined in (3.26), such that Vωγ ψ, ψl ≤ 2Eψ, ψl , per γ where Vω (x) = Vω (x + γ ), and ·, ·l := 2l | · |2 d x.

(4.8)

Proof. Consider ϕ ∈ Ran pq (θ ) a normalized eigenfunction of the operator rq,n,ω (θ ) corresponding to an eigenvalue smaller than E. Then we have Vω ϕ, ϕ L ≤ Eϕ, ϕ L .

(4.9)

Whenever necessary, we extend ϕ by magnetic periodicity (i.e. the periodicity with respect to the magnetic translations) to the whole plane R2 . Note that Kq,b (x, x ; θ )ϕ(x )d x = eiθ(x −x) K q,b (x, x )ϕ(x )d x ϕ(x) = ϕ(x; θ ) = 2L

R2

686

F. Klopp, G. Raikov

with x ∈ 2L (see (3.31) and (3.32) for the definition of K q,b and Kq,b respectively). Evidently, ϕ ∈ L ∞ (R2 ), and since it is normalized in L 2 (2L ), we have

1/2 |Kq,b (x, x ; θ )|2 d x ϕ L ∞ (R2 ) ≤ sup 2L

x∈2L

⎛ ⎜ ≤ sup ⎝

⎛ ⎝

2L

x∈2L

where ˜ q (y) :=

⎞2

⎞1/2

˜ q (x − x + α)⎠ d x ⎟ ⎠

≤ C, (4.10)

α∈2L Z2

b 2 b |y| q , 2π 2

y ∈ R2 ,

(4.11)

and C depends on q and b but is independent of n and θ . Fix C1 > 1 large to be chosen later on. Consider the sets 2 l 1 2 2 2 L+ = γ ∈ 2lZ ∩ 2L ; |ϕ(x)| d x ≥ |ϕ(x)| d x , C1 L γ +2l 2L 2 l 1 2 2 2 L− = γ ∈ 2lZ ∩ 2L ; |ϕ(x)| d x < |ϕ(x)| d x . C1 L γ +2l 2L The sets L− and L+ partition 2lZ2 ∩ 2L . Fix C2 > 1 large. Let us now prove that for some γ ∈ L+ , one has per 2 Vω (x)|ϕ(x)| d x ≤ C2 E |ϕ(x)|2 d x. γ +2l

γ +2l

Indeed, if this were not the case, then (4.9) would yield per −E |ϕ(x)|2 d x≤ Vω (x)|ϕ(x)|2 d x − E γ ∈L− γ +2l

γ ∈L−

γ +2l

E ≤ γ ∈L+

γ +2l

≤−E(C2 − 1)

|ϕ(x)| d x − 2

γ ∈L+

γ +2l

γ ∈L− γ +2l

≤ ≤

γ ∈L+ γ +2l

γ ∈L+ γ +2l

γ +2l

γ ∈L+ γ +2l

|ϕ(x)|2 d x

per Vω (x)|ϕ(x)|2 d x

|ϕ(x)|2 d x.

On the other hand, the definition of L− yields |ϕ(x)|2 d x = |ϕ(x)|2 d x + 2L

γ +2l

(4.12)

(4.13)

|ϕ(x)|2 d x

1 l 2 |ϕ(x)|2 d x C1 L 2L γ ∈L− 1 |ϕ(x)|2 d x + |ϕ(x)|2 d x. C1 2L |ϕ(x)|2 d x +


687

Plugging this into (4.13), we get

1 E 2 |ϕ(x)| d x ≥ E(C2 − 1) 1 − |ϕ(x)|2 d x C1 2L C1 2L

(4.14)

which is clearly impossible if we choose (C2 − 1)(C1 − 1) > 1. So from now on we assume that (C2 − 1)(C1 − 1) > 1. Hence, we can find γ ∈ 2lZ2 ∩ 2L such that one has per Vω (x)|ϕ(x)|2 d x ≤ C2 E |ϕ(x)|2 d x, γ +2l

γ +2l

γ +2l

2 l 1 2 |ϕ(x)| d x ≥ |ϕ(x)|2 d x. C1 L 2L

Shifting the variables in the integrals above by γ , we may assume γ = 0 if we replace per γ Vω by Vω . Thus we get Vωγ (x)|ϕ(x)|2 d x ≤ C2 E |ϕ(x)|2 d x,

2l

2l

2l

2 l 1 2 |ϕ(x)| d x ≥ |ϕ(x)|2 d x. C1 L γ +2L

Due to the magnetic periodicity of ϕ, we have |ϕ(x)|2 d x = |ϕ(x)|2 d x γ +2L

2L

which yields

2l

Vω (x)|ϕ(x)|2 d x ≤ C2 E 1 |ϕ(x)| d x ≥ C1 2

2l

|ϕ(x)|2 d x,

(4.15)

2 l |ϕ(x)|2 d x. L 2L

(4.16)

2l

Let us now show that roughly the same estimates hold true for ϕ replaced by a function ψ ∈ Pq L 2 (R2 ). Set ψ := Pq χ− eθ ϕ where eθ (x) := eiθ x , x ∈ R2 , and χ− denotes the characteristic function of the set {x ∈ R2 ; |x|∞ < L}. Note that ϕ − eθ ψ = eθ Pq χ+ eθ ϕ, where χ+ is the characteristic function of the set {x ∈ R2 ; |x|∞ ≥ L}. Let us estimate the L 2 (2L )-norm of the function ϕ − eθ ψ. We have 2 2 2 iθ x ϕ − eθ ψ L := ϕ − eθ ψ L 2 ( ) = K q,b (x, x )χ+ (x )e ϕ(x )d x d x 2L

2L

R2

2

≤ sup |ϕ(x )| x ∈R2

×

2L

R2 R2

˜ q (x − x )χ+ (x )χ+ (x )d x d x d x, ˜ q (x − x ) (4.17)

688

F. Klopp, G. Raikov

˜ q being defined in (4.11). Bearing in mind estimate (4.10), and taking the function ˜ at infinity, we easily find that (4.17) implies the into account the Gaussian decay of existence of a constant C > 0 such that for sufficiently large L we have ϕ − eθ ψ2L ≤ e−L/C . As ϕ is normalized in L 2 (2L ), this implies that, for sufficiently small E, ψ L ≥

1 ϕ L and ϕ − eθ ψ L ≤ e−L/C ψ L . 2

(4.18)

per

As Vω is uniformly bounded, it follows from our choice for L and l and estimate (4.18) that, for E sufficiently small, 2 l 1 |ψ(x)|2 d x ≥ |ϕ(x)|2 d x − Cϕ − eθ ψ2L C1 L 2l 2L 2 l 1 ≥ |ψ(x)|2 d x, ˜ C1 L 2L per per 2 2 2 ˜ Vω (x)|ψ(x)| d x = Vω (x)|ϕ(x)| d x + Cϕ − eθ ψ L ≤ C2 E |ψ(x)|2 d x. 2l

2l

2l

Hence, we obtain inequalities (4.15)–(4.16) with ϕ replaced by ψ ∈ Pq L 2 (R2 ). Now, we write ψ = j≥0 a j e j (see (3.26)). Using the fact that {e j } j≥0 is an orthogonal family on any disk centered at 0 (this is due to the rotational symmetry), we compute 2 2 2 |ψ(x)|2 d x ≤ |ψ(x)| d x = |a | j √ √ |e j (x)| d x, (4.19) 2l

and

|x|≤ 2l

|ψ(x)|2 d x ≥

2L

|x|≤L

|x|≤ 2l

j≥0

|ψ(x)|2 d x =

|a j |2

j≥0

|x|≤L

|e j (x)|2 d x.

(4.20)

Fix m ≥ 1 and decompose ψ = ψ0 + ψm , where ψ0 =

m

ajej,

ψm =

j=0

ajej.

(4.21)

j≥m+1

Our next goal is to estimate the ratio √ |e (x)|2 d x j,q |x|< 2l , 2 |e |x| bL 2 .

Note that the function f is increasing on the interval (0, 1). Since j ≥ m + 1, and C, the 2 constant in (4.6), is greater than one, we have blj < 1. Therefore,

bl 2 /j

e( j−q) f (s) ds ≤

0

bl 2 ( j−q) f (bl 2 /j) e . j

(4.25)

On the other hand, using a second-order Taylor expansion of f , we get f (s) ≥ f (( j)) +

s − ( j) 1 − , s ∈ (( j)/2, ( j)). ( j) 2

Consequently,

( j)

e( j−q) f (s) ds ≥

0

( j) ( j)/2

e( j−q) f (s) ds ≥

( j) ( j−q)( f (( j))−1)) e . 2

(4.26)

Putting together (4.24)-(4.26), we obtain

√ 2l

|x|
0. Pick η > 2/(κ − 2), and ν0 > max κ1−2 , ν where ν = ν(η) is the number

defined in Corollary 3.1. Finally, fix an arbitrary κ > κ and set n ∼ E −ν0 ,

L = (2n + 1)a/2, l = E

− κ 1−2

2

, m ∼ E − κ −2 .

Then the numbers m, l, and L, satisfy (4.6) – (4.7) provided that E > 0 is sufficiently small. Further, for any γ0 ∈ 2lZ2 ∩ 2L we have 1 −κ γ0 2 ωγ u(x − γ )|ψ(x)| d x ≥ l ωγ |ψ(x)|2 d x Vω ψ, ψl ≥ C3 2l 2l |γ |≤l

|γ |≤l

(4.29) with C3 > 0 independent of θ and E. Hence, the probability that there exists γ ∈ 2lZ2 ∩ 2L and a non-identically vanishing function ψ in the span of {e j }0≤ j≤m such that (4.8) be satisfied, is not greater than the probability that l −2

ωγ ≤ C3 El κ−2 = C3 E

κ −κ κ −2

.

(4.30)

|γ |≤l

Applying a standard large-deviation estimate (see e.g. [15, Subsect. 8.4] or [22, Sect. 3.2]), we easily find that the probability that (4.30) holds, is bounded by

κ −κ 2 κ −κ 2 −2 −2 −2 κ κ κ exp C4 l ln P(ω0 ≤ C3 E ) = exp C4 E ln P(ω0 ≤ C3 E ) with C4 independent of θ and E > 0 small enough. Applying our hypothesis that P(ω0 ≤ E) ∼ C E κ , E ↓ 0, with C > 0 and κ > 0, we find that for any κ > κ, θ ∈ T∗2L , and sufficiently small E > 0, we have 2 (4.31) P(rq,n,ω (θ ) has an eigenvalue less than E) ≤ exp −C5 E κ −2 | ln E| with C5 > 0 independent of θ and E. Putting together (3.39), (4.5) and (4.31), and taking into account that area ∗2L = π 2 L −2 , we get Nb (2bq + E) − Nb (2bq) ≤

2 b exp −C5 E κ −2 | ln E| + exp (−E −η ) 2π


691

which implies lim inf E↓0

ln | ln Nb (2bq + E) − Nb (2bq)| 2 ≥ | ln E| κ −2

for any κ > κ. Letting κ ↓ κ, we get (4.1). Assume now the hypotheses of Theorem 4.1 ii). In particular, we suppose that u(x) ≥ β Ce−C|x| , x ∈ R2 , C > 0, β > 0. Put β0 = max {1, 2/β}. Pick an arbitrary β > β and set

l = | ln E|1/β , m ∼ | ln E|β0 . Then (4.6)–(4.7) are satisfied provided that E > 0 is sufficiently small, and similarly to (4.29), for any γ0 ∈ 2lZ2 ∩ 2L we have 1 −c6 l β γ0 Vω ψ, ψl ≥ e ωγ |ψ(x)|2 d x C6 2l |γ |≤l

with C6 > 0 independent of θ and E. Arguing as in the derivation of (4.31), we get P(rq,n,ω (θ ) has an eigenvalue less than E) ≤ exp −C7 | ln E|1+2/β ln | ln E| (4.32) with C7 > 0 independent of θ and E. As in the previous case, we put together (3.39), (4.5) and (4.31), and obtain the estimate Nb (2bq + E) − Nb (2bq) ≤

b exp −C7 | ln E|1+2/β ln | ln E| + exp (−E −η ) 2π

which implies lim inf E↓0

2 ln | ln (Nb (2bq + E) − Nb (2bq))| ≥1+ ln | ln E| β

for any β > β. Letting β ↓ β, we get (4.2). Finally, let us assume the hypotheses of Theorem 4.1 iii). In particular, we assume that u(x) ≥ C1{x∈R2 ;|x−x0 | 0, x0 ∈ R2 , and ε > 0. Due to τx0 H0 τx∗0 = H0 and τx0 1{x∈R2 ;|x−x0 |

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