Additive and cancellative interacting particle systems

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

724

David Griffeath

Additive and Cancellative Interacting Particle Systems

Springer-Verlag Berlin Heidelberg New York 1979

Author David Griffeath Dept. of Mathematics University of Wisconsin Madison, Wl 53706 USA

AMS Subject Classifications (1970): 60 K35 ISBN 3-540-09508-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09508-X Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under £354 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publishel © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

Preface These notes are based on a course given at the University of W i s c o n s i n in the spring of 1978.

The subject is (stochastic) interacting particle systems, or

more precisely, certain continuous time M a r k o v processes with state space S = {all subsets of Z d } .

This area of probability theory has been quite active

over the past ten years : a list of references, by no m e a n s comprehensive, found at the end of the exposition.

m a y be

In particular, several surveys on related

material are already available, a m o n g them Spitzer (1971), D a w s o n

(1974b),

Spitzer (1974b), Sullivan (1975), Georgii (1976), Liggett (1977) and Stroock (1978). There is rather little overlap between the present treatment and the above articles, and where overlap occurs our approach is s o m e w h a t different in spirit. Specifically, these notes are based on 9raphical representations of particle systems, an approach due to Harris (1978).

The basic idea is to give explicit

constructions of the processes under consideration with the aid of percolation substructures.

While limited in applicability to those systems which admit such

representations, Harris' technique m a n a g e s to handle a large number of interesting models.

W h e n it does apply, the graphical approach has several advantages over

alternative methods.

First, since the systems are constructed from "exponential

alarm clocks, " the existence problem does not arise.

Also, the uniqueness problem

can be handled with m u c h less difficulty than for more general particle systems. Another appealing feature is the geometric nature of the representation, which leads to "visual" probabilistic proofs of m a n y results. coupling.

Finally, there is the matter of

O n e of the basic strategies in studying particle systems is to put two or

more processes on a joint probability space for comparison purposes.

Graphical

representations have the property that processes starting from arbitrary initial configurations are all defined on the s a m e probability space, in such a w a y that natural couplings are often e m b e d d e d in the construction. conceptual simplification in m a n y arguments.

This is a major

Altogether, Harris' approach makes

the material easily accessible to a gifted graduate student having a familiarity with the elementary theory of M a r k o v chains and processes. The development is divided into four chapters.

Chapter I contains basic

notation, general concepts and a discussion of the major problems in the field of interacting particle systems.

It also includes a description of the percolation

substructures which are used to define the processes w e intend to study. is devoted to additive systems. Harris (1978).

Chapter II

The "lineal" additive systems were introduced by

%Ve also consider "extralineal" additive systems.

and pointwise ergodic theorems are proved.

General ergodic

A m o n g the specific models treated in

some detail are contact processes, voter models and coalescing random walks. Chapter Ill deals with cancellatlve systems, a second large class of models which admit graphical representation.

There are analogous general ergodic theorems for

IV

this class.

S p e c i f i c t o p i c s i n c l u d e an a p p l i c a t i o n to the s t o c h a s t i c I s i n g m o d e l ,

a n d l i m i t t h e o r e m s for g e n e r a l i z e d v o t e r m o d e l s a n d a n n i h i l a t i n g r a n d o m w a l k s . C h a p t e r IV w e d i s c u s s t h e u n i q u e n e s s p r o b l e m for a d d i t i v e a n d c a n c e l l a t i v e W e h a v e c l o s e n to p r e s e n t t h i s m a t e r i a l l a s t ,

In

systems

since uniqueness questions seem

r a t h e r e s o t e r i c in c o m p a r i s o n with the important p r o b l e ms of e r g o d i c t h e o r y .

The

g r a p h i c a l a p p r o a c h s h o w s how n o n u n i q u e n e s s can a r i s e w h e n t h e r e is " i n f l u e n c e from

oo .

"

A g r e a t d e a l o f t h e m a t e r i a l i n t h e s e n o t e s h a s a p p e a r e d in r e c e n t r e s e a r c h p a p e r s by m a n y a u t h o r s .

At t h e e n d o f e a c h s e c t i o n i s a p a r a g r a p h e n t i t l e d " N o t e s "

w h i c h i d e n t i f i e s t h e s o u r c e s of t h e r e s u l t s c o n t a i n e d t h e r e i n .

All r e f e r e n c e s a r e t o

t h e B i b l i o g r a p h y w h i c h f o l l o w s C h a p t e r IV. I w o u l d l i k e t o a c k n o w l e d g e my g r a t i t u d e to many m a t h e m a t i c i a n s for t h e i r contributions, T. H a r r i s ,

especially

R. H o l l e y ,

M. Bramson,

H. K e s t e n ,

D. D a w s o n ,

T. L i g g e t t ,

Sheldon Goldstein,

L. G r a y ,

F. S p i t z e r a n d D. S t r o o c k .

Let me

a l s o t h a n k t h e v a r i o u s S o v i e t m a t h e m a t i c i a n s w h o s e p i o n e e r i n g work o n c l o s e l y r e l a t e d d i s c r e t e t i m e s y s t e m s w a s a m a j o r s o u r c e of i n s p i r a t i o n for t h e c o n t i n u o u s time theory. Finally,

A s a m p l i n g o f t h e i r p u b l i c a t i o n s i s i n c l u d e d in t h e B i b l i o g r a p h y .

my t h a n k s go out to R i c h a r d A r r a t i a ,

S t e v e G o l d s t e i n and Arnold N e i d h a r d t

f o r t h e i r m a n y c o m m e n t s and c o r r e c t i o n s a s t h e s e n o t e s w e r e t a k i n g s h a p e .

David Griffeath Madison, Wisconsin A u g u s t , 1978

CONTENTS Page iii

Preface

CHAPTER I : INTRODUCTION. 1.

Preliminaries

Z.

Percolation substructures

CHAPTER

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

II : A D D I T I V E

......................

1 9

SYSTEMS.

i.

The general construction . . . . . . . . . . . . . . . . . . . . . . .

14

Z.

Ergodic t h e o r e m s for extralineal additive s y s t e m s . . . . . . . . . .

19

3.

Lineal additive s y s t e m s

Z6

4.

Contact systems:

basic properties

5.

Contact systems:

limit t h e o r e m s in the nonergodic c a s e . . . . . .

6.

C o n t a c t s y s t e m s in several d i m e n s i o n s

7.

Voter m o d e l s

8.

Biased voter m o d e l s

9.

Coalescing random walks

I0.

CHAPTER

....................... .................

...............

............................. ......................... ......................

Stirring a n d exclusion s y s t e m s

III : C A N C E L L A T I V E

...................

Z9 38 44 46 55 58 63

SYSTEMS.

I.

The general construction

......................

Z.

Extralineal cancellative s y s t e m s with pure births . . . . . . . . . .

71

3.

Application to the stochastic Ising m o d e l

74

4.

Generalized voter m o d e l s

......................

76

5.

Annihilating r a n d o m w a l k s . . . . . . . . . . . . . . . . . . . . . .

80

..............

66

CHAPTER IV : UNIQUENESS AND N O N U N I Q U E N E S S . 1.

Additive and cancellative pregenerators . . . . . . . . . . . . . . .

89

Z.

Uniqueness theorems

9Z

3.

Nonuniqueness

........................

examples

......................

98

Bibliography

101

Subject index

107

CHAPTER

I.

I: I N T R O D U C T I O N

Preliminaries. Throughout the exposition w e wlll use the following notation: Z d = the x,y,z S=

d-dimensional

c Zd

integer lattice

(d >- i) ;

are called sites.

{all subsets of Z d } ,

S O = {all finite subsets of Z d } , S

= {all infinite subsets of Z d } . oo

A, B, C ¢ S are called confi~uratlons.

A

A(x) = i

if

x ~ A,

= 0

if

x~/A.

Write

will always be a finite confi~uratlon,

IAI

Is the cardinality of A

i.e.

A ~ SO ;

.

Important finite configurations are tile n - b o x

bn(X ) centered at x ~ Z d :

bn(X) = {Y = (YI' "'" 'Yd ) : IY2 - x~l -< n for 1 -< 2 -< d} , and the block

[x,y]

C Z

, x,y ¢ Z :

[x,y] = { z : x - < z ~ < y } O n e useful abuse of notation is to write

x

instead of

{x}

for the singleton

configuration at site x ; w e will do this w h e n e v e r it is convenient. T = [0, co) is the (continuous) r,s,t,u

~ T

time parameter set;

aretimes.

Our objects of study will be certain continuous time processes,

or particle processes. A (£t)t~ T '

Here

A ~t

l.e.

A t0 = A •

Such a process will be written as

or simply

(~A) .

is the configuration of the process at tlme We

S-valued M a r k o v

t ,

say that there is a particle at site x

and

A

at time

is the initial state, A t if x c ~t "

Other notations for particle processes are

(~]A)and

(~A) .

of particle processes will be called a .particle system.

A family

P and

{(%A); A e S}

E will be the

probability l a w and expectation operator respectively governing such a system. S o m e additional notation: = {all probability m e a s u r e s on tL, v ~ ~ 8A ~ M

S} ;

are often called dlstributions.

is the m e a s u r e concentrated at A ( S •

Bernoulli product m e a s u r e such that ~t0 = 6)~ ,

0 ?

Basic contact systems.

In thls case

{(~A)} is a spin system

with flip rates

xcA

Cx(A ) = 1 : klAn

{x-l,x+l} I

x/A.

A ' and k > 0 is a parameter. W e m a y think of the site x as infected w h e n x ~ ~t A healthy w h e n x / ~t " Thus infected sites recover at constant rate 1 , while healthy sites are infected at a rate proportional to the number of infected neighbors. the parameter k is an infection index. representations. {x} × T •

Contact processes also admit graphical

N o w three types of graphical device are attached to each

First, a sequence of 6's is put d o w n at rate 1 (i.e. with independent

exponential mean-i times between successive killing infection if it is present.

6's ).

These will have the effect of

Next, a sequence of directed arrows :

Is put d o w n at rate k , and finally a sequence of arrows at rate X .

Thus

> x-I x is also put d o w n

< x x+l The resulting percolation substructure wlll look something like figure ii.

Defining ~A by (I.Z) , in terms of thls second substructure, w e obtain the basic

contact system.

(i. 6)

Problem.

infected.

Let

(~)

be the basic contact process starting with only the origin

S h o w that for all sufficiently small positive k , the infection dies out

with probability one. These notes will be devoted exclusively to particle systems which can be constructed from exponential random variables with the aid of percolation substructures.

In thls w a y w e circumvent the first major problem in the theory of random

interacting particle systems : I. Existence : W h e n is there a system

{(~A)} with given jump rates c ?

A great m a n y systems do not admit graphical representations in terms of percolation substructures, and for these the existence problem is nontrivial.

A second funda-

mental question is :

If. Uniqueness :

W h e n is there a unique particle system

{(~A)} with given jump

rate s ? Even for the models w e will study, a precise formulation and treatment of this problem requires technical machinery ; w e therefore defer uniqueness questions until Chapter 4. O n c e the system is well-defined, interest centers on:

III. Ergodic theory :

W h a t is the asymptotic behavior of the processes

(~A) as

t~oo? W e n o w discuss the broad outlines of problem Ill.

A familiar property of M a r k o v

processes is their "loss of memory" under appropriate assumptions on the transition mechanism.

Starting from measure

~ , it is c o m m o n for ~pt to converge to an

equilibrium, or Invariant measure v as t ~ notion is that of w e a k convergence:

lira ~t({A: A N A =

. For particle systems the appropriate

lim ~tt = v

A0} ) = v({A : A n A =

(t c T o__[ t = 0,I, --.) if

A0} ) VA 0 C A,

By inclusion-exclusion, this last is equivalent to :

S0 .

7 l i m ¢ ~ t ( A ) = CV(A) t~o0 Say t h a t

v

VA ~ S O

i s i n v a r i a n t for t h e s y s t e m

{(~A)}

if v P t = v

p a r t i c l e s y s t e m s we study will a l m o s t a l w a y s be F e l l e r , ~ pt

~pt

as

~

~

for e a c h f i x e d

Any F e l l e r s y s t e m h a s a t l e a s t o n e e q u i l i b r i u m . mea

P- C t

that

ft

1 =T

p C t' - - v

as

{(~A)} .

t ~ T •

define the Cesaro

~pS d s

~ ~ ~ ,

v

for s o m e s u b s e q u e n c e

is invariant.

We have seen that

equilibrium.

t ~ T •

0

t' --oo ,

s y s t e m is c a l l e d e r g o d i c

~ •

Choose

v

such

U s i n g t h e F e l l e r p r o p e r t y , it

Let ~ b e t h e s e t o f a l l i n v a r i a n t m e a s u r e s

~ /9

if $ = {v}

t' .

so is

in the c a s e of a Feller s y s t e m .

for s o m e

v c ~ ,

i.e.

The

if it has a unique

This i s e q u i v a l e n t t o

(1.7)

3v ~ N : lira ~ C t = v t~oo

Say t h a t

The

in the sense that

To s e e t h i s ,

S i s c o m p a c t (in t h e d i s c r e t e p r o d u c t t o p o l o g y ) ,

is e a s y to c h e c k that for

tE T •

7

sure s

Since

for e a c h

{(~A)}

V~ ~

is s t r o n g l y e r g o d i c if

(1.8)

3 v ~ N : lira

~apt= V

¥~

~ •

•

t~co

Clearly strong ergodicity implies ergodicity. will invariably derive

(1.8)

rather than

(1.7)

When proving ergodic theorems we in t h e s e n o t e s .

no k n o w n e x a m p l e of a p a r t i c l e s y s t e m w h i c h s a t i s f i e s

(1.7)

H o w e v e r , t h e r e is but not

(1.8).

For

c o n v e n i e n c e we will u s u a l l y omit the word "strong" in the s t a t e m e n t of e r g o d i c i t y results.

(1.9)

Problems.

Prove that

(1.7)

i s e q u i v a l e n t to e r g o d i c i t y .

n e e d o n l y b e c h e c k e d for d e l t a m e a s u r e s

p = 6A ,

ergodicity.

x E S}

Find a Feller family

an equilibrium

v,

such that

{(~);

(1.7)

h o l d s but

A( S ,

Show that

to e n s u r e strong

on a c o m p a c t s t a t e s p a c e

(1.8)

(1.8)

does not.

S ,

and

8

(l.10)

Problem.

Let

{(Xt)}

be a Feller family on a c o m p a c t state space

invariant me as ur e for the family.

S h o w that the stationary process

ergodic if there is a set of states

Sv C

lira

6 ct=

t~oo

S h o w that

(~)

{(cA)}

ifwhenever

extremals, measures. v ~ ~

is Birkhoff

such that v(Sv) = 1 and Vx

~ S V

X

6 pt= v x

Vx

( S

v

nonergodic if it has more than one equilibrium,

v = cv0 + (l-c)v I for s o m e

v0 = v = vI , invariants.

v

an

is mixing if V(Sv) = 1 and lira t~oo

Call

S

(~)

S , v

i.e. if v

v 0 , V l ~ ~9 and

0 < c
0 ,

say there is a p a t h u p

to (x,t) .

M o r e generally,

(y, s) to (x,t)

(y,s)

without

( y , s) ~ D 1 t o s o m e

D 1 to

DZ ,

(x,t)

~ Dg •

set

f~t = { ( y ' s )

For our construction

labelled

of particle

[~ i n

systems,

P,

to

By convention there is always a path

there is a path up from

if t h e r e i s a p a t h u p f r o m s o m e

from

(= increasing in T ) and

edges w h i c h lead from

6 on the interior of an u p w a r d edge•

D 1, D Z ~ Z d X T For

(1.5),

if there is a chain of alternating "upward"

"directed horizontal" having a

(i.i), (1.3) and

0 < s -< t} ,

a key ingredient

f~ =

U t>O

f~t

will be the quantities

,

11

(z. 7)

N~t(B) = the n u m b e r of paths up from

(A,O) U f~t to (B,t) in f~ . AA

Given any

@(k; V , W )

A

A

k i , x -= k i , x ,

V i , x --- V i , x

A

A

@(A;V,W)

, there is a dual substructure

d e f i n e d by

and

A

z c W i , x ( y )

y~ Wi,x(Z)

Thus the dual substructure reverses the directions of all arrows. consider

~t = the restriction of ~ A

time run "down" from Pt '

('En, x" n -> i) .

i.e. letting

0 = t to t = 0 , and reversing the direction of all arrows in ~t

A

A

= ~

A

restricted to Z d x [0, t ]

on the s a m e

This follows from the time reversibility of the sequences Evidently

{3

path up from

(y,s)

(z.8)

to (x,t)

in @t }

a

=

By reversing time,

and

A

w e obtain a realization of

probability space.

to Z d x [0, t] •

Fix t < co ,

{3

path d o w n f r o m

A

(x,0)

to (y,t-s)

A

in ~t }

a n o b s e r v a t i o n w h i c h w i l l be c r u c i a l for t h e a n a l y s i s to c o m e .

P - a.s.,

The d u a l s u b s t r u c t u r e s

/k

Pto

c o r r e s p o n d i n g to t h e

~to Of f i g u r e s

i and

i i a r e s h o w n in f i g u r e s

iii and

iv

respectively. (2.9)

Problems.

Let e

be the extralineal substructure with

I -z {I,Z} , X

XI,x = Kx ' XZ,x = kx' otherwise.

Vl,x : {x} ,

A

1 , k x=

x

Z

(b)

:

{([A)} .

for all t -~ 0

the M a r k o v chain

configuration Problem.

flip rates

W i , x ( y ) = {y}

Now

consider the special case where

d = i,

For this model s h o w that if A ~ S O , then

A (a) It ~ SO

(Z.10)

and

{x ~(x)>0}

Describe the particle system X

~ ' W z , x (x) =)Z

Put

~t :

K

VZ,x:

([A) t

P - a.s., m a k e s only instantaneous visits to each

A c SO .

The basic voter model

(d : I) is the spin system

{([A)}

with

12

_%. 6

6 6

--

6 ; 6

-6 -3

-Z

Z figure iii.

--9,6

6

6

,6

-4

-3

-Z

3 figure iv.

13

1 IA n Cx(A) = 5-

{x-i, x+l)} I

x / A

= ~ -1 i AC n { x - l , x+l} I

x c A

^ S h o w t h a t t h e v o t e r m o d e l may b e d e f i n e d a s i n ( 1 . Z ) , but in t e r m s of ~ , s u b s t r u c t u r e for t h e

(Z.11)

Notes.

@ of E x a m p l e

the dual

(1.1).

L i n e a l p e r c o l a t i o n s u b s t r u c t u r e s w e r e i n t r o d u c e d b y H a r r i s (1978);

w e r e f e r t h e r e a d e r t o t h a t p a p e r for more d e t a i l s o f t h e f o r m a l c o n s t r u c t i o n .

The

i d e a b e h i n d t h i s t y p e o f r a n d o m g r a p h g o e s b a c k t o t h e p i o n e e r i n g work o n p e r c o l a t i o n by B r o a d b e n t a n d H a m m e r s l e y (1957). tions with particle systems,

s e e C l i f f o r d a n d S u d b u r y (1973),

S h a n t e a n d K i r k p a t r i c k (1971), c h a i n i n P r o b l e m s (Z. 9)

For more o n p e r c o l a t i o n t h e o r y a n d i t s c o n n e c -

Toom (1968) a n d V a s i l e v (1969) •

is due to Blackwell

are a t the end of Section II. 7.

H a m m e r s l e y (1959),

(1958).

The i n s t a n t a n e o u s

R e f e r e n c e s for t h e v o t e r m o d e l

CHAPTER II: ADDITIVE SYSTEMS 1.

The g e n e r a l c o n s t r u c t i o n . Let @= @(X;V,W) b e a p e r c o l a t i o n s u b s t r u c t u r e .

For t-> 0 ,

A~ S ,

with

NA(B) as i n ( I . 2 . 7 ) , define ~tA = {x : NA(x) > 0}

(i.i) Then

{(~A)}

is an

system induced by

S-valued M a r k o v family, called the (canonical.) additive particle A ~ . If ~t = B and the (i, x) clock goes off, then according to

(i.I), configuration

B jumps to ~/i,x(B)

(cf. (I.Z.Z)).

An additive system is

called lineal, extralineal, local a n d / o r translation invariant if the underlying of the corresponding type.

(i.2)

Proposition.

A particle system that A C

{(%A)}

Proof.

then

A,B c S ,

A ;- N t (x) > 0

t -> 0

B o__[r N t (x) > 0 .

(additivity)

[3

is called m o n o t o n e if for every pair A, B c S

Corollary.

B ~t

such

for all t-> 0 .

Every additive system

By additivity,

if B D A

{(~A)}

is monotone.

then

B A B-A ~t = £t U ~t D

A ~t

for all t-> 0 .

[]

In order to apply s o m e of the basic facts from Chapter I, w e want be Feller.

To guarantee this, one needs a very mild hypothesis on

has influence from times

.

B there is a joint probability space on which A ~t C

(1.3)

is an additive system,

A B = ~t U ~t

NA U B t (x) > 0
...

co t__oo(x, t) if there are n o n - e m p t y such that

sets

~ .

{( A)}

to

Say that

A I, A Z , • • • and

15

(i) for each yea

n >-- i,

there is a path up from

(Y,tn)

to (x,t)

for all

n"

and

lira lYnl :

(ii)

oo

for s o m e

Y

n~oo

If, in addition,

lira n--oo

(iii)

then

P

the

A n

IAnl

n

~ A

n

can be c h o s e n so that

= oo ,

is said to have strong influence from

co to (x, t) .

Influence from

co to

(x, t) w h i c h is not strong is called w e a k influence.

(1.4)

Proposition.

If e

is a substructure such that

P(strong influence from

then the additive

Proof. as

Write

A n-

A .

Bn ~ Z d

A {(~t)}

system

A A ~0t (A) = P(~t N A If A n - - A ,

as

n~oo

A {~t n N

.

A

oo to (x,t)) : 0

~ Zd ,

t -> 0 ,

induced

by

= ~) .

It suffices to s h o w that

An A ~0t (A)--~0t(A )

B n~

Bn= AN

then there are

P

Vx

is Feller.

SO

such that A n N

B n and

Now

= J ~ } A {C A N

A = Jg} C

= {3

path up from exactly one of (A n N or

(A A B e O) n ' c

As

n--co,

{3

to

B n , 0)

(A,t)]

path up from

c (B n ,

O) t o

(A, t)}

.

these last events converge to C

{3

path up from C

(B n, 0) to (A,t) V n }

{strong influence from

co to (x,t)

Thus the claim follows from the hypothesis.

We the models

will discuss

substructures

in this chapter

x~

A}.

[]

with influence

and the next will have

for s o m e

from

co in Chapter

no influence

from

oo .

IV,

but all

In fact,

if

16

(1.5)

sup

~

k.

y c zd

Y ~ ~/('i, ''?()ZI ~ d_y)

=

M


0 : 3 p a t h u p f r o m ( : oo

S = SO U A .

t

(B,O)

to

~t

in

i ~}

e x i s t s ).

^B A {(~t ); B ~ S 0} of S-valued M a r k o v chains, called the

dual processes for {(~)} , and given by AB ~t = {X : 3 path up from (B,0) to (x,t) Jn ~} = A

AB t < TA

AB t_>TA

Finally, introduce "C~ = inf{t >--0 : ~tB = ~ }

AB

Note that ~ finite.

and

A

(= co if no such

are both traps for

t exists).

AB A B AB (~t) ' so at most one of T ~ and T~

is

Our first theorem will be the main tool in the study of additive systems.

17 To s t a t e i t ,

we i n t r o d u c e t h e n o t a t i o n Ct~ =

~ ~

~

(1.8)

,

Ac

S,

Theorem

induced by

P,

A~

SO ,

S,

B~

~ pt ,

Bc

A

(Pt

=

6APt (P

et(A)=~(

hA= ~),

SO . A

( A d d i t i v e d u a l i t y equation.) L e t {(~)} be t h e a d d i t i v e s y s t e m AB {(~t ); B ~ S O } t h e c o r r e s p o n d i n g d u a l s y s t e m . For e a c h t -> 0

(pt(B) = (Pt (A)

(1.9)

M o r e generally,

A E

if

(1.10)

•

A

is the expectation

t~(B)= E[ ^- ~ (~S)]

o p e r a t o r for t>- 0,

For ~8 = Bernoulli product measure with density we

(1.11)

^

I ~ 0 , A ~

VA

{¥ = A} •

¥

On o c c a s i o n s w h e r e w e c o n s i d e r s u c h a p r o c e s s

and set (~

) ,

it

will be assumed without further comment that this construction has been carried out.

(i.17)

Notes.

Lineal additive processes are studied by Harris (1978);

Berteln and Galves

(1978).

Graphical duality has appeared in one form or another

in Broadbent and H a m m e r s l e y Toom

(1957), Clifford and Sudbury

(1968) and Vasilev (1969).

Leontovich

Z.

(1970).

Holley and Liggett (1975), Holley and

Stroock and Williams

Monotone

(1973), Harris (1978),

For another more analytical approach to duality,

the reader is referred to Harris (1976), Stroock (1976d), Holley,

see also

(1977) and Vasershtein and

(= attractive) systems are discussed by Holley (197Zb).

Ergodic theorems for extralineal additive systems. In this section w e derive general ergodic theorems for extralineal additive

systems. measure

v c ~

(z.i) where

A particle s y s t e m and a constant

{(~tA)} ~ > 0

~IPt({A : A n A = A0} ) cA

-

is called exponentially er@odic if there is a such that, for every

~ ~ • , A 0 O A ~ SO ,

v({A: A n A = A0}) I -< c A e - ~ t

is a positive constant depending only on

A .

,

20

(Z .Z)

Theorem.

e(k;V,W)

•

A

Let

{(~t)}

be an extralineal

additive

system

with substructure

If

(Z.3)

inf Y ~ Zd

~ (i,x):

ki

~X

=

~ > 0

Y~ % , X then the system is exponentially ergodic.

Proof.

Condition

at each

site

y

dual process

(Z.3)

states

that a

with rate at least

goes to

A

By duality, for any

[3 a p p e a r s

K > 0 .

Thus,

with rate at least

A A B AB P(T~ A T A

(Z.4)

In fact,

A(

S,

~ .

(Z.l) holds with

in the substructures from any non-empty It f o l l o w s

¢~ = K

f~ a n d finite

that

-~t

> t) --< e

B c SO , t ~ T •

B ~ SO ,

A AAB AAB ~t(B) = P(T~ --< t) + P(~t N A =

AB A B )Z , t < ~)~ A T A ).

Rearranging,

@A(B)

A/XB P(T~

-

=P

Now

apply

(Z.4)


0 . x x C h e c k the various assertions m a d e about proximity systems.

In particular, verify that every additive spin s y s tem is a proximity system. by e x a mp le that distinct substructures additive system, s a m e j u m p rates. a substructure

i.e. that distinct

~i and

(k I ; V I,WI)

~Z

can give rise to the s a m e

and

(k Z ; V Z , W Z) can induce the

Prove that any additive system has a representation in terms of

~(X;V,W)

such that either V i , x :

If {(~ At )] is a proximity system, coalescing branching processes.

~

or W i , x ( y ) = {y} V y

A •

.

on ~

are

a particle in the dual tries to

C i, x ~ $0 "

Whenever

attempt to o c c u p y the s a m e site they coalesce into one. sends the whole process to

^B (~t)

then its dual processes

At rate ki, x

replace itself with particles situated on

x

Show

two particles

At rate

In keeping with Corollary

Kx

a particle at

(1.15), ergodicity

25

of the proximity system is equivalent to eventual absorption of the corresponding coalescing branching system at either @ (Z. 15)

Problem.

+X X

with probability one.

be an extralineal proximity process such t h a t

{(~)}

Let

or A

and

> 0 forall x , X

K x

inf x

> 0

Kx+Xx S h o w by example that the convergence

Prove that the system is (strongly) ergodic. need not be exponential. A

(2.16)

Problem.

Let { ( ~ ) }

be a (one dimensional) basic voter model w i t h

spontaneous birth at the origin, i.e.

c0(A) = K(I-A(0))

+

A(0)

the e x t r a l i n e a l proximity system with flip r a t e s

+

(F1 -A(0))IAn {-I,i}[,

1

Cx(A)= A(x)+ (~--A(x))IAD {x-l,x} I for some

K > 0 .

. x>~0,

Prove that the system is (strongly) ergodic.

The final result of this section is a correlation inequality for proximity systems. (2.17)

Theorem.

K

{(~A)}

cA(Bu C)-

Proof.

By d u a l i t y ,

CA(B) cA(c)

A~

S,

B, C c

SO , t e T •

it s u f f i c e s to c h e c k t h e e q u i v a l e n t i n e q u a l i t i e s

AB

(Z.I8)

i s a p r o x i m i t y system, then

[J

C

~t

AB

/"C

(A) -> ~t(A)~t (A) •

To do this, w e use a strategy similar to the one which proved Theorem (Z. 6). w e fix B and

C,

and construct independent copies of (~B) and A

independent substructures

Namely,

(?tC) by using

A

~I and @? to define them.

But n o w w e introduce a

different representation of process interpretation. from

(~zxtB-U C ) , by making use of the coalescing branching AB Namely, whenever a particle from (~t) collides with one

AC (~t) ' the former survives and the latter dies.

mechanism is indistinguishable from coalescence,

Since this collision

w e do in fact obtain a copy of

26 AB U C (It-)

with the key property AB U C AB AC It C It U ~t

(Z.19)

In terms of our construction,

Vt ~ T

.

(Z.18) is equivalent to

AB U C AB AC P([t • A = ~) -> P(([t U It ) ~ A = ~) ,

an immediate consequence of (Z .Z0)

Problems.

(Z.19) •

[]

S h o w by example that the correlation inequalities of the last

theorem do not hold for all additive systems.

For which additive

{(~A)}

other than

proximity systems are the inequalities valid? (Z.ZI) Notes.

A result closely related to Theorem (Z.Z) m a y be found in

Schwartz (1977).

For versions of

(Z.Z) in the spin system setting, see Holley and

Stroock (1976d) and (in discrete time) Vasershtein and Leontovich (1970).

The

discrete time analogue of Theorem (Z. 6) is proved by Bramson and Griffeath (1978a); similar but more sophisticated inequalities for the stochastic Ising model (cf. (III.3)) have been obtained by Holley and Stroock (1976b). R. Arratia (private communication) has s h o w n that v

satisfies a strong form of exponential mixing

w h e n the hypotheses of (Z. 6) are satisfied.

Pointwise ergodic theorems for

particle systems were first obtained by Harris (1978);

w e note that Theorem (Z. 8)

can also be proved by generalizing the criterion he gives for lineal additive systems. Lineal proximity systems and coalescing branching processes were introduced by Holley and Liggett (1975).

Problems

(Z.15) and

(Z.16) are adapted from Holley

and Stroock (1976a) and Schwartz (1977) respectively.

Harris (1977) has proved a

much more general version of Theorem (2.17) by an entirely different method.

3.

Lineal additive systems. If the percolation substructure ~(k; V , W )

so that no

~'s appear, then w e abbreviate

is lineal, i.e. if Vi, x-:

~ = {~(X,W) •

Additive systems induc-

ed by lineal substructures have the important property that spontaneous creation is impossible.

In other words,

)~ is a trap so that 6@

is invariant.

In

27

biological contexts such systems might be termed "biogenetic" (as opposed to "abiogenetic").

Ergodicity is therefore equivalent to w e a k convergence to

6@ from

any initial state, and the ergodic theory of lineal systems turns out to be m u c h more delicate than that of extralineal ones.

The remaining sections of this chapter will

be devoted to the study of specific lineal additive systems voter models,

coalescing random walks) in some detail.

(e.g.

contact processes,

But first, w e note a few

simplifications which take place in the duality theory for the lineal case,

and prove

an ergodic theorem for lineal proximity processes.

(3 .i)

Theorem.

ture ~ ( k , W ) , substructure systems.

Let

{([A)}

AB {(It ); B ~ S} ~(k,W)

For each

.

Let

t e T ,

(3.1)

be the lineal additive system induced by a substructhe lineal additive system induced by the dual

tA

~0

and

-'B ~ot

be the zero functions

of the respective

A, B e S ,

~tA(B) = ~~tB (A)

There is an extreme invariant measure

v I c h~

such that

8 d-- vl as Z

t~oo .

Moreover,

{(~A)}

ergodic k,

are nonergodic.

k *.+ be defined similarly in terms of the one-sided systems

{({x'A)} , the analogous assertions hold.

Proof.

Property

0 -< k I < k Z < oo . define

[ 0 ; call

{ ( [ A • t) } can be represented in terms

O b s e r v e that

~A c kI, t in the joint realization,

[A

A~

k Z , t

which yields

S

t~ T '

(4.3).

Hence

'

PX

is increasing in k .

If

31

k < k¢ ,

then

invariant,

Pk-- V k , l

so are the

( {0

is infected}) = 0 .

6Z P ~ , t ~ T,

is infected}) > 0 ,

this case.

whence

(3 .i) .

v I / 69 .

W i t h p k , k ¢ , PX and k~

Proposition.

2k - 2 pk--< 2 k _ 1 =

whence k¢ -> 1 ,

0

Thus

k+ > 2

Proof.

We

are translation v k,l = 69

If k > i¢ , then

Clearly the system is nonergodic in []

d e f i n e d a s in Proposition (4.1) ,

X-Z,

=0 whence

@

k >I,

+_< k - 3 Pk k 1

X- 0 , R k , t m o v e s one unit to the right at rate k ,

least one unit to the left at rate

1 .

one unit to the right at rate

Thus

creases by at least

1 .

1 at rate

Z ,

Lk,t

m o v e s one unit left at rate k ,

D t increases by

whenever

D t -> 1 .

1 at rate 3X , w he r e a s the process dies out at rate

PX -< P0(Xn -> 0

Yn)

anda_!t

,

1 .

1 at rate ZX , From value

0 ,

It follows that

at least

and deD t goes to

32

where

(Xn) i s a d i s c r e t e t i m e M a r k o v c h a i n o n t h e s t a t e s p a c e

{-1,0,1,

Z, . . . }

with transition probabilities

Px x+l

k - l+k

POI

Zk - l+Zk

P-I-I

=

i Px x-i - l + k

'

'

PO-I-

x >~ 1 ,

1 l+Zk

i

the t o t a l probability equation :

Consider

Vn)

P o ( X n _> 0

(4.5) =

Since

X n

Z__~X [Pl(Xn > 0 Vn) + PI(Xn = 0 for some n)Po(Xn ->0 I+ZX

i s a r a n d o m w a l k w h e n r e s t r i c t e d to

x ~ t ,

¥n)]

the famous gambler's ruin

formula implies that

PI (Xn = 0

S u b s t i t u t e in

(4.5)

for P X "

(4.6)

for s o m e

a n d s o l v e for

n)

I

- k

k > i

=

k-- 0 Vn) ,

the desired upper bound

[]

Problem.

D e r i v e t h e b o u n d s on

PX+ a n d

X .+

g i v e n in P r o p o s i t i o n ( 4 . 4 ) .

W e n o w t u r n t o o n e of t h e d e e p e s t r e s u l t s i n t h e t h e o r y o f p a r t i c l e s y s t e m s : t h e p e r m a n e n c e o f i n f e c t i o n for c o n t a c t s y s t e m s w i t h s u f f i c i e n t l y l a r g e no k n o w n p r o o f t h a t

X* < co w h i c h i s t r u l y e l e m e n t a r y ,

H o l l e y a n d L i g g e t t (1978)

comes the closest.

k .

There is

but a r e m a r k a b l e method of

We sketch their approach,

referring

t o t h e i r p a p e r for m o s t of t h e d e t a i l s .

(4.7)

Theorem.

With

PX ' X , ,

i

~I

whence

X, ~ Z,

and

+ X,

defined as in Proposition

i 4

PX - > ~ + ~

PX

Zk

k > Z ,

and +

I

[I

P~ ->g+ 4 7 - T

I

k>4

,

(4.1) ,

33

whence

k + < 4 The basic and one-sided cases are analogous,

Sketch of proof. former.

The idea is to find a translation invariant

[~({A : 0 ¢ A}) > 0 A ~ S0 .

and

~,t(A)

= P([~,t N A = @)

This clearly proves nonergodicity;

(4.8)

Vk,l({O

such that

is decreasing in t for all

in fact

is infected}) ~> 1 - ~M(0) > 0 .

For the remainder of the discussion notation.

~ = ~k

so w e discuss the

k

will be fixed, and often suppressed from the

By self-duality of the basic contact process

(cf. (3.5))

and

(i.i0) ,

(A) = E [ ~ ( E t A ) ] ¢t

It t h e r e f o r e s u f f i c e s to c h e c k t h a t d E[~0[~ ( [A) ] d-~

(4.9)

t

-< 0 =

Unfortunately, however,

VA

~ SO .

0

no product m e a s u r e

~0

satisfies

(4.9)

a renewal m e a s u r e w h i c h works provided

k

for all A .

There i__ss, The

is large enough. oo

renewal m e a s u r e

~f ~ ~

is determined by a probability density

f = (fk)k= 1

co

such that m =

~ kf k < co . k=l abilities given by

~f is translation invariant,

~f({A : A(x) = A(X+Yl) . . . . .

A(z) = 0 for all other

A(X+Yl+...+yn)

with cylinder prob-

= I,

z ~ [x,x+Yl+O..+yn])

n

= m -I

~ f 2,=i Y2,

The m e t h o d of H o l l e y and L i g g e t t i s to c h o o s e equality in case

(4.9)

A = [x,y]

for a r b i t r a r y

A with

for s o m e

x -~ y ,

~ = ~{ so c h o s e n .

(fk)

so t h a t

(4.9)

holds with

and t h e n to p r o v e t h e i n e q u a l i t y The a l l - i m p o r t a n t s e c o n d part of

34

the program is rather involved, and Liggett (1978). contact process k ¢ [ 0, n-l]

so w e will omit it, and refer the reader to Holley

To find the desired

f,

note that w h e n

grows one unit at either end with rate

recovers at rate

i.

k,

Thus, equality in

A = [ 0, n-l] , the while an infected site

(4. 9 ) is equivalent to the

equation n-I

[~b([0,n-l]

- {k}) - ~ ( [ 0 , n - 1 ] ) ]

k=0

(4.10)

=

k[~bc([0,n-1]) - ~ ( [ 0 , n ] ) ]

+ X[~a([0,n-1])

- ~([-l,n-l])]

.

co

Put

Fn

~

fk

Then

°

(4. i0 ) b e c o m e s

k : n+l (4.11)

ZkF n =

n-i ~

FkFn. k,

n -> 1

(F 0 =

i).

k:0

To

the

find

Fn ,

introduce the generating function

E(x) =

~ n=0

Fn xn .

(4.11)

is

equivalent to

z x(r(×)

- l) : x r Z ( x )

,

or

r ( x ) --

! F n = n!(2n) (n+l)!

O n e can solve for F to get k -> 2

.

Over

this

k - 4 k Z - Zkx

parameter

(Zk)-n

which

is s u m m a b l e

for

range, co

k Since

b ( { A : 0 ~ A}) = m -I ,

(4.1Z)

Problem.

0 (4.8)

yields the lower bound on

[]

PK "

S h o w that analogous computations for the one-sided systems give

(Note that + both the upper and lower bounds for Pk are precisely the s a m e as those for PZX " + + It is an intriguing and open question as to whether X, > ZX, , X, < Zk, , or rise to the inequality for

perhaps

X%+ = z k . . )

p~

w h i c h is stated in T h e o r e m

(4.7)

°

35

2X-2 .5

i

X,

2

3 X figure v.

•

.

'

o

7

1

,

-

-

•

.

.

.

.

.

7

.5

J

2

X~ 3

4

5

I

6

I

!

7 X

figure vi.

36

To s u m m a r i z e ,

we

h a v e seen that

pk= lim P(O ~ ~Z is increasing in k ,

0

x ' t ) = p(~ ' t ~

t--oo

equals

0

for k -< 1 ,

Vt)

and is strictly positive for k >-Z .

In

fact,

PX

is s a n d w i c h e d b e t w e e n the two curves s h o w n in figure v .

While we have

drawn

PX

to be continuous at X = k~ , there is no k n o w n rigorous basis for this.

The analogous graphs for the one-sided s y s t e m s are s h o w n in figure vi. Theorem

(4.7)

gives the best k n o w n upper b o u n d s for X%

and

k +% •

In

contrast, there is a technique for improving the lower b o u n d s of Proposition (4.4). We

illustrate this with our next result.

Proposition.

(4.13)

Let k,

and

k +, be the critical values for the basic and one-

sided contact s y s t e m s respectively.

k, > 1 + ~ - 7 -

Proof.

~ 1.16

6

Then

and

X+ > ~ *-

'

~

Z.41

W e derive the first b o u n d ; the s e c o n d is left as an exercise.

By self-

duality, it suffices to prove that 0 P ( T X , ] ~ = co) = 0 w h e n e v e r

Set

( ~ ( A ) = P ( T,A ..@ =~~ ).

and note that

time M a r k o v chain obtained by looking at Also, by translation invariance, (~({x, x+Z)}) --- (7(. -- .), etc.

(A)

ZX o(.) - I + ZA

(B)

~('') : ~

(c) (D)

1

o

1 +437 6

is a harmonic function forthe discrete

{(~X, t)} at its j u m p times

w e can write

or(x) -~ (~(.) ,

The following total probability equations are obtained

k

~(') +i-%--f ~(" "') ' 1 1 +zx

~('")-

2 3 + zx

T I, T Z , • • •

(~({x,x+l]) =- o('-) ,

o(..) ,

~('-')-

O b s e r v e next that

k -
x by using

to (z,t) . R xt instead.

Hence

x

z ~ St .

The s a m e argument

This completes the proof of (5.6).

We

remark that the nearest neighbor nature of the infection m e c h a n i s m is crucial in constructing the composite path, for otherwise paths can "jump over one another. " Turning to the proof of (5.7), w e introduce

A

•N = min{t ~ T : For N -> IAI , A TN < =

I~AI

= N}

(= co if no such

t exists),

N -> 1 .

since ]~ is a trap, a standard M a r k o v process argument s h o w s that

P - a.s.

on

A {T~=

~ } •

Hence

40

P(lim inf RtA < co, ~ =

= 7's~

co)

I~IB

(0,~o) B:

A

t , T~=~) ~ ds, ~ A A : B)P(liminfRB--~--

P(O¢

Z+ N t (0) > 0

S i n c e p a t h s c a n n o t jump o v e r o n e a n o t h e r , 0 i m p l y N t (0) > 0 . Hence

lira i n f

way down to the critical

We conclude and one-sided

and

Theorem

(5.1)

The present result is of interest constant.

this section

one-dimensional

lastthree theorems hold forthe for any

+ k > k. ,

We remark that for

P(O ~ I t ) _ s 2 •

. 0 + Nt (Z)

> 0 together

Z P ( 0 ~ [+~) > i _

lira i n f

as desired.

Yt

yields because

[]

with a brief survey of further results contact •

+,A}

i(%k,t)

systems. .

First,

Intact,

for the basic

we note that none of the

if A ~

S O then

~UAPt--6)~

k : in the nonergodie case the set of infected sites wanders off to the right

if it does not die out.

By taking A

to be a countable disjoint union of larger and

larger blocks which are farther and farther apart, and by taking k > k +, , get examples

where

along one subsequence Liggett only

all the

it applies

6f{ and

(1978)

6APt

does not converge

and to

69

as

t ~

one

, but rather converges

can

to

v1

along another.

has shown that both the basic and one-sided

systems

v I as extreme invariant measures for al__/lparameter values

convergence to v I from "nice" initial measures

Z

have

k •

Also,

takes place in both systems for

all parameter values, as a special case of a result to be mentioned in the next section.

The questions of convergence and pointwise ergodicity for the basic

systems with X

just above

(5.16)

Theorem

Notes.

X. , and starting from arbitrary Z ~ ~ , remain open.

(5.1) is proved in Griffeath (19T8a).

A similar discrete

time result w a s obtained by Vasilev (1969), using the contour method of percolation theory. cation).

The proof of (5.1) which w e give here is due to Liggett (private c o m m u n i There is a sketch of this proof in the introduction of Liggett (1978) ; it has

the advantage of leading to (a) Liggett's theorem that ~

is one-dimensional in the

44

nonergodic case,

6.

and

(b) T h e o r e m

(5.13)

(which is taken from Griffeath (1979)).

C o n t a c t s y s t e m s i n several dimensions. There are m a n y w a y s to generalize the basic contagion model studied in the

last section. d >- 1 .

consider here only the most natural generalization to Z d ,

By the basic

system on i >- i,

We

Zd

and

with

d-dimensional

contact system w e m e a n the lineal proximity

Ix -: {0,i, .-- ' Zd} , k 0 ,x -: I '

C i , x = { x , y i}

for i -> 1 ' where

mediately adjacent to site x .

In words,

C 0 ,x a @ , k i , x =- k

the Yi are the

Zd

for

sites im-

infected particles recover at rate 1 ,

while infection takes place at a rate proportional to the n u m b e r of infected neighbors.

(x

and

y

in Z d

are neighbors if Ix-yl = 1 .)

constant for the rate of infection is given by the parameter Less is k n o w n about several-dimensional of a critical k d case.

in each dimension

d

The proportionality

X .

contact systems,

but the existence

can be proved just as in the one-dimensional

In fact, using s o m e of the methods already discussed,

one obtains the

following results.

(6.1)

Theorem.

parameter

k ,

d If X~ = S U p { X

d k > k~ ,

Let and set

PX =

Vl( 0

d-dimensional

is infected).

Then

PX

contact system with is increasing in k •

: PX = 0} , then the system is ergodic for k < k

,

nonergodic for

and

(6.g)

Proof.

{(~A t)} be the basic

I d < g gd-i -< k~ The argument for everything except

(6.2)

is very similar to the proof of

Proposition (4.1), so w e omit it.

To get the left hand inequality in (6.Z),

the m et ho d of Proposition (4.13).

Namely,

suffices to prove that A g(A) = P(Tx,j~ = co) ,

(A)

Zdk g({0}) = l + Z d k

(B)

~({0, el})=

since

0 P ( T k , 9 = ~) = 0 w h e n e v e r and note that

°(t0'elJ)[l

{(~A,t )},~ 1 k < "Zd-i

is self-dual, "

it

Let

(~ satisfies the total probability equations

' Zd

l+(Zd-l)kl

apply

a({O})+

I+(Zd-I)XX

~, (~({0,el,eZ]) j=z

45

(Here w e have m a d e use of the translation invariance of subadditivity,



k.d < 2

Problem.

By p u s h i n g t h e " s t r o n g subadditivity m e t h o d " f a r t h e r ,

show that

Z X. -> .359.

Virtually all of the k n o w n dimension-independent results for nonergodic contact systems are due to Harris (1976, 1978) ; unfortunately his methods require regularity assumptions on the initial state. (n= 0,I, ...) if A~] b n ( X ) / 9 it is n - d e n s e for s o m e

lira n~ Note that

6A

n •

Say that A e

for all x ~ Z d

A measure

~ ~ ~

So is n-dense

(bn(X) as in (I.l)).

A

is dense if

is called regular if

sup [ ~ ( b n ( X ) ) - [L(0~})] : 0 x~ Z d

is regular if A

is dense,

and that any translation invariant it is

46

regular.

The convergence theorem of Harris states that for any parameter value

if {([A)]

is a basic

d-dimensional

contact system

k ,

(or any of a large class of

contact systems which includes the one-sided system on

Z) , and if ~ is

regular, then pt

-~({#})6~+(l-~({~})v

I

as

t-~

.

This implies that the only translation invariant equilibria are mixtures of vI ,

but it is not k n o w n whether there are additional nontranslation invariant

equilibria w h e n dimension

d -> Z .

d -> Z

very large k . on

6]~ and

Zd

Pointwise ergodic convergence to v I has been proved in

only for initial measures

6A

with

A

dense,

and then only for

Finally, there is a growth rate theorem : for basic contact systems

with sufficiently large infection rates

k :

0

l~x,tl

P(limt_o~inf - - t

0

> 0 I T~ = ~) = i .

O n e can s h o w that the growth is of order at most open for d -> Z .

(6.4)

Notes.

(6.Z)

but the exact order remains

This concludes our discussion of contact processes.

The lower b o u n d in (6.Z)

Liggett (19 ?5).

td ,

is due to Harris (1974)

It improves a result of Dobrushln

is from Holley and Liggett (1978).

in Griffeath (1975).

(1971).

Problem (6.3)

and Holley and

The upper bound in is based on a computation

The rest of the results mentioned in this section m a y be found

in Harris (1974, 1976,

1978), except for s o m e refinements in Griffeath (1978).

Similar techniques were applied to discrete time systems by Vasershtein and Leontovich

?.

(1970).

Voter models. This section is devoted to the study of lineal proximity systems with the

property that

IC i, xl = 1 for all i ( I x ,

the so-called voter models.

According to

(Z.IZ), such a system has flip rates which can be written in the form

Cx(A) = kxA(X) + (I-ZA(x))

~ z ( A

(Xz, x -> 0 ).

k z,x

To simplify matters , w e will treat only translation invariant voter

47

models,

w h o s e f l i p r a t e s c a n b e w r i t t e n i n t h e n o r m a l i z e d form

(7.1)

Cx(A) = k[i(x) + (I-ZA(x))

for some probability density

~ z~ A

p = (Pz; z ~ Z d) and

"voter model, " w e think of the sites of Z d

is influenced by voter y

k > 0 .

, To explain the n a m e

as occupied by persons w h o are either

in favor of or opposed to some proposition (say "voter" at x

pz_x ]

1 = "for" , 0 = "against" ).

with weight

Py-x '

The

and changes opinion

at a rate proportional to the s u m of weights of voters with the opposite opinion. particular, the "total consensus" states

6@

and

In

6zd are both traps for the system.

Since w e are interested in asymptotic behavior of the model,

and the factor k

may

be removed by a change of time scale, w e will a s s u m e henceforth that k = I . foremost question for the voter models is : independence, " i.e.

a product measure

The

Starting from a state of "individual Z(~ , does the interaction lead to

"eventual unanimity" or not? The dual systems for voter models are coalescing branching systems in which each branching tries to replace a particle by another single particle. these are coalescinq random walks.

Particles attempt to execute independent

continuous time random walks with mean-I density

In other words,

exponential holding times and transition

p , but coalesce upon collision.

In particular, the one-particle dual

process is merely a random walk with density

p .

We

say that p is recurrent or

transient according to which property this random walk enjoys.

The basic

d-dimensional voter model is the system such that 1 Pz = z-~ =

i.e.

Izl = l

0

otherwise ,

the voter model whose one particle dual is simple

d-dimensional --

Given a density that if (XtI) and

then

(

times.) if ~

-

p , define the symmetrization (XZ)

-p of p by

Pz -

Pz

random walk. +

P-z

Z

are independent continuous time walks with density

) is a random walk with density

Note p ,

-p ( a n d m e a n ~- e x p o n e n t i a l h o l d i n g

The f u n d a m e n t a l r e s u l t for v o t e r m o d e l s i s t h a t e v e n t u a l u n a n i m i t y o c c u r s

is recurrent,

but disagreement

persists

if p

is transient.

Thus the basic

48

voter model b eh av es one w a y in dimensions one and two, dimension three or

more.

(7.Z)

Let

but entirely differently in

These assertions are m a d e precise as follows.

A Theorem.

with flip rates of p .

{(It)}

(7.1) for s o m e irreducible density

If -p is recurrent,

Zd

be the (translation invariant) voter model on p .

Let ~

be the symmetrization

then

pt (7.3)

~

for any initial measure

-- (I- 8) 6)~ + ~6zd [, such that ~

as

t~

({z}) ~ 1 - ~ .

If p

is transient,

then

corresponding to each ~ c (0, I) there is a distinct translation invariant equilibrium v v 8 , with ~ 8 ( { z } ) -= I - (~ butno___tta mixture of 6}~ and 6zd, suchthat

(?.4)

~pt

Moreover,

Proof.

each

v8

as

t -~o

.

v(} is mixing with respect to translations in Z d .

If ~[~({z}) = I - ~

for all z c Z d ,

i /kx Ct~({X}) = E[¢Z(~ t )] = 1 - @

for all t ( T ,

one particle dual is a r a n d o m walk.)

lim

P([t~(x)/

then by since

To prove

(I.I0),

iX ~t = {z}

(?.3)

[~(y)} = 0

for s o m e

z .

(The

it suffices to s h o w that

~

Vx,y

Zd .

t~co

E q u i v a l e n t l y we c h e c k t h a t (7.5)

lim t~co

~t~({x,y})=

i- e

Vx,y

A key fact about coalescing

random walks is that

t~

I if A / ~ ,

, and always at least

Nt A.

so that

¢ Zd .

= I =

t *i i s

nonincreasing

as

1 exists

P - a.s.

lim t~oo

Thus A{X, ~({x,y))= AEL~[r(~t Y})]

(7.6) "1

A ~{X,y}

= (-8) P(Nt Assume

-p recurrent.

density

p until a collision occurs,

Let t~oo

in (?.6)

Since

to get

(It{x' Y})

(?.5).

A

= i) + E L ~

(It

~

t{x,y}

= 2]

acts as two independent r a n d o m walks with

w e have Next,

~A(X,y}),

AP(I'JQo .'h.{X, y} = I) = 1

assume

~

transient.

for all x , y ( zd . Letting

t--~

49 in

(1.11), AA

~0 0)N ~ lira ~t (A) = ~[(i] t~co so tx0 pt

converges

to a measure

v0

such that AA

v0 (7.7)

Clearly

~

v0

To s e e t h a t

is a translation v0

~[(I-0)

(A) =

invariant

is not a mixture of

d o e s no_._~th o l d f o r ~ = ~0"

]

equilibrium 6~f a n d

But f o r

vo

N

such that

6zd

^ ^ {x, y}

(7.5)

^ ^ {~, y} = i)+ (I-0)ZP(N

(i-0)- 0(I-0) P ( N ~

9 ( ~ { x , y}

0 ~ (0,i) and

( { z } ) =- 1 - 0 •

we need only check that

,, ^{x, y}

provided

vo

x/y,

({x,y})= (I-0) P(N

=

~

= Z) > 0 .

= Z)

= Z) / 1 - 0

This last probability is positive

since p is transient.

To finish the proof, it remains only to s h o w that v 0 is

spatially mixing, i.e.

for each

(7.8)

lim

v0

[9

B,C

~ S O - {)~} ,

(B U (z+C)) - ~V0(B)~

v 0

(C)] = 0

]zl-~ By duality, the quantity in brackets equals AB C ~[(I_o)~B U (z+C) N ] ] E[(I-0) N ~ ]

E[(1-O)

Recall from Theorem (Z.17) that copies of

^B

(~t) and

~c

(

--

(~B U C) can be constructed from independent

) in such a w a y that

u

if

~sn~s

--

For the remainder of the proof w e will be referring to that construction. Thus w e AB U C AB AC can assume that N = N + N if the two independent processes never interact.

Hence

50

ve

[~p

v8 (BU C)-¢p -< P( -
~ y~A

IA I = k , V f~ z~ A

£j yc A c >~ z~A

and

A AX P([t /}~)

t h e c a r d i n a l i t y of t h e v o t e r p r o c e s s b e c o m e s I

Pz-y

Pz-y = q(A) .

and b e c o m e s

Thus

k +l

k - i at rate

(l~tl) jumps like at simple random walk with

c

absorption at 0 after exponential holding times with rate at least Z . A

AZ

P(T~ < ~o ) :

at rate

1,

and the proposition follows.

Zd Since the distribution of It whether an individual site,

walks at arbitrarily large times,

[]

converges to

the origin say,

Hence

6~ ,

i t i s n a t u r a l to a s k

is visited by the coalescing

or w h e t h e r t h e r e i s a l a s t v i s i t .

random

This "recurrence"

question is settled by our next result. (9 .Z)

Theorem.

If

{([t)}a

is the coalescing

r a n d o m w a l k s o n Zd

with density

p ,

then Zd P(lim sup It

(0) = 1) = 1 .

t~oo

The p r o o f r e l i e s o n a l e m m a , w h i c h w i l l i m p l y t h a t t h e e x p e c t e d a m o u n t of Zd time that 0 is occupied by (It ) is infinite. T h i s p r e l i m i n a r y r e s u l t , s t a t e d for the voter model,

i s of i n t e r e s t i n i t s o w n r i g h t .

(9.3)

Let

t--> 0 ,

Lemma.

{ (AA I t )}

b e t h e v o t e r m o d e l on Zd w i t h d e n s i t y

p .

T h e n for

60 A A0 )-I P ( [ t /JZ)_> ( l + t Proof.

AS already noted,

A0 It = A , I -< IAI < ~ ,

when

the voter process

increases or decreases by one particle at the s a m e exponential rate 2 2 Pz-y " Clearly q(A) < I-A_I . Let (Zt)t >_ 0 be a birth and y~ A c z~ A death process on {0, I, Z, • • • } with absorption at 0 , and with transition from

q(li) =

k to k-I or k+l

(k-> I) at the s a m e exponential rate

note that since this process jumps at least as fast as more quickly.

_>

(Hint:

(Zt) at 1,

(Problem

Problem.

(9.4).)

Hence

-I u(t)= (l+t) ,

S h o w that the function

(Zt) is a G a l t o n - W a t s o n

Then

u(0) = I and

completing the proof.

u(t) defined above satisfies

du Z ~ - = -u

process.)

Zd Proof of T h e o r e m (9.Z). Let T t = m i n { s -> t : 0 ~ ~s } , and note that Zd {lira sup ~t (0)= I} = lira lira {Tt~ It,u]} . For 0 -< t < u , b y t h e t ~ ~ t~ u--~

Markov

property and monotonicity, u

s[J

~

Sd(o )

t

ds] =

]u

f

P(T t

dr, ~ Tt zd ~ dA) E[ fo u - r ~2(0) ds]

t

u

t

Zd

7 P( t dr.

Thus

P(q:t * I t , u ] )

last l e m m a ,

t,

[t'u])E[~ 0

u

dAl u

= P(Tte

For each fixed

and

p(zt/o).

Let u(t) denote the right side of this last inequality.

(9.4)

Start

Thus

P(i~t°/s) du Z d--t- = -u

k .

A0 (%t) ' it will be absorbed

°

Zd ~s (0) ds]

Zd

(01 ds]

.

u Zd E[]" ~s (0) ds] t -> u Zd E[]0 ~s (0) ds]

the left side tends to 1 as u ~ o

•

since by duality and the

61 =

E[;

zd

Zd

~o

~s (0)ds] = fo

P ( O e ~s ) ds

~o

~

j£ Zd Thus P(limsup ~t (0)=i)= t ~

lim

~

lim P(Tt~ [t,u])= i,

a s = oo

and the proof is

U ~¢o

t~

finished.

(l+s)- 1

P(~: / ~ ) d s >- JO

[]

W i t h more c a r e o n e c a n e x t e n d T h e o r e m ( 9 . 2 ) t o c o a l e s c i n g r a n d o m w a l k s s t a r t i n g from a n y d e n s e c o n f i g u r a t i o n A . the case where

p is transient;

The r e a l c o n t e n t o f t h e t h e o r e m l i e s in

t h e r e i s a m u c h s i m p l e r a n d more g e n e r a l r e s u l t

if p is recurrent: (9.5)

Problem.

Show that for p irreducible recurrent, A

P ( l i m s u p ~t (0) = 1 ) = 1

VA/

t~co

(p

i r r e d u c i b l e m e a n s t h a t t h e g r o u p g e n e r a t e d by

{y e Zd : py > 0}

i s a l l o f Zd

o)

W e c o n c l u d e t h i s s e c t i o n w i t h c l u s t e r s i z e r e s u l t s for c o a l e s c i n g r a n d o m walks on Z .

To k e e p m a t t e r s s i m p l e ,

we start from

d i s t a n c e from t h e o r i g i n t o t h e f i r s t n o n - n e g a t i v e s i t e prove

a

d i s t r i b u t i o n l i m i t t h e o r e m for

(9.6) Theorem.

5z x

Let such that

D+(~ Z ) x c ~(

be the .

We

D+ (~Z)

With D+ defined as above, Z D+ (~Z) P( __ -< ~ ) 4t

lira t--= Proof.

Write

n = n(t) = L a Q ' - t - 3

1 -

c~

-

(

ks3

e

f

4~

s 4

ds .

0 i s t h e g r e a t e s t i n t e g e r in

s •)

By duality,

D+(~)

_

P(

_


0 , for s u c h

(i,x) ,

the labels

Vi,x/~ ~

and

are called

Wi,x(y

pure births.

) = {y}

Vy

;

To d e f i n e /k

introduce the modification

~(~,

V, ~ r )

of t h e d u a l s u b s t r u c t u r e

/k

/k

P(X,V,W)

such

that ~i,x

= Zk i,x = k.

1pX

if

(i,x)

has pure births in

~

,

otherwise.

Define "~t = { ( x , s ) ,

0 < s -< t :

~B ~A = i n f { t -

0:

3

a pure birth occurs at

odd n u m b e r of p a t h s u p from

Now let ~B=

( nAB t,

e )

0 ~ t < --A *

(1.4) =

A

~B

Td~ --< t < ¢ °

(x,s) (B,O)

in 7} to ~ t

,

in 7 )

68 where AB q t = { x : 3 odd n u m b e r of p a t h s up from B

s t --- n u m b e r of p a t h s up from

(Recallthat

St=

{(x,s),

(B,0)

0 < s-< t :

to

(B,0)

to

St i n ~

(x,t)

in ~}

,

(rood Z) .

a birth occurs at

(x,s) in ~ }

The

l)

duality equation for cancellative systems m a y be stated as follows. (1.5)

Let

Theorem.

{(~B),B~ s o } t>O,

{ (T]B)} b e t h e c a n c e l l a t i v e

t h e c o l l e c t i o n of d u a l p r o c e s s e s

A~ S ,

Let

defined by

(1.4) .

£

J

T h e n for e a c h

B~ S O ,

P(I~]An B I even)-- ~(l~tB N AI+ s B

Proof.

system induced by

Pt(k ; V,W)

and

Pt(k ; V,W)

even,

~TB~

> t)

+ ~1 ~P ( ~ ~B

_< t )

be the forward and reverse percola i

tion substructures Acopyof in

~t

Pt'

on

Zd × [ 0 , t ]

:

Zd x [ ~ , t ]

t h e r e s t r i e t i o n of P ( X ;

as follows : at each location

V,W) ( x , s)

constructed to

zd×

(i.e.

the

[~0's

are ignored),

(~Asl0 -< s -< t and

and only the

~l'S

t} : {l~t B N AI + eB

0 or w i t h a

a r e p u r e b i r t h s i n Pt Thus

Moreover,

even, ~TBA > t} P - a . s . ,

((A, 0) U 8t ) and

(B,t),

use the fact that

I n O n B t even

I {Y ~ B : N A(y) NA(B)

> P ( ~ B _< t) = 0

1 .

equivalent to

{ a n e v e n n u m b e r of p a t h s b e t w e e n

If

[3 w i t h a

are r e a l i z e d on the joint s u b s ~ u c t u r e

{ llqtA N Bleven, N TB>

To s e e t h i s ,

can be embedded

t h e n we o b t a i n t h e d e s i r e d v e r s i o n of Pt "

(~sB/0 _< s-< t

since both events are a.s.

[0,t],

w h e r e a pure b i r t h o c c u r s i n -~t '

f l i p a f a i r c o i n t o d e c i d e w h e t h e r to s u b s c r i p t t h e l a b e l If a l l of t h e c o i n f l i p s a r e i n d e p e n d e n t ,

as in Chapter I.

we're done.

odd}l even

even

AB B I~]t n A I + st even,

Otherwise,

write

P - a.s.

AB ~t, > t} .

69

{IT]At N B l e v e n ,

~TBA -< t} : (En F N G) O (EN F C n G c)

where

E : {T B--< t} ,

F=

{~Bl'S occur at T AB}

,

G = {odd number of paths to (B,t) from ((A,0) U ~t) - (zd,TAB)} •

The key observations, which follow from the construction ~ are that i are conditionally independent given E , and that P(F IE) = 2- " P ( E N F N G) + P ( E N F c N G c) = I [ p ( E N

F

and

G

W e conclude that

G) + P ( E N Ga)]

i

= 2- s (s) • []

The d e s i r e d d u a l i t y e q u a t i o n f o l l o w s i m m e d i a t e l y .

The quantities

- t, number of paths up from (B,0) to (x,s) in ~

P - a.s.

sinceif

State

hitting times;

~

is also a trap.

~

W e let T0B,

dearly at most one is finite.

~B

"cI

and

iseven ~B

TA

Vx~

Zd

be the respective

For convenience, we also put

70

?B

= ~BA TO

~IB T A ~TAB .

Two e a s y c o n s e q u e n c e s

of T h e o r e m (1.5) c o n c l u d e t h i s

section. (1.8)

Corollary.

v ( r~

such that

Let

A

{O]t" )}

be cancellative.

pt ILl

as

V

~

There is an invariant measure

['P(T 0 < ~ ) + ' P ( ~

t ~°°

where

CV(B)=

Proof.

Integrating the duality equation with respect to

,

= co)] and using the fact 2

~l 1 E (B) = ~-

whenever

) ~ / B e SO ,

~t1

•t

g (B) : E[~I_ ( { A :

we g e t

IAn ~'B!

+ 8B even}), ~ B > t]

2

4~- P(T A --< t)

--< t ) + F

= ~('~

Let

t - - o~

(1.9)

for each

(I.11)

Proof:

--< t )

a n d do s o m e a l g e b r a to f i n i s h t h e p r o o f .

Corollary.

(1.10)

> t) + FP(TA

tim t ~°~

Let

{ (A)

}

be cancellative.

(

= ( A , 0 ) ) - P(

•

[] If

= (A,I))] = 0

A

B ~ S O , then

{ (~]A)}t is ergodic.

~ ( ~ B < co) = 1

The d u a l i t y e q u a t i o n c a n b e r e w r i t t e n a s

In particular, ergodicity holds if

VB ~ S O

71

q~ ( B ) = ~ (

- t)+

~

-

•

[ #(~B = (A, a), IA n A I + s even)

a=0,1

lAnAI

_ ~p(rl t~B = (A , ~ ) ,

%ssuming

(1.10),

the last

Bxists for each

B ,

~ote that condition

(1.1Z)

Notes.

systems

zancellative

Z.

(1.11)

"Q : 0

(1.9)

Extralineal

the dual to

(2.I)

"

A •

t~=

•

Thus

This proves

odd)]

.

A ~ t (B)

lira

ergodicity.

Finally,

[]

systems

due to Hoiley

generalizes

and Stroock

the theory of spi

(1976a).

The graphical

In particular,

approach

to

is new.

systems

The analogue

Let

(2.Z)

of

of cancellative

to t h e a d d i t i v e

Theorem.

as

(1.10) .

inf A~ SO

{(A)

with pure births.

case,

need not be ergodic. A .

0

on one of their results.

cancellative

In contrast {ire systems

implies

duals,

is based

systems

to

and is independent

Our treatment

with

Corollary

sum tends

+ s

translation

One needs

of Theorem

}

the presence

(II • Z . 2 )

be cancellative.

2 purebirth (i,z):

invariant

extralineal

cancella-

of pure births to send

in the present

setting

is

If

k. = 1,z

~ > 0 ,

Ia n Vi, zlOdd then the system

Proof.

Condition

rate at least ~B T < =

is exponentially

(Z.2)

ensures

2 K from any state

P - a.s.

,

ergodic.

In f a c t ,

(II.2.1)

that the dual process other than

and the duality

equation

(@,0), yields

( ~ t ~) (@,1)

Q=ZK

holds with

goes to and

A .

A Thus

with

•

72

: I 0{B) 7( C
0 .

at e a c h

to be of the form

l,X

Wi,x(Z) : {x,z}

(2.4)

{y}

[3

One can assume

site x ,

{ O] A) }

with pure births is

n o w determine the general form of the flip rates for cancellative spin

systems.

before,

system

ergodic.

Pick

c a n find a

for

[]

X I

x

= JJ

x¢

z/x

Ci,x

to get the m o s t general cancellative and

V, = {x} l~X '

by r e m o v i n g

i= 0 .

llx : {i ~ Ix : Vi, x : {x} } ,

i -> I . Then,

As

denoting

the flip rates for

so i n d u c e d h a v e the form

C x ( i ) = Kx +

~

hi. x i0 : x I(AO C i , x ) h { x } [ o d d i~

+

E

hi, x i1 • x " I ( A • Ci, x ) A { x } l e v e n i~

73

T o ensure that there is n o influence from

~x

=

~

co ,

k. l,X

let

,

i• Ix assume

(2.5)

sup ki, x < ~o , x

and also

(II. 1.6).

T h e n T h e o r e m (Z.1) a p p l i e s i f

(g.6)

inf X The duals

(~]t)

branching processes

for t h e s e s p i n s y s t e m s

with parity.

replace itself with particles already occupied, This describes 0

and

Finally, (Z.7) Ci,x/

With rate

located at

"annihilation"

t h e e v o l u t i o n of

may b e t h o u g h t of a s a n n i h i l a t i n g

ki

Ci, x

"

AB

a particle at

At e a c h s i t e of

takes place, (~t)

,

,x

(a)

Cj, x

Show that one can assume i ~ Ix0 ,

if

j ~ I x1'

Ci,x

/ {x}

ic

ixl e f f e c t s

A

at rate

Vi ,

and

in t h e g e n e r a l r e p r e s e n t a t i o n

D e r i v e t h e f o l l o w i n g r e s u l t s for c a n c e l l a t i v e

modifying the additive versions

x¢

AB nt

ZK x

(Z.4)

of

If

~ ~ •

If ( Z . Z )

is spatially mixing and

holds and

unique equilibrium

systems by

proved previously.

s p a t i a l l y m i x i n g for e a c h (ii)

which is

spin systems.

Problem.

(i)

Ci, x

t r i e s to

simply flips back and forth between

tB

a flip occurring each time a clock indexed by AB a particle at x • n t s e n d s t h e e n t i r e p r o c e s s to Problem.

AB x c nt

so the site becomes unoccupied.

1,

cancellative (Z.8)

Kx > 0

P v

f~

is local,

then

bpt

is

t < is local and translation

for

{ ( A)}

invariant,

then the

has exponentially decaying

correlations.

(Z.9)

Notes.

For a n o t h e r a p p r o a c h to t h e t h e o r y of c a n c e l l a t i v e

Holley and Stroock S t r o o c k (1976a).

(1976d).

Theorem

(Z.1)

generalizes

spin systems

see

a r e s u l t from H o l l e y a n d

74

3.

Application

to the stochastic

The basic

Isin@ m o d e l .

d-dimensional

stochastic

Isinq model is the spin system

on

Zd

with flip rates

Cx(A ) : [I + exp {- 0Ux(A)}] -I ,

(3.1) where

Ux(A ) = 4 ( 2 A ( x ) - l)(d - ] AN Nx[) (N x = {Y ~ Z d : ly-xl = I}) .

0 ~- 0

is a parameter.

This system

{(~]tA )}

is

one of the simplest and most widely studied models for the evolution of a physical system with two possible states per site (e.g. solid or liquid, "spin up" or "spin down" in a piece of iron, etc.). Background and motivation for the choice (3.1) will be found in the papers mentioned in the Notes of {(nO) } here.

for arbitrary values of

0

(3.3) .

The construction

requires methods which will not be discussed

For certain parameter values, however, the stochastic Ising model has a

cancellative representation, and in these cases our methods apply. with flip rates

The systems

(3.i) are important because the Gibbs measures with potential U

are equilibria for them.

~ c ~

is such a measure if

~

has positive cylinders

and

(3 .z)

~([A, {x}] I [ A , A ] ) = [I + exp{0Ux(A)}] -I ,

where Given

[A,A] : {Be S: such a

and put

~:

[~ , A U x,

B• A:

if we write

A• A} ,

forall finite

xA = A A x

then for

A

A : N

for the configuration

C A C Z d - {x} . X

"A f l i p p e d

at

as above,

~ ( [ A , X ] ) C x ( A ) : b ( [ A , ~ ] ) ~([xA,{x}]

I

[A,A])

: ~([xA,K]) ~([A, {x}] l [xA,A]) = [L([xA,A]) Cx(xA) Roughly, then, the flow from

0

to

1

equals

site w h e n the stochastic Ising model is started in

the flow from .

1 to

0

at each

This suggests

that the

x"

75 A

Gibbs measure

~

is invariant for

{(~]t)} ,

a fact w h i c h can be proved rigorously•

It turns out that the stationary process starting from

~

is time reversible, i.e.

has the s a m e d y n a m i c s whether time runs b a c k w a r d s or forwards. S u p p o s e n o w that for every

~ - 0 .

this, take Cz,x (3.1).

d = 1 .

T h e n the stochastic Ising m o d e l is cancellative

Indeed, it {s simply a voter m o d e l with pure births•

• x ~- (I + e4~) -I ,

I0x = {I,Z} ,

Ix = ~ '

= {x + I} ' X 1 ,x = k 2 , x = ~1 - (i + e 48)-I . Moreover,

since

C l , x = {x - I} ,

Then

K = (I + e48) -I > 0 ,

(2.4)

{( A)}--

In particular, there is only one equilibrium for the m o d e l , G i b b s state v

with potential

decaying correlations. transition w h e n

U .

By Problem (Z. 8 it) ,

so there is only one v

has exponentially there is no p h a s e

d = 1 . d = Z

is m u c h more interesting.

O n s a g e r asserts that there is more than one Gibbs m e a s u r e Q >

coincides with

is exponentially ergodic.

In the language of statistical physics,

The situation w h e n

and only if

To see

Q~

= arc sinh 1 ~ .88 .

m o d e l is therefore nonergodic.

For

~ >

A f a m o u s result of

~

~,

with potential

U

if

the stochastic Ising

This is one of the simplest e x a m p l e s of a translation

invariant local spin s y s t e m with strictly positive flip rates w h i c h is nonergodic. It is not k n o w n whether such a s y s t e m exists in one dimension. computation s h o w s that the representation if and only If mechanisms

Z-dimensional stochastic Ising m o d e l has a cancellative e -
0

and

lation invariant

m < 1 ,

then the system

is exponentially

case the system is also ergodic if

m :

1

ergodic. and

Ci,x

In t h e t r a n s : C i = j~

for

i .

some

A particularly simple family of systems without pure births to which (4.1)

does not apply consists

models,

where

to take

I

x

ICi,x I = 1

= Zd .

of the

(translation

for all

i .

invariant)

generalized

As i n t h e a d d i t i v e

After some manipulation,

case,

Theorem

voter

it is convenient

the flip rates for generalized

voter

models can be written in the form

2 Cx(A) : Zk--(l+ (I-2A(x)) [ 2 Pz-x +If) N A Pz-x ] ) ze (x+I 0) r] A ze (x

(4.2)

for s o m e and

k > 0 ,

I1 of Z d

probability density

Problems.

I 1 = J~ .

that

( I I . 7)

(4.Z)

from

(2..4) .

has a cancellative

What other systems

I0

By a constant change

X = I .

Show how to get

voter models of section with

and disjoint subsets

such that I0 U II = support p = {z : Pz > 0} .

of time scale, one can a s s u m e

(4.3)

p = (Pz ; z ( Z d) ,

Show also that any of the

representation

have both additive

of the form

and cancellative

(4.Z)

represents-

tions ? The generalized Problem (1.3).

When

"voter component" on

Z1 ,

{ ( n tA) }

where

voter models for which I0/J~

and

and an "anti-voter

A0 = { e v e n i n t e g e r s } are both traps.

{ 0]tA)}

should result.

irreducible,

"

For the basic

(4.2)

We now prove this,

i.e.

and sufficient

to be ergodic.

the group spanned

density,

anti-voter

model

it is clear that

A I = {odd i n t e g e r s }

does not have traps,

we give a necessary

with flip rates

and

models of

the configurations

If w e m o d i f y t h e m o d e l b y t a k i n g

then the resulting

In fact,

component.

This is because

are the anti-voter

we can think of the model as having a

p is the simple random walk transition

is not ergodic.

ergodicity

I1/Jg

I0 = ~

1 P-1 = P2 = Z- ' so it seems

plausible

as an application

condition

we assume

p is all of

that

of Corollary

for a generalized

To a v o i d t r i v i a l i t i e s , by the support of

for example,

Zd .

(3.4).

voter model that

p is

78

(4.4)

Theorem.

some irreducible m

,

Let

{( A)}

density

z c I0 U I1 ,

be cancellative,

p .

with flip rates

of the form ( 4 . 3 )

for

T h e s y s t e m i s e r g o d i c if t h e r e a r e i n t e g e r s

only finitely many non-zero,

such that

Z

(4.5)

~ m z.z : 0

and

~ m z is odd. z ( I1

the s y s t e m i s n o n e r g o d i c .

Otherwise

i P-I = PZ : Z-

Note that the system having

is ergodic, since w e can apply

the theorem with m_l = Z , m Z : 1 . Proof.

The dual processes

for generalized voter models are annihilating

(~B)

r a n d o m walks with parity~

w e will verify (1.10) for these processes.

The argument

is based

(

s -> 0 ,

on comparison

of

which ignore the annihilation state space occupancy

and basic of sites.

) with processes rule after time

probability

) :

s •

) ,

Naturally we must enlarge the

s p a c e to a l l o w for m u l t i p l e

Using a more elaborate

, sa

(but finite)

graphical representation,

this can be

done in such a way that

(4.6)

~B ~B s~]t = ~]t

(4.7)

AB ~t

Let

~

behaves

be the extended

PBA(t):

P(~

for

s,

like independent

state

space for the

1

: (A,0)),

t s

PBA(t):

~(

random walks after time

AB (s~t)

: (A,1))

. .

s •

Write Define

0

sPBA(t)

and

s

pl

BA

v

analogously to

(i.10)

(4.8)

s~~ B t ,

in terms of

where

A

is a generic element of

S .

we want to show that

lira t--~

1

~ f{/A~

I p % A ( t )- pBA(t) l : 0 So v

The s u m may be e x t e n d e d

to

A e S ,

and majorized by

VB~

SO

According

79

+ 7A ~ ~s(t )

Is P B° A(t)

1 - sPBA(t) I

+~ s (t)+2s (t)

1

Z

3

To estimate the first two sums w e use the "fundamental coupling inequality" : if X1 a n d measure

XZ a r e ( g e n e r a l ) r a n d o m v a r i a b l e s g o v e r n e d b y a j o i n t p r o b a b i l i t y P,

a n d if

~1

and

lib- ~all -~ P ( x l / x z ) . s

~Z

are their respective

using ( 4 . 5 ) ,

+ ~ g ( t ) -~ Z • P ( t h e d u a l h a s a c o l l i s i o n b e t w e e n t i m e s

dual has a collision after

s ) •

AB (~t)

Since

which disappear with each collision,

apply the M a r k o v property at time

(4.9)

i.e.

lim t--~

i f t h e a n a l o g u e of

AB 10~]t ] = IBI

forall

lira s~

s to

s

t •

and

t>

s,

t ) -~ g • P ( t h e

h a s f i n i t e l y m a n y p a r t i c l e s t w o of s s s u p ()il(t)~_ + ~, (t)) = 0 . Next, t->s Z

s ~'3 (t) •

It follows that (I.i0) holds if

0 ]~ 7 A ¢ ~ [ 0PBA(t) - 0p IB ACt) [ = 0

(4.8)

then

t h e c o n c l u s i o n i s t h a t for

ms

~l(t)

laws,

h o l d s for t h e t o t a l l y i n d e p e n d e n t

B ~ ~,

process.

Clearly

Also,

o

0 P B A (t) = ~({0

= A} f] E0)

and 1 v iB 0 P B A ( t ) = P({0~t = A} N E1 ) ,

where

E0

(and E l)

are respectively the events that the total number of displace-

ments from I1 through time (and odd). in

t

by the

IBI independent random walks is even

Using these observations and

(4.9) is majorized by

(4.7), it is not hard to see that the sum

80

(4.i0)

0 E I P o z (t) - p l o z ( t ) z ~ Zd

]BI!

Consider

A = Zd x

coordinate.

{0,1}

as an additive

The one particle

walk which

starts

at

dual performs

( x , s) c A b y

1 ~0 i) p 0 z ( t ) = Pr(X ' = z) .

and ]B

be t h e s u b g r o u p o f

to

( 0 ~ i ) c ]B .

~

variation

abelian in

~,

norm in

B ,

group. so

yo

But

(4.11)

(4.10) tends

the proof of ergodicity (0,1)/

")-~/~

in the same irreducible

on a countable communicate

{(i,0)

when

then any

A 0 = { z : e z = 0} ,

we note that

A1 :

0

(4.5)

b ~ B

")11-o

Problem.

as

set of states (0,1) c B

holds for

to

Denoting '

: i ~ 11} , (4.5)

the = z)

and l e t

is equivalent

as

holds.

of a continuous

says that

x=

t--~

t--~

(0,0)

,

(0,0) y:

for each If

(4.5)

does

has a unique representation

{z : s z : 1} .

It i s e a s y

to check

time random walk and

(0,1)

(0,1)

.

B c SO .

Thus the This completes

not hold,

i.e.

if

b = ( z , Sz) . that

A0

and

Define A1 are

The proof is finished.

[]

S h o w that an anti-voter model is ergodic if and only if its density

p has odd period in Z d , i.e.

(4.13)

.

p o z ( t ) = Pr(X

The h y p o t h e s i s

traps for {( A)} , so the system is nonergodic.

(4. IZ)

~

in the second

0

: i c I 0} U { ( i , l )

generated by T •

II~(×t~ x, y

a random walk on

(X x , a)),

]Let ~ =

mod Z

Now it is well known that

(4.11) for any

group with addition

Notes.

zd/grp(I I -'I I) = an odd positive integer.

This section is adapted from Griffeath (1977).

Our approach to

generalized voter models is based on the treatment of anti-voter models of Holley and Stroock (1976d). Anti-voter models were first studied by Matloff (1977).

5.

Annihilating

random walks.

For lineal

(e $) to

~,

cancellative

of the duals

(Zts)

systems

are identically

and we get the symmetric

duality

P(q n Bleven)=

{(~tA)}

0,

the second

coordinate

so we can discard t h e m

processes

~

reduces

equation

Aleven)

S.

S0

81

[ u s t a s in t h e a d d i t i v e l i n e a l s e t t i n g , on a l l o f

(5.1)

S •

W e s t a t e t h e a n a l o g u e o f T h e o r e m (II • 3.1) ,

Theorem.

structure

P ,

t h e d u a l c a n b e e x t e n d e d to a M a r k o v f a m i l y

Let

{(TIt ); AB

substructure

•

{iDA)}

be the c a n c e l l a t i v e s y s t e m i n d u c e d by a l i n e a l s u b -

B { S}

For e a c h

(5.Z)

the l i n e a l c a n c e l l a t i v e s y s t e m i n d u c e d by the dual t > 0 ,

~A(B)

There is an i n v a r i a n t m e a s u r e

but omit the e a s y proof.

A,B c S,

at l e a s t one f i n i t e ,

AB = $ t CA)

•

v { ~

such that

~!

pt

~

v

as

t ~o

Moreover,

ergodic

{( A)}






X

(ii)

sup x,A: x~A

Cx(A) = K < ~

,

and (iii)

v P (weak explosion

then there is a unique system

Proof:

It suffices to check

vB T )= 0

at

{([A)}

(Z .4) .

VB ~ S O ,

with pregenerator

G

induced

by

P •

An appropriate decomposition yields

vB

E'[UA

(~vB)'

n

-< t ]

Tn

AvB -< P ( ' c n

(a n )

c [t-6,

t])

(b n)

A vB vB + P ( T n < t- 6, I ~ v B 1 -> M). sup ~cn B:IBI-> M

(c n)

+ P(T n

< t,

I[vB T

for arbitrary

6 e (O,t]

,

M

~

0 .

Now

n

I < M)

sup u Z ( B ) s -> 6

97

lira s u p n--=

an

VT.. B

Let

=f

vB

P(~n ~ [t-6,t]

for infinitely

many

n):'P('~Se

v B (~t)

be the time of the first jump by

[t-6,t])

v B vB vB m : T¢ + ( ' ~ B _ m..) i s t h e

Since

independent s u m of an exponential variable and the remaining time, of

VB T

that

is absolutely continuous on .
0
- 0 ,

arrows arrive at

x

from every site in [0, x+l ] at rate I00 x

It is e a s y to c h e c k the h y p o t h e s e s of T h e o r e m e x a m p l e exhibits an unusual p h e n o m e n o n w h i c h merits a brief discussion. G

of

G

(Z.7), so u n i q u e n e s s holds.

This

in the theory of M a r k o v i a n semigroups,

G i v e n pregenerator

G,

one defines the closure

by B

graph ( G ) = Thus if

G

has domain

suchthat

llh-fnll--

is u n i q u e n e s s for

~9(-G) and

0

G,

graph (G)

and

h c ~9(-G) ,

II~h-Gfnil--

t h e n "G

(in C x C) .

t h e n t h e r e are f u n c t i o n s

0

as

is t h e g e n e r a t o r of

the c a s e w h e n e v e r t h e H i l l e - Y o s i d a T h e o r e m a p p l i e s . however,

if G e : ~ ( G e) -- C

~ ~ ~ ( G e)

such that w h e n e v e r

1 llGe~ - G fll > i-~ ' extending

(Z.9)

G

is not

Notes.

i.e. G

is the generator of fc ~

and

3.

Nonuniqueness

ifthere

In p a r t i c u l a r t h i s is

For t h e p r e s e n t e x a m p l e , {([ A )} , one can find a function , then

Thus the unique generator

in this case.

T h e o r e m (Z.3)

and E x a m p l e (Z.8)

Asarule,

{(%A)} .

II~- fll < ~1

9 c ~ ( G e) - ~(G) .

and C o r o l l a r y (Z.5)

a d d i t i v e s e t t i n g of r e s u l t s from H o l l e y , (2.7)

n--~

fn ~ ~

are a d a p t a t i o n s to t h e

S t r o o c k and W i l l i a m s (1977).

Theorem

are t a k e n from Gray and G r i f f e a t h (1977).

examples.

In this final section w e briefly discuss n o n u n i q u e n e s s possibilities for particle systems. considered.

To k e e p matters simple, only the additive setting will be

O n e of the simplest n o n u n i q u e n e s s e x a m p l e s w a s encountered already

in Problem (1.4) . in

~

For those flip rates, the presence of w e a k influence from

gives rise to distinct s y s t e m s defined by (II.l.l) and

there is a c o n t i n u u m of s y s t e m s with the flip rates of (1.4) . ture P

with w e a k influence from

~

(1.3) . Indeed,

In fact, a n y substruc-

gives rise to an infinite family of Feller

additive systems.

(3. i) Theorem. substructure

Let

f~ •

If

G

be the additive pregenerator induced by a percolation

99

~.vB /~[

< ~ ,

vB I~B

_] < = ) > 0

then there is a continuum

of distinct

More precisely,

be the maximal set in

let

A

for some

Feller additive

B~ S O ,

systems

with pregenerator G • vB such that A C ~vB ~ - a.s.

Zd

37

Then to each probability system

{ ( ~ A t) }

.Izd_

Sketch of proof: t ~ (0,~)

isolated

~

(Z d U {co})× T A t~

SO ,

w-distributed to

Zd

G

and semigroup

independent

If

draw arrows from

x ~ %A

S O U {A} ,

measure

and extend

as follows.

allowing for "influence

on

-

there corresponds ( P tw) ,

a Feller

where

W lzd_A

Given a probability be

point

w

with pregenerator

4/if A t,

measure

At=

(A t , t )

through

~

co . "

on

S O U {A} ,

random variables.

to a percolation

A ,

to

~

label

(~,t)

(co,t)

.

let Adjoin an

substructure

with a

~ ,

~

on

while if

We may think of these

arrows as

Now say that

if

(i)

there is a path up to

(x,t)

from

(A,O) ,

(ii)

there is a path up to

(x,t)

from some

possibly

"through

co , "

or

y ~ Zd U { ~ } ,

the path again possibly

( y , s)

labelled

"through

~ ,

~ , "

or m

(iii)

there is strong influence

(iv)

there is a path d o w n f r o m

from

=

to

( x , t)

or

visits to

A path only enters at time

s .

We

o~

Zd × T

(x, t) in the reverse substructure w h o s e

h a v e an accumulation

from

(~ , s)

point.

if the reverse path " w a n d e r s off to ~

leave the precise formulation of the effects involving

as the details of the construction,

to the interested reader.

~

,

"

as well

O n e c a n c h e c k that

100

the system

hypotheses

{(%A

t)}

so defined is Feller with pregenerator

of the theorem,

different

Tr's

G,

and that under the

g i v e r i s e to d i f f e r e n t s y s t e m s .

[3

D~

(3.Z)

Problems.

(1.Z)?

Which measure

Which

~

Problem (1.4), are traps,

~

g i v e s r i s e to t h e s y s t e m

yields the system

{(~t)}

of

(1.3)?

{(It+)}

For t h e f l i p r a t e s of

construct a translation invariant system such that both

and one such that neither

of T h e o r e m ( 3 . 1 ) ,

]~

nor

Z

find additional nonuniqueness

is a trap.

defined by

}~

and

Z

Under the hypotheses

examples which are not covered

by the construction sketched above.

(3.3)

Problem.

c

for s o m e

Let

x

be a spin system on

Z

with flip rates x--< -i,

(i) : 0

rx > 0 ,

additive.

{( A)}

: r0[A(0 ) + (I-ZA(0))A(1)]

x=

: rx[A(x ) + (I-ZA(x)) (PxA(X+l) +qxA(X-l))]

x-> i,

0 < Px < 1 ,

with

Describe the dual processes

qx : i - Px • vA (%t) '

S h o w that

{(~A)}

O,

is

in particular the one-particle duals

Vx

(It) '

X c Z •

ness e xa mp le s

For general

as you can find.

Dynkin and Yushkevich

(3.4)

Notes.

r's and

p's ,

discuss as m a n y kinds of nonunique-

(You m a y want to m a k e use of Chapter IV of

(1969).)

The material of this section is b a s e d on Gray and Griffeath (1977),

although the graphical approach is new.

The simple nonuniqueness

(1.4) first appeared in Gray and Griffeath (1976). m a y be found in Holley and Stroock (1976a).

example of

Another nonuniqueness

example

Bibliography O m e r A d e l m a n (1976).

S o m e use of some "symmetries" of some random processes,

.

I

Ann. Inst. Henri Polncare IZ, 193-197. %

F. Bertein and A. Galves (1978). association,

Z. Wahrscheinlichkeitstheorie Verw. Geb. 41, 73-85.

D. Blackwell (1958). states,

U n e classe de systemes de particules stable par

Another countable M a r k o v process with only instantaneous

Annals of Mathematical Statistics Z9, 313-316.

M . Bramson and D. Griffeath (1978a). model,

Annals of Probability,

Renormalizing the 3-dimensional voter

to appear.

M . Bramson and D. Griffeath (1978b).

Clustering and dispersion rates for s o m e

interacting particle systems on Z , Annals of Probability, to appear. S. Broadbent and ]. H a m m e r s l e y (1957). mazes,

Percolation processes I. Crystals and

Proc. Cambridge Phil. Soc. 53, 6Z9-645.

I. Chover (1975).

Convergence of a local lattice process,

Stochastic Processes

and their Applications 3, 115-135. P. Clifford and A. Sudbury (1973).

A model for spatial conflict,

Biometrika 60,

581-588. D. D a w s o n

(1974a).

Information flow in discrete M a r k o v systems,

~ournal of

Applied Probability ii, 594-600. D. A. D a w s o n no. i0,

(1974b).

Discrete M a r k o v Systems,

Carleton Math. Lecture Notes

Carleton University, Ottawa.

D. D a w s o n

(1975).

Synchronous and asynchronous'reversible M a r k o v systems.

Canadian Mathematical Bulletin 17, 633-649. D. A. D a w s o n

(1978).

The critical measure diffusion process,

Z. Wahrscheinlichkeitstheorie Verw. Geb. 40, IZ5-145. D. D a w s o n and G. Ivanoff (1978).

Branching diffusions and random measures,

Advances in Probability 5, ed. A. yoffe, P. Ney,

Dekker, N e w York.

R. L. Dobrushin (1971).

Markovian processes with a large number of locally

interacting components,

Problems of Information Transmission 7, 149-164 and

Z35-Z41. R. Durrett (1978).

An infinite particle system with additive interactions, to appear.

E. B. Dynkin and A. A. Yushkevich (1969). Problems,

Plenum Press, N e w York.

P. Erdos and P. N e y (1974). particles,

M a r k o v Processes. Theorems and

Some problems on random intervals and annihilating

Annals of Probability Z, 8Z8-839.

102

I. Fleischmann

(1978).

T . A . M . S . Z39,

353-389.

Limit theorems for critical branching random fields,

H. O. Georgii (1976).

Stochastische Prozesse F{~r Interaktionssysteme.

Heidelberg. R. Glauber (1963).

The statistics of the stochastic Ising model,

Iournal of

Mathematical Physics 4, 294-307. L. Gray

(1978).

Controlled spin systems,

L. Gray and D. Griffeatb (1976). systems,

953-974.

O n the uniqueness of certain interacting particle

Z. Wahrscheinlichkeitstheorie verw. Geb. 35, 75-86.

L. Gray and D. Griffeath (1977). processes,

Annals of Probability 6,

O n the uniqueness and nonuniqueness of proximity

Annals of Probability 5, 678-69Z.

D. Griffeath (1975).

Ergodic theorems for graph interactions,

Advances in Applied

Probability 7, 179-194. D. Grlffeath (1977). .

An ergodlc theorem for a class of spin systems,

Ann. Inst.

I

Henri Pozncare 13, 141-157. D. Griffeath (1978a).

Limit theorems for nonergodic set-valued M a r k o v processes.

Annals of Probability 8, 379-387. D. Griffeath (1978b).

Annihilating and coalescing random walks on Z d .

Z. Wahrscheinlichkeitstheorie verw. Geb. 46, 55-65. D. Griffeath (1979). Probability 7,

Pointwise ergodicity of the basic contact process,

Annals of

13 9-143 o

R. Griffiths (1972).

The Peierls argument for the existence of phase transitions,

Mathematical Aspects of Statistical Mechanics,

~.C.T. Pool (ed.), S I A M - A M S

Proceedings, Providence, Amer. Math. Soc. 5, 13-Z6.

~. Hammersley

/

(1959).

processus de filtration.

process,

.

.

s"

Le Calcul des Probabilites et ses Applications.

National de la Recherche Scientifique, T. E. Harris (1960).

.

Bornes superzeures de la probablhte critique dans un Centre

Paris, 17-3 7.

Lower bound for the critical probability in a certain percolation

Proc. Cambridge Phil. Soc. 56, 13-Z0.

T. E. Harris (197Z).

Nearest neighbor M a r k o v interaction processes on multi-

dimensional lattices, Advances in Mathematics T. E. Harris (1974).

9, 66-89.

Contact interactions on a lattice,

Annals of Probability 2,

969-988. T. E. Harris (1976). Probability 4, 175-194.

O n a class of set-valued M a r k o v processes,

Annals of

103

T. E. Harris (1977).

A correlation inequality for Markov processes in partially

ordered state spaces,

Annals of Probability 5, 451-454.

T. E. Harris (1978). Additive set-valued Markov processes and percolation methods, Annals of Probability 6, 355-378. L. L. Helms

(1974). Ergodic properties of several interacting Poisson particles,

Advances in Mathematics IZ, 3Z-57. Y. Higuchi and T. Shiga (1975). Some results on Markov processes of infinite lattice spin systems, R. Holley (1970).

Journal of Mathematics of Kyoto Universit Y 15, ZII-ZZ9.

A class of interactions in an infinite particle system,

Advances

in Mathematics 5, Z91-309. R. Holley (1971).

Free energy in a Markovian model of a lattice spin system,

Communications in Mathematical Physics Z3, 87-99. R. Holley (197Za).

Markovian interaction processes with finite range interactions,

Annals of Mathematical Statistics 43, 1961-1967. R. Holley (197Zb). interactions,

An ergodic theorem for interacting systems with attractive

Z. Wahrscheinlichkeitstheorie verw. Geb. Z4, 325-334.

R. Holley (1974). Recent results on the stochastic Ising model, Rocky Mountain ~ournal of Mathematics 4, 479-496. R. Holley and T. M. Liggett (1975). Ergodic theorems for weakly interacting infinite systems and the voter model,

Annals of Probability 3, 643-663.

R. Holley and T. M. Liggett (1978). The survival of contact processes,

Annals of

P[obability 6, 198-206. R. Holley and D. Stroock (1976a). interacting processes,

A martingale approach to infinite systems of

Annals of Probability 4, 195-ZZ8.

R. Holley and D. Stroock (1976b). Applications of the stochastic Ising model to the Gibbs states,

Communications in Mathematical Physics 48, 249-266.

R. Holley and D. Stroock (1976c).

L Z theory for the stochastic Ising model,

Z. Wahrscheinlichkeitstheorie verw. Geb. 35, 87-101. R. Holley and D. Stroock (1976d). Dual processes and their application to infinite interacting systems,

Advances in Mathematics,

R. Holley and D. Stroock (1978).

to appear.

Nearest neighbor birth and death processes on the

real llne, Acta Math. 140, 103-154. R. Holley, D. Stroock and D. ~Villiams (1977). Applications of dual processes to diffusion theory, Proc. Sympos. Pure Math. 31, 23-36. Providence, R.I.

Amer. Math. Soc.,

104

F. P. Kelly

(1977).

The a s y m p t o t i c b e h a v i o r of an i n v a s i o n p r o c e s s ,

Journal of

.Applied Probability 14, 584-590. W. C. Lee

(1974).

Random stirring of the real l i n e ,

.Annals of Probability Z,

58O-592. T. M. Liggett (1972).

Existence theorems for infinite particle systems,

T.A.M.S.

165, 471-481. T. M. Liggett (1973).

A characterization of the invariant measures for an infinite

particle system with interactions, T. M. Liggett (1974).

T.A.M.S.

A characterization of the invariant measures for an infinite

particle system with interactions If, T. M. Liggett (1975). process,

T.A.M.S.

179, 433-453.

T.A.M.S.

198, Z01-ZI3.

Ergodic theorems for the asymmetric simple exclusion

ZI3, Z37-Z61.

T. M. Liggett (1976).

Coupling the simple exclusion process, Annals of Probability

4, 339-356. T. M. Liggett (1977). particles,

The stochastic evolution of infinite systems of interacting

Lecture Notes in Mathematics 598, 187-Z48.

T. M. Liggett (1978).

Attractive nearest neighbor spin systems on the integers,

Annals of Probability 6, K. Logan

(1974).

Springer-Verlag, N e w York.

6Z9-636.

Time reversible evolutions in statistical mechanics,

Cornell

University, Ph. D. dissertation. I. C. Lootgieter (1977).

Probl~mes de r~currence concernant des mouvements

aleatolres de particules sur Z avec destruction,

Ann. Inst. Henri Pomcare 13,

IZ 7-139. V. A. Malysev

(1975).

The central limit theorem for Gibbsian random fields,

Soviet

.Math. Dokl. 16, 1141-1145. N. Matloff (1977).

Ergodicity conditions for a dissonant voter model, Annals of

Probability 5, 371-386. D. Mollison

(1977).

Spatial contact models for ecological and epidemic spread,

]. Royal Statistical Soc. B, 39, 283-326. C. 7. Preston (1974). D. Richardson 74, 515-5Z8.

(1973).

Gibbs states on countable sets, R a n d o m growth in a tesselation,

S. Sawyer

(1976).

genetics,

Annals of Probability 4, 699-728.

S. Sawyer

(1978).

model,

to appear.

Cambridge University Press. Proc. Cambridge Phil. Soc.

Results for the stepping stone model for migration in population

A limit theorem for patch sizes in a selectively-neutral migration

~05

D. S c h w a r t z deaths,

(1976).

Ergodic t h e o r e m s for a n i n f i n i t e p a r t i c l e s y s t e m w i t h b i r t h s a n d

A n n a l s of P r o b a b i l i t y 4 ,

D. S c h w a r t z

(1977),

783-801.

A p p l i c a t i o n s of d u a l i t y to a c l a s s of M a r k o v p r o c e s s e s ,

Annals of Probability 5, 5ZZ-53Z. D. Schwartz

(1978).

O n hitting probabilities for an annihilating particle model,

Annals of Probability 6, 398-403. V. K. Shante and S. Kirkpatrick (1971). Introduction to percolation theory,

Advances

in Physics ZO, 325-357. F. Spitzer (1970).

Interaction of M a r k o v processes,

Advances in Mathematics 5,

Z46-zgo. F. Spitzer (1971). R a n d o m fields and interactingparticle systems, Seminar Notes, Williamstown, F. Spitzer (1974a).

M.A.A. Summer

Mass.

Recurrent random walk of an infinite particle system, T . A . M . S .

19__~8, 191-199. F. Spitzer (1974b).

Introduction aux processus de M a r k o v ~ parametre" s dans

Lecture Notes in Mathematics 390,

Z

v

Springer-Verlag, N e w York.

F. Spitzer (1976).

Principles of R a n d o m W a l k ,

F. Spitzer (1977).

Stochastic time evolution of one dimensional infinite particle

systems,

,

Znd ed. , Springer-Verlag, N e w York.

B . A . M . S . 83, 880-890.

O. N. Stavskaya

(1975).

Sufficient conditions for the uniqueness of a probability

field and estimates for correlations,

Matematicheskie Zametki 18, 609-6Z0.

O. N. Stavskaya and I. I. Pyatetskii-Shapiro (1968). spontaneously active elements, D . Stroock (1978).

O n h o m o g e n e o u s networks of

Problemy kibernetiki, N a u k a M o s c o w

ZO, 91-106.

Lectures on Infinite Interacting Systems, Kyoto University

Lectures in Math. no. Ii. W.

G. Sullivan (1974).

of random fields,

A unified existence and ergodic theorem for M a r k o v evolution

Z. Wahrscheinlichkeitstheorie verw. Geb. 31, 47-56.

W . G. Sullivan (1975).

M a r k o v processes for random fields, Communications Dublin

Inst., "Advanced Studies A, No. 23. W . G. Sullivan (1976).

Processes with infinitely m a n y jumping particles, P . A . M . S .

54, 3Z6-330. A. L. T o o m 9,

(1968).

A family of uniform nets of formal systems, Soviet Mathematics

1338-1341.

A. L. T o o m

(1974).

Nonergodic multidimensional systems of automata, Problemy

Peredachi Informatsii i0, Z39-Z46.

106

L. N. Vasershtein

(1969).

Markov processes over denumerable products of spaces,

describing large systems of automata,

Problemy Peredachi Informatsii 5 (3), 64-73.

L. N. Vasershtein and A. M . Leontovich

(1970).

Invariant measures of certain

Markov operators describing a homogeneous random medium,

Problemy Peredachi

Informatsil 6 (i), 71-80. N. B. Vasilev

(1969).

Limit behavior of one random medium,

Problemy Peredachl

Informatsii 5 (4), 68-74. N. B. Vasll'ev, L. G. Mityushin,

I. I. Pyatetskii-Shapiro and A. L. T o o m

(1973).

Stavskaya Operators (in Russian), Preprint no. IZ, Institute of Applied Math. , A c a d e m y of Sciences,

U.S.S.R. , M o s c o w .

T. Williams and R. Bjerknes

(1972).

Stochastic model for abnormal clone spread

through epithelial basal layer, Nature Z36, 19-ZI.

Subject Index Additive pregenerator 89 additive system 14 annihilating branching processes with parity annihilating random walks 5, 81 anti-voter model 67 Biased voter model box 1

55

Cancellative pregenerator 89 cancellative system 66 coalescing branching processes 24 coalescing random walks 3, 47, 58 configuration 1 contact systems 5, 44 critical phenomenon 30 cylinder function 2.2. Dense configuration 45 distribution Z domain of attraction 8 dual processes 16, 67 dual substructure ii duality equations 17, 68 Edge 5Z equilibrium 6 ergodic 7 exclusion system (additive) 64 explosion 90 exponentially decaying correlations exponentially ergodic system 19 extralineal substructure i0 extralineal system 14 extreme invariant measure 8 Feller system

7

Generalized voter models 77 Gibbs measures 74 graphical representation 3 Influence from oo 14 invariant measure 6, 7 [ump rates

Z

Lineal substructure I0 lineal system 14 local substructure i0 local system 14 Monotone system 14 minimal dual processes Neighbor

91

44

One-sided contact systems

29

20

73

108

Particle process 1 particle system Z path up 3, i0 percolation substructure pointwis e ergodicity 8 proximity system Z4 pure births 67

5,9

R a n d o m stirring 63 recurrent density 47 regular distribution 45 Self-dual substructure Z9 self-dual system Z9 site 1 spin system Z stochastic Ising model 74 strong correlations 53 strong explosion 96 strong influence from o0 14 strongly ergodic 7 substructure 5 T a m e function ZZ time reversible system 14 transient density 47 translation invariant substructure translation invariant system 14 Unique system Voter model Weak weak weak weak

89

ii, 46

convergence 6 correlations 53 explosion 96 influence from co 15

i0