Log-Gases and Random Matrices (LMS-34) (London Mathematical Society Monographs)

Log-Gases and Random Matrices

London Mathematical Society Monographs Editors: Martin Bridson, Terry Lyons, and Peter Sarnak Editorial Advisers: Mikhail Gromov, Jean-Francois Le Gall, and Richard Taylor The London Mathematical Society Monographs Series was established in 1968. Since that time it has published outstanding volumes that have been critically acclaimed by the mathematics community. The aim of this series is to publish authoritative accounts of current research in mathematics and high-quality expository works bringing the reader to the frontiers of research. Of particular interest are topics that have developed rapidly in the last ten years but that have reached a certain level of maturity. Clarity of exposition is important and each book should be accessible to those commencing work in its field. The original series was founded in 1968 by the Society and Academic Press; the second series was launched by the Society and Oxford University Press in 1983. In January 2003, the Society and Princeton University Press united to expand the number of books published annually and to make the series more international in scope.

LMS-34. Log-Gases and Random Matrices, by P. J. Forrester LMS-33. Prime-Detecting Sieves, by Glyn Harman LMS-32. The Geometry and Topology of Coxeter Groups, by Michael W. Davis LMS-31. Analysis of Heat Equations on Domains, by El Maati Ouhabaz

Log-Gases and Random Matrices

P.J. Forrester

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright © 2010 by Princeton University Press Requests for permission to reproduce material from this work should be sent to Permissions, Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved

Library of Congress Cataloging-in-Publication Data Forrester, Peter (Peter John) Log-gases and random matrices / P.J. Forrester. p. cm. -- (London Mathematical Society monographs) ISBN 978-0-691-12829-0 (hardcover : alk. paper) 1. Random matrices. 2. Jacobi polynomials. 3. Integral theorems. I. Title. QA188.F656 2010 519.2--dc22 2009053314

British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. f Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Preface

Often it is asked what makes a mathematical topic interesting. Some qualities which come to mind are usefulness, beauty, depth and fertility. Usefulness is usually measured by the utility of the topic outside mathematics. Beauty is an alluring quality of much of mathematics, with the caveat that it is often something only a trained eye can see. Depth comes via the linking together of multiple ideas and topics, often seemingly removed from the original context. And fertility means that with a reasonable effort there are new results, some useful, some with beauty, and a few maybe with depth, still awaiting to be found. More than fifteen years ago I embarked on a project to write in monograph form a development of the theory of solvable log-gas systems in statistical mechanics. As a researcher in the field, I had personally witnessed and experienced some of the interesting qualities of this topic, and I was keen that these be recorded in a form which could serve as a reference for researchers in related fields. Little did I realize that in the ensuing years these related fields would be the subject of intense research activity, requiring a revision of both the focus of the book, and my own research directions, to properly reflect these developments. Although my focus thus evolved away from the statistical mechanics of log-gas systems, this subject still proved to be a unifying theme in the presentation of the subject matter. And as a further give away as to my own research origins, there is a fairly strong flavor of the language of classical equilibrium statistical mechanics throughout, although a similar background of the reader can hardly be expected. More likely the motivation of the reader will come from the topics of random matrices, Painlev´e systems, stochastic growth processes, or Jack polynomials. These are some of the the related fields referred to above, which have been the subject of much recent activity, and which promise to remain interesting topics into the future. Of these it is random matrices which appears along side log-gases as the unifying theme of the book. This marriage of topics has a fine historical pedigree, with the log-gas picture of eigenvalues of random matrices being used to great advantage in the pioneering work of Dyson [147]. While providing a directed logical framework, a development of the intersection between log-gases and random matrices necessarily excludes substantial portions of each of the topics taken separately. However, the latter is necessary in order to achieve a mostly self-contained presentation. Seeking the common intersection of two topics can then be seen as a way of achieving this in a fairly democratic manner. In addition there is intersection with a third topic at work, keeping a further bound on the content, but also being responsible for much of the richness of the mathematics. This third topic is integrable systems. In general the exact calculation of correlations and probability distributions for interacting statistical mechanical systems is an intractable problem; however, underlying integrable structures make log-gases and random matrices an exception. The development of this topic leads to the study of determinantal and Pfaffian processes and the corresponding orthogonal polynomials, as well as Painlev´e systems and Jack polynomials. The quality of usefulness marked the beginning of the study of random matrices and log-gases in mathematical physics. As already mentioned, log-gases were introduced as a tool by Dyson to study random matrices, or as expressed in [201], to liberate the mathematics where none yet exists. Random matrices themselves were introduced by Wigner as a model for the statistical properties of the highly excited energy levels of heavy nuclei. Many of the early works on this theme (up to 1965) are conveniently collected together in the work of Porter [447], along with an introductory review. Long before their occurrence in physics, random matrices appeared in mathematics, especially in relation to the Haar measure on classical groups. Perhaps the first work of this type is due to Hurwitz, who computed the

vi

PREFACE

volume form of a general unitary matrix parametrized in terms of Euler angles [301]. The book of Weyl [540] contains the Haar volume form written in terms of eigenvalues and eigenvectors for the classical groups, and the book of Hua [300] inter-relates these forms to similar measures relating to spaces of Hermitian matrices. In mathematical statistics Wishart [547] gave the volume form of a rectangular matrix X in terms of the volume of the corresponding positive definite matrix XT X. Two other early mathematical works of lasting importance to the field are those of Dixon [137] and Selberg [483], both of which relate to multidimensional integrals with integrands which can be interpreted as probability measures associated with random matrices. The historical development of random matrices is well documented. Two recent informative accounts are [226], [75]. However, as already stated, the present work addresses only the intersection of the topics of log-gases, random matrices and integrable systems, and so a more extensive historical introduction beyond that already given does not serve as as an informative introduction to the content. Instead it is perhaps worth isolating some of our major topics, giving them some context and providing commentary on how they are to be developed. Jacobians All the works referenced above in relation to how random matrices appear in mathematics relate to Jacobians. To gain insight into the prevalance of Jacobians throughout random matrix theory, consider, for example, the problem of studying the eigenvalues of an N × N real symmetric random matrix, in the situation that the joint distribution on the space of the independent elements is given. The dimension of this space is N (N + 1)/2. The eigenvalue/eigenvector decomposition provides a change of variables from the independent elements of the matrix to its N eigenvalues and N (N − 1)/2 variables associated with its eigenvectors. A strategy then to study the eigenvalues is to perform this change of variables, and an essential ingredient for this task is the computation of the corresponding Jacobian. In the case of real symmetric matrices, complex Hermitian and quaternion real Hermitian matrices, these Jacobians are computed in Chapter 1. Chapter 1 also contains the computation of the Jacobian for a change of variables from the independent elements of an N ×N real symmetric tridiagonal matrix to its eigenvalues and a further N − 1 independent variables relating to its eigenvectors, and a Jacobian relating to the Householder transformation. In Chapter 2 Jacobians are computed in relation to spaces of unitary matrices, including orthogonal and symplectic unitary matrices, which have dimensions O(N 2 ). Jacobians are also computed for the change of variables from the elements to the eigenvalues and variables relating to the eigenvectors for certain unitary and real orthogonal Hessenberg matrices. In these latter circumstances the underlying spaces are of dimension O(N ). The singular value decomposition of rectangular matrices (or equivalently certain decompositions of positive definite matrices), the block decomposition of unitary matrices, and positive definite matrices formed from bidiagonal matrices are some of the settings which give rise to calculations of Jacobians undertaken in Chapter 3. Jacobians of a different sought appear in Chapter 4. Here rational functions with random coefficients in their partial fraction expansion are encountered, and we seek to change variables from a description in terms of these coefficients to one in terms of the zeros. For this purpose use is make of tools already known from the computation of Jacobians in Chapters 1–3, in particular the calculus of wedge products, and also the classical Vandermonde and Cauchy determinants. In Chapter 11 Jacobians are encountered in the change of variables of differential operators given in terms of the elements of parameter-dependent random matrices, to the differential operators given in terms of corresponding eigenvalues and variables relating to the eigenvectors. Finally, in Chapter 15, a task similar to that addressed in Chapter 1 is undertaken, namely the change of variables from the description of N × N real, complex, or quaternion real matrices in terms of the independent elements, to one in terms of the eigenvalues (which are typically complex) and an appropriate number of other variables. Also computed are some Jacobians relating to the change of variables of a random polynomial from its coefficients to its zeros. Determinantal point processes and orthogonal polynomials of one variable A determinantal point process is a statistical system of many particles (points) in which the k-point corre-

PREFACE

vii

lation function is a k × k determinant for each k. The study of eigenvalues of random matrices with complex entries, and also of log-gas systems at the special coupling β = 2 (in the cases considered in this work, the former are mostly special cases of the latter, due to our subject matter being typically restricted to the intersection of the two fields) gives rise to determinantal point processes. Furthermore, the corresponding determinants are determined by just one quantity, referred to as the correlation kernel. To exhibit this fact an essential role is played by orthogonal polynomials. It turns out that in the cases of interest it is the classical orthogonal polynomials which are required. Because full information on the asymptotic properties of these polynomials is known in the existing literature, it is possible to proceed and calculate scaling limits. A generalization of a determinant point process is a Pfaffian point process, in which the k-point correlation function is a 2k × 2k Pfaffian (or equivalently a k × k quaternion determinant) for each k. The eigenvalues of matrix ensembles studied in Chapters 1–3 in which the matrices are diagonalized by real orthogonal or symplectic unitary matrices are examples of Pfaffian point processes. These eigenvalues can be interpreted in terms of log-gas systems at the particular coupling β = 1 and β = 4 respectively. In the theory of Pfaffian processes skew orthogonal polynomials play a role analogous to that played by orthogonal polynomials in the theory of determinantal processes. For the particular skew inner products encountered from the random matrix problems of Chapters 1–3, the required skew orthogonal polynomials can be expressed in terms of classical orthogonal polynomials, and moreover the elements of the Pfaffian are determined by a single 2 × 2 block, the elements of which can be expressed in a summed form suitable for asymptotic analysis. In Chapter 15 non-Hermitian Gaussian random matrices are studied, with real, complex, and real quaternion entries. The eigenvalues in the complex case form a determinantal point process, while in the other two cases a Pfaffian point process results. The Selberg integral, Jack polynomials and generalized hypergeometric functions Familiar in the theory of the Gauss hypergeometric function is the Euler integral, which has the feature that it can be evaluated in terms of gamma functions. The Selberg integral can be considered as an N -dimensional generalization of the Euler integral. In a random matrix context, it appears as the normalization of various ensembles considered in Chapters 1–3. In a log-gas context, it gives the partition function for general β > 0. When written in a trigonometric form, extra parameters can be interpreted as providing the full distribution of certain linear statistics in the circular β-ensemble. In the case β = 2, and in the limit N → ∞, this ties in with the Fisher-Hartwig asymptotic formula from the theory of Toeplitz determinants, covered in Chapter 14. One of the structures underlying the Selberg integral is a further multidimensional integral referred to as the Dixon-Anderson integral. Like the Selberg integral, it can arrived at by the consideration of a problem in random matrix theory, and it too can be evaluated in terms of gamma functions. The many free parameters in the Dixon-Anderson integral allow for an interpretation giving an inter-relation between the distribution of every second eigenvalue in classical matrix ensembles at β = 1, and the joint distribution of the eigenvalues for a related classical matrix ensemble at β = 4. The integrand of the Selberg integral and its various limits is, up to normalization, the eigenvalue probability density function of the various classical β-ensembles given in Chapters 1–3. Theory linking the Selberg integral with the Dixon-Anderson integral can also be used to provide stochastic three-term recurrences (in the degree N ) for the corresponding characteristic polynomials. The integrand of the Euler integral is the weight function for the classical Jacobi polynomials (when defined on the interval [0, 1]). Likewise, the integrand of the Selberg integral, and its various limiting forms, can be used to define inner products which permit complete sets of orthogonal polynomials with special properties. The most fundamental are the Jack polynomials, which relate to the integrand of the Selberg integral in trigonometric form, specialized to correspond to the eigenvalue probability density function for the circular β-ensemble. Using the Jack polynomials as a basis, generalized classical Hermite, Laguerre and Jacobi polynomials, which are multivariable counterparts of the one-variable classical orthogonal polynomials of the same name, can be studied. Another viewpoint of the Jack polynomials is as the polynomial (in complex exponential variables) portion of the eigenfunctions for the Fokker-Planck operator of the Dyson Brownian motion model of the log-gas on a circle. This topic is developed in Chapter 11. A crucial feature is an alge-

viii

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braic theory of the Fokker-Planck operator, in which it is decomposed into fundamental commuting operators relating to the degenerate Hecke algebra of type A, and involving exchange operators. The presentation of the theory of Jack polynomials given in Chapter 12 begins from a study of the simultaneous nonsymmetric polynomial eigenfunctions of these commuting operators. The underlying degenerate Hecke algebra allows these polynomials to be constructed inductively using two fundamental operations (transposition and raising), and these operations allow for the explicit evaluation of associated scalar quantities such as various normalizations. The operations of symmetrization and antisymmetrization also play an important role. It is well known that the Euler integral can be extended to provide the solutions of the Gauss hypergeometric differential equation. Likewise, weighting the Selberg integrand by an appropriate factor gives rise to multidimensional integrals which relate to multidimensional hypergeometric functions based on Jack polynomials. These are studied in Chapter 13. With the parameters specialized, these integrals can be interpreted as correlations for log-gas systems. Duality formulas, in which multidimensional integrals of this type are expressed as other multidimensional integrals, this time of dimension independent of N , provide the basis for the asymptotic analysis of the corresponding correlations for all even β at least. Furthermore, Jack polynomial theory can be used to compute the bulk dynamical two-point density-density correlation for the Dyson Brownian motion model perturbed from its equilibrium state for all values of rational values of β. Painlev´e transcendents The Painlev´e differential equations are a distinguished family of second order nonlinear equations. In applied mathematics they are perhaps best known for their role in soliton theory, and thus the study of integrable partial differential equations. Certain solutions of the Painlev´e differential equations — the Painlev´e transcendents — appear in the calculation of gap probabilities for classical random matrix systems corresponding to log-gas systems with β = 1, 2 and 4 (although the latter two are restricted to those instances in which their is an inter-relation with a β = 2 log-gas system; one way the latter comes about is by superimposing two β = 1 ensembles, and integrating over every second eigenvalue, while in the bulk the β = 1 gap probability is transformed by making use of an evenness symmetry). The viewpoint taken in Chapter 8 on these calculations is an algebraic theory of Painlev´e systems based on a Hamiltonian formulation, due mainly to Okamoto, which has the feature of using the Toda lattice equation to inductively construct determinant solutions from a seed solution (the latter relating to an underlying linear second order equation). These determinants can be identified with the gap probabilities of certain log-gas systems at β = 2. Moreover (formal) scaling of the differential operators gives analogous characterization of the gap probabilities in various scaling limits. As a consequence of these characterizations, high precision calculation of the gap probabilities can be undertaken. In Chapter 9 additional viewpoints on these results are considered. One is the study of function theoretic properties of the gap probabilities expressed as Fredholm determinants. Indeed, starting with the Fredholm determinant form seems necessary to account for the scaled limit rigorously. Instead of using function theoretic properties, this starting point can also be developed from a Riemann-Hilbert viewpoint, which in turn is closely related to studying isomonodromic deformations of linear second order differential equations. The Fredholm determinant evaluations allow the high precision calculations of the gap probabilities initiated from the Painlev´e evaluations to be extended. Macroscopic electrostatics and asymptotic formulas Averaging a linear statistic against the eigenvalue spectrum of a random matrix gives a mean value proportional to N (the number of eigenvalues), but a variance of order unity. In applications, this effect shows itself in the study of the statistical properties of the conductance of a quantum wire, noted in Chapter 3. It can be anticipated, and a precise formula for the variance formulated, by hypothesizing that for large length scales the log-gas behaves like a macroscopic conductor, and then using linear response arguments based on the predictions of two-dimensional electrostatics. Moreover, this hypothesis leads to the prediction that the full distribution of the linear statistic will be a Gaussian. For some log-gas systems this has been rigorously established, one of these being that corresponding to the Szeg¨o asymptotic formula from the theory of Toeplitz determinants.

PREFACE

ix

One of the most basic predictions from macroscopic electrostatics is the leading form of the density profile for random matrix ensembles. It can be used too to predict the O(1/N ) correction to this form. Another application is to gap probabilities, for which the large gap size asymptotics can be predicted to the leading two orders, and at the soft edge the large deviation forms of the left and right tails can be computed. When available, the exact results agree with these predictions. Non-intersecting paths and models in statistical mechanics The generating function for non-intersecting paths on an acyclic directed graph is well known to be given in the form of a determinant. In a number of cases of interest this determinant can be evaluated, revealing that the joint probability density function for the paths possessing prescribed coordinates is of a log-gas form, with β = 1 or 2. Non-intersecting paths underly a number of statistical mechanical models, in particular the polynuclear growth model and the Hammersely model of directed percolation. To understand how this comes about requires a study of the Robinson-Schensted-Knuth correspondence from bijective combinatorics. This in turn leads naturally to the study of Schur polynomials, which are in fact examples of Jack polynomials. It is shown that fluctuations of the primary observable quantities in the polynuclear growth model and the Hammersely model of directed percolation (the height of the profile and length of the path, respectively) can be expressed as random matrix averages over the unitary group, and that these matrix averages can be rigorously analyzed in the appropriate scaling limits. Various symmetrizations of the Hammersely model of directed percolation are particularly natural. Examples of these relate to averages over random matrices from the orthogonal and symplectic groups. Transformations of these averages to relate to gap probabilities in Laguerre random matrix ensembles with β = 1 and 4 allows the the rigorous analysis of the scaling limits. Applications of random matrix theory All the random matrix ensembles introduced in Chapters 1–3, for β = 1, 2 and 4 at least, can be associated with problems in quantum physics. The work of Wigner and Dyson relates the Gaussian ensembles to quantum Hamiltonians; the circular ensembles relate to scattering from a disordered cavity; Verbaarschot has given an interpretation of chiral random matrices in terms of the Dirac equation as it relates to QCD; and quantum transport problems lead to the Jacobi ensemble. For general values of β > 0 the eigenvalue p.d.f.’s of the β-ensembles appear as the ground state wave function of a class of quantum many-body problems with the 1/r2 pair potential. The eigenvalue p.d.f. for the complex random matrices of Chapter 15 has the interpretation as the absolute value squared of the ground state wave function for spinless fermions confined to a plane in the presence of a perpendicular magnetic field. An application of the GOE to the statistics of high-dimensional random energy landscapes is given in Chapter 1. In Chapter 3 features of Wishart matrices and the Jacobi ensemble relating to multivariate statistics are discussed, as is the application of Wishart matrices to wireless communication, numerical analysis and quantum entanglement (the latter requires a further constraint on the trace). In Chapters 5 and 14 an account is given too of the application of both the GUE and CUE to the study of statistical properties of the zeros of the Riemann zeta function. The applications to statistical mechanics, as summarized under the previous heading, is given in Chapter 10. It is clear from the above descriptions that the chapters have not been organized according to these headings. Instead the ordering has been determined by the desire to first define and motivate the various classical random matrix ensembles and their generalizations (for example, β extensions, minor processes), to give the mathematics leading to the determination of the corresponding eigenvalue p.d.f.’s, and to relate the latter to log-gases. This accounts for Chapters 1–4. Chapters 5–7 are about the calculation of correlations for the p.d.f’s encountered in Chapters 1–4 when the former can be expressed in terms of determinants (Chapter 5) or Pfaffians (Chapter 6). Chapters 8 and 9 give the theory leading to the computation of gap probabilities and spacing distributions in some of the systems for which the correlations were computed in the previous three chapters. With knowledge of the evaluation of gap probabilities and related random matrix averages in terms

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of Painlev´e transcendents thus established, we proceed in Chapter 10 to show how this can be put to use in the analysis of certain models in statistical mechanics relating to non-intersecting paths. The generalization of the Gaussian ensembles to be parameter dependent, or equivalently to have Brownian-motion valued entries, is introduced in Chapter 11, leading to the Calogero-Sutherland quantum many-body system and families of commuting operators. The polynomial eigenfunctions of these commuting operators are studied in Chapters 12 and 13, culminating in the computation of correlation functions for general β. Theory from Chapter 4 on the Dixon-Anderson integral again appears in Chapter 13, for its relevance to the computation of correlations for general β (or more precisely, for the inter-relations it provides), while theory from Chapters 1–3 relating to the β-ensembles is developed to give characterizations of the general β bulk and edge states in terms of stochastic differential equations. Continuing the general β theme, the study of fluctuation formulas is taken up in Chapter 14. The topic of the log-gas in a two-dimensional domain (which is in fact where my own studies began), and the corresponding random matrix ensembles in which the eigenvalues are complex, is the theme of final chapter of the book. After this introduction to the content and organization, a few words about the presentation are appropriate. As already remarked it has been my desire to give enough detail so that the development is self-contained. A large portion of the necessary working is carried out in the body of the text, but use too has been made of an exercises format which is both more space efficient and less laborious. I have aimed to structure the exercises with sufficient intermediate results so that they can reasonably be worked through, without the need to consult the original references. Generally it has been my intention to keep the subject matter moving. Consequently, there are a small number of results requiring a technical working beyond the main stream of the book, which necessarily have been omitted. I have been most fortunate to have had research fellowships from the Australian Research Council for the duration of this project. This has freed up time and energy for me to follow, and to be part of, many of the developments which have taken place since I began writing. Both being an active researcher in the field and following the developments have been necessary for writing this monograph. While rewarding, studying the research literature is often difficult and inefficient. My own learning was most efficient when studying instead monographs, in particular those of Gupta and Nagar [279], Haake [284], Hua [300], Mehta [398], Macdonald [376] and Muirhead [410]. Similarly it is my hope that this work will prove itself to be an efficient learning resource in preparation for future researches. There are a number of individuals who have over the years lent their assistance to this project, both directly and indirectly. My wife Gail places value on the worth of such academic pursuits, and provided a home environment to make it possible. For getting me started in research, and teaching me some fundamentals, I thank R.J. Baxter, B. Jancovici and (the late) E.R. Smith. Collaborations with K. Aomoto, T.H. Baker, P. Desrosiers, N.E. Frankel, T. Nagao, E.M. Rains and N.S. Witte have been of great value. E. Due˜nez provided some critical comments on my earlier writing on the circular ensembles which were of much help, and P. Sarnak saw enough potential in these earlier writings to recommend the work to Princeton University Press. Most recently F. Bornemann has provided me with high precision numerical data calculated from Fredholm determinants for use in Chapters 8 and 9, and A. Mays provided some help in relation to the proofreading.

Peter Forrester Melbourne, Australia January 2010

Contents

Preface

v

Chapter 1. Gaussian matrix ensembles 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Random real symmetric matrices The eigenvalue p.d.f. for the GOE Random complex Hermitian and quaternion real Hermitian matrices Coulomb gas analogy High-dimensional random energy landscapes Matrix integrals and combinatorics Convergence The shifted mean Gaussian ensembles Gaussian β -ensemble

Chapter 2. Circular ensembles 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Scattering matrices and Floquet operators Definitions and basic properties The elements of a random unitary matrix Poisson kernel Cauchy ensemble Orthogonal and symplectic unitary random matrices Log-gas systems with periodic boundary conditions Circular β -ensemble Real orthogonal β -ensemble

Chapter 3. Laguerre and Jacobi ensembles 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12

Chiral random matrices Wishart matrices Further examples of the Laguerre ensemble in quantum mechanics The eigenvalue density Correlated Wishart matrices Jacobi ensemble and Wishart matrices Jacobi ensemble and symmetric spaces Jacobi ensemble and quantum conductance A circular Jacobi ensemble Laguerre β -ensemble Jacobi β -ensemble Circular Jacobi β -ensemble

Chapter 4. The Selberg integral 4.1 4.2 4.3

Selberg’s derivation Anderson’s derivation Consequences for the β -ensembles

1 1 5 11 20 30 33 41 42 43

53 53 56 61 66 68 71 73 76 81

85 85 90 98 106 110 111 115 118 125 127 129 130

133 133 137 145

xii

4.4 4.5 4.6 4.7 4.8

CONTENTS

Generalization of the Dixon-Anderson integral Dotsenko and Fateev’s derivation Aomoto’s derivation Normalization of the eigenvalue p.d.f.’s Free energy

Chapter 5. Correlation functions at β = 2 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Successive integrations Functional differentiation and integral equation approaches Ratios of characteristic polynomials The classical weights Circular ensembles and the classical groups Log-gas systems with periodic boundary conditions Partition function in the case of a general potential Biorthogonal structures Determinantal k-component systems

Chapter 6. Correlation functions at β = 1 and 4 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Correlation functions at β = 4 Construction of the skew orthogonal polynomials at β = 4 Correlation functions at β = 1 Construction of the skew orthogonal polynomials and summation formulas Alternate correlations at β = 1 Superimposed β = 1 systems A two-component log-gas with charge ratio 1:2

Chapter 7. Scaled limits at β = 1, 2 and 4 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

Scaled limits at β = 2 — Gaussian ensembles Scaled limits at β = 2 — Laguerre and Jacobi ensembles Log-gas systems with periodic boundary conditions Asymptotic behavior of the one- and two-point functions at β = 2 Bulk scaling and the zeros of the Riemann zeta function Scaled limits at β = 4 — Gaussian ensemble Scaled limits at β = 4 — Laguerre and Jacobi ensembles Scaled limits at β = 1 — Gaussian ensemble Scaled limits at β = 1 — Laguerre and Jacobi ensembles Two-component log-gas with charge ratio 1:2

Chapter 8. Eigenvalue probabilities — Painleve´ systems approach 8.1 8.2 8.3 8.4 8.5 8.6

Definitions Hamiltonian formulation of the Painlev´e theory σ -form Painlev´e equation characterizations The cases β = 1 and 4 — circular ensembles and bulk Discrete Painlev´e equations Orthogonal polynomial approach

Chapter 9. Eigenvalue probabilities — Fredholm determinant approach 9.1 9.2 9.3 9.4 9.5

Fredholm determinants Numerical computations using Fredholm determinants The sine kernel The Airy kernel Bessel kernels

156 160 165 172 180

186 186 193 197 200 207 212 217 223 229

236 236 246 251 263 269 274 278

283 283 290 297 298 301 308 312 316 319 323

328 328 333 349 363 372 375

380 380 385 386 393 399

CONTENTS

9.6 9.7 9.8 9.9 9.10

Eigenvalue expansions for gap probabilities The probabilities Eβsoft (n; (s, ∞)) for β = 1, 4 The probabilities Eβhard (n; (0, s); a) for β = 1, 4 Riemann-Hilbert viewpoint Nonlinear equations from the Virasoro constraints

Chapter 10. Lattice paths and growth models 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Counting formulas for directed nonintersecting paths Dimers and tilings Discrete polynuclear growth model Further interpretations and variants of the RSK correspondence Symmetrized growth models The Hammersley process Symmetrized permutation matrices Gap probabilities and scaled limits Hammersley process with sources on the boundary

Chapter 11. The Calogero–Sutherland model 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Shifted mean parameter-dependent Gaussian random matrices Other parameter-dependent ensembles The Calogero-Sutherland quantum systems The Schr¨odinger operators with exchange terms The operators H (H,Ex) , H (L,Ex) and H (J,Ex) Dynamical correlations for β = 2 Scaled limits

Chapter 12. Jack polynomials 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Nonsymmetric Jack polynomials Recurrence relations Application of the recurrences A generalized binomial theorem and an integration formula Interpolation nonsymmetric Jack polynomials The symmetric Jack polynomials Interpolation symmetric Jack polynomials Pieri formulas

Chapter 13. Correlations for general β 13.1 13.2 13.3 13.4 13.5 13.6 13.7

Hypergeometric functions and Selberg correlation integrals Correlations at even β Generalized classical polynomials Green functions and zonal polynomials Inter-relations for spacing distributions Stochastic differential equations Dynamical correlations in the circular β ensemble

Chapter 14. Fluctuation formulas and universal behavior of correlations 14.1 14.2 14.3 14.4 14.5 14.6

Perfect screening Macroscopic balance and density Variance of a linear statistic Gaussian fluctuations of a linear statistic Charge and potential fluctuations Asymptotic properties of Eβ (n; J) and Pβ (n; J)

xiii

403 416 421 426 435

440 440 456 463 471 480 487 492 495 500

505 505 512 516 521 524 530 540

543 543 550 553 555 558 564 579 583

592 592 601 613 627 633 634 640

658 658 663 665 672 680 688

xiv

14.7

CONTENTS

Dynamical correlations

Chapter 15. The two-dimensional one-component plasma 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11

Complex random matrices and polynomials Quantum particles in a magnetic field Correlation functions General properties of the correlations and fluctuation formulas Spacing distributions The sphere The pseudosphere Metallic boundary conditions Antimetallic boundary conditions Eigenvalues of real random matrices Classification of non-Hermitian random matrices

698

701 701 706 711 718 725 729 738 744 747 752 760

Bibliography

765

Index

785

Chapter One Gaussian matrix ensembles The Gaussian ensembles are introduced as Hermitian matrices with independent elements distributed as Gaussians, and joint distribution of all independent elements invariant under conjugation by appropriate unitary matrices. The Hermitian matrices are divided into classes according to the elements being real, complex or real quaternion, and their invariance under conjugation by orthogonal, unitary, and unitary symplectic matrices, respectively. These invariances are intimately related to time reversal symmetry in quantum physics, and this in turn leads to the eigenvalues of the Gaussian ensembles being good models of the highly excited spectra of certain quantum systems. Calculation of the eigenvalue p.d.f.’s is essentially an exercise in change of variables, and to calculate the corresponding Jacobians both wedge products and metric forms are used. The p.d.f.’s coincide with the Boltzmann factor for a log-gas system at three special values of the inverse temperature β = 1, 2 and 4. Thus the eigenvalues behave as charged particles, all of like sign, which are in equilibrium. The Coulomb gas analogy, through the study of various integral equations, allows for the prediction of the leading asymptotic form of the eigenvalue density. After scaling, this leading asymptotic form is referred to as the Wigner semicircle law. The Wigner semicircle law is applied to the study of the statistics of critical points for a model of high-dimensional energy landscapes, and to relating matrix integrals to some combinatorial problems on the enumeration of maps. Conversely, the latter considerations also lead to the proof of the Wigner semicircle law in the case of the GUE. The shifted mean Gaussian ensembles are introduced, and it is shown how the Wigner semicircle law can be used to predict the condition for the separation of the largest eigenvalue. In the last section a family of random tridiagonal matrices, referred to as the Gaussian β-ensemble, are presented. These interpolate continuously between the eigenvalue p.d.f.’s of the Gaussian ensembles studied previously.

1.1 RANDOM REAL SYMMETRIC MATRICES Quantum mechanics singles out three classes of random Hermitian matrices. We will begin our study by specifying one of these—Hermitian matrices with all entries real, or equivalently real symmetric matrices. The independent elements are taken to be distributed as independent Gaussians, but with the variance different for the diagonal and off-diagonal entries. D EFINITION 1.1.1 A random real symmetric N × N matrix X is said to belong to the Gaussian orthogonal ensemble (GOE) if the diagonal and upper triangular elements are independently chosen with p.d.f.’s 2 1 √ e−xjj /2 2π

and

2 1 √ e−xjk , π

respectively. The p.d.f.’s of Definition 1.1.1 are examples of the normal (or Gaussian) distribution 2 2 1 √ e−(x−μ) /2σ , 2 2πσ

denoted N[μ, σ]. With this notation, note that an equivalent construction of GOE matrices is to let Y be an N × N random matrix of independent standard Gaussians N[0, 1] and to form X = 12 (Y + YT ).

2

CHAPTER 1

The joint p.d.f. of all the independent elements is P (X) :=

N  j=1

N  2 1 −x2jj /2  1 −x2jk √ e √ e = AN e−xjk /2 π 2π 1≤j J of

J2 . 4c

c D X |yj |2 E , N j=1 λ − λj N

1=

(1.138)

where each |yj |2 has mean unity, and {λj } are the eigenvalues of a member of the specified Gaussian ensemble but with H0 = 0. We know that the density of the {λj } is then given by the semicircle law (1.129). Hence for N large Z J N “ DX 2N λ “ J 2 ”1/2 ” |yj |2 E ρb (y) dy = ∼ 1− 1− 2 , 2 λ − λj J λ −J λ − y j=1 where the first relation follows from the fact that the eigenvalues and eigenvectors are independently distributed and |yj |2  = 1, while the equality, which requires that λ > J, follows from (1.132). Substituting this in (1.138) and solving for λ gives the stated result.

1.9 GAUSSIAN β-ENSEMBLE The p.d.f. (1.28) is realized by the eigenvalues of the GOE, GUE and GSE for the values of β equal to 1, 2 and 4 respectively. In this section a family of random tridiagonal matrices, referred to as the Gaussian βensemble, with (1.28) as their eigenvalue p.d.f. for general β > 0, will be studied. They can be motivated by the reduction of GOE or GUE matrices to tridiagonal form. 1.9.1 Householder transformations A familiar technique in numerical linear algebra is the similarity transformation of a real symmetric matrix to tridiagonal form using a sequence of reflection matrices, referred to as Householder transformations. Explicitly, let A be a real symmetric matrix [aij ]i,j=1,...,N . Then one can construct a sequence of symmetric real orthogonal matrices U(1) , U(2) , . . . , U(N −2) such that the transformed matrix U(N −2) U(N −3) · · · U(1) AU(1) U(2) · · · U(N −2) =: B(N −2) is a symmetric tridiagonal matrix. These matrices have the structure   0j×N −j 1j (j) (j) (j)T , = U = 1N − 2u u 0N −j×j VN −j×N −j

(1.139)

(1.140)

44

CHAPTER 1

where u(j)T u(j) = 1 and VN −j×N −j is symmetric real orthogonal. Geometrically U(j) corresponds to a reflection in the hyperplane orthogonal to u(j) . (1) Consider first the construction of U(1) . Choosing the components ul of u(1) as 1 a12 1/2 a1l (1) (1) (1) 1− u1 = 0, u2 = , ul = − (l ≥ 3), (1.141) (1) 2 α 2αu2 (1)

where α = (a212 + · · · + a21N )1/2 , we then have u(1)T [al1 ]l=1,...,N = (a12 − α)/2u2 . This in turn implies that B(1) := U(1) AU(1)

(1.142)

has b11 = a11 ,

b1k = bk1 = 0 (k ≥ 3)

b12 = b21 = α,

and is thus tridiagonal with respect to the first row and column. The matrices U(j) , j = 2, 3, . . . in order are (j) (j) (j) now defined by the formulas (1.141), but with u1 = u2 = · · · = uj = 0, and the analogue of the entries a1l replaced by the elements in the first row of the bottom right (N − j + 1) × (N − j + 1) submatrix of B(j−1) . A number of works (see [157] and references therein) posed the question as to the form of B(N −2) when A is a member of the GOE. It was found that like A itself, the elements of B(N −2) are all independent (apart from the requirement that B(N −2) be symmetric) with a distribution that can be calculated explicitly. P ROPOSITION 1.9.1 Let N[0, 1] refer to the standard normal distribution as defined below Definition 1.1.1, and let χ ˜k denote the square root of the gamma distribution Γ[k/2, 1], the latter being specified by the p.d.f. (1/Γ(k/2))uk/2−1 e−u , u > 0, and realized by the sum of the squares of k independent Gaussian √ 2 distributions N[0, 1/ 2]. (The p.d.f. of χ ˜k is thus equal to (2/Γ(k/2))uk−1 e−u , u > 0.) For A a member of the GOE, the tridiagonal matrix B(N −2) obtained by successive Householder transformations is given by ⎤ ⎡ N[0, 1] χ ˜N −1 ⎥ ⎢ χ ˜N −2 ⎥ ⎢ ˜N −1 N[0, 1] χ ⎥ ⎢ χ ˜ N[0, 1] χ ˜ N −2 N −3 ⎥ ⎢ ⎥. ⎢ . . . .. .. .. ⎥ ⎢ ⎥ ⎢ ⎣ χ ˜2 N[0, 1] χ ˜1 ⎦ χ ˜1 N[0, 1] Proof. Let GOEn denote the ensemble of n × n GOE matrices. From the Householder algorithm, the first row and column of B(N−2) are he same as those of B(1) in (1.142), and thus from (1.141) we have (N−2)

b11

= N[0, 1],

(N−2)

b12

=χ ˜N−1 ,

where use has been made of the assumption that A is a member of GOEN , and the definition of χ ˜2N−1 as a sum of squares of Gaussians. To proceed further we must compute the distribution of the bottom N −1×N −1 block of B(1) . In general, (1) denoting such a block of the matrix X by XN−1 , it follows from (1.140) that BN−1 = VN−1 AN−1 VN−1 . Since the elements of the real orthogonal matrix VN−1 are independent of the elements of AN−1 , which itself is a member of (1) GOEN−1 , it follows immediately from the general invariance of the GOE under orthogonal transformations that BN−1 (1) is also a member of GOEN−1 . Applying the Householder transformation to BN−1 , we thus get (N−2)

b22

Continuing inductively gives the stated result.

= N [0, 1],

(N−2)

b23

=χ ˜N−2 .

45

GAUSSIAN MATRIX ENSEMBLES

1.9.2 Tridiagonal matrices The result of Proposition 1.9.1 suggests investigating the Jacobian for the change of variables from a general real symmetric tridiagonal matrix ⎡ ⎤ an bn−1 ⎢ bn−1 an−1 bn−2 ⎥ ⎢ ⎥ ⎢ ⎥ bn−2 an−2 bn−3 ⎢ ⎥ (1.143) T=⎢ ⎥, .. .. .. ⎢ ⎥ . . . ⎢ ⎥ ⎣ b2 a2 b1 ⎦ b1 a1 to its eigenvalues and variables relating to its eigenvectors. First, for each eigenvalue λk and correspond(1) ing eigenvector vk , it is easy to see by direct substitution that once the first component vk =: qk of vk is specified, all other components can be expressed in terms of λk and the elements of T. To make the eigendecomposition unique we specify that qk > 0, and furthermore note that T, being symmetric, can be orthogonally diagonalized, and so doing this we have n

qk2 = 1.

(1.144)

k=1

The Jacobian for the change of variables from b := (bn−1 , . . . , b1 ),

a := (an , an−1 , . . . , a1 ),

(1.145)

to λ := (λ1 , . . . , λn ),

q := (q1 , . . . , qn−1 )

(1.146)

can be calculated using the method of wedge products. However, one must first establish some auxiliary results. P ROPOSITION 1.9.2 Let (X)11 denote the top-left hand entry of the matrix X. We have n

qj2 . λj − λ

(1.147)

n−1 2i bi (λi − λj ) = i=1 n 2 . i=1 qi

(1.148)

((T − λ1)−1 )11 =

j=1

Also 

2

1≤i<j≤n

Proof. Now

((T − λ1)−1 )11 = e1 · (T − λ1)−1e1 ,

where e1 := (1, 0, . . . , 0) . Since {vj } is an orthonormal set, T

e1 =

n X

(e1 · vj )vj =

j=1

n X

qj vj ,

(1.149)

j=1

and substituting into the above equation gives (1.147). This derivation makes no explicit use of T being tridiagonal, rather just that (1.149) holds, for which it is sufficient T be real symmetric. To derive (1.148) [140] we begin by recalling that in general for X an n × n nonsingular matrix, (X−1 )11 =

det Xn−1 , det X

(1.150)

46

CHAPTER 1

where Xn−1 denotes the bottom right n − 1 × n − 1 submatrix of X. Hence we can rewrite (1.147) to read Qn−1 n (n−1) X qj2 ) i=1 (λ − λi Qn , = λ − λj i=1 (λ − λi ) j=1 (n−1)

where {λi

(1.151)

} denotes the eigenvalues of Xn−1 . It follows from this that qj2 =

Pn−1 (λj ) , Pn (λj )

Pk (λ) :=

k Y (k) (λ − λi ),

(1.152)

i=1 (k)

where Pk (λ) is the characteristic polynomial of the bottom right k × k submatrix of T, say Tk , and {λi } the corresponding eigenvalues. Hence Qn n Y |Pn−1 (λi )| qi2 = Q i=1 . (1.153) 2 1≤i<j≤n (λi − λj ) i=1 Next, by expanding along the first row of λ1k − Tk , one obtains the three-term recurrence Pk (λ) = (λ − ak )Pk−1 (λ) − b2k−1 Pk−2 (λ) and it follows from this that

k−1 Y

(k−1)

|Pk (λi

2(k−1)

)| = bk−1

i=1

Since

k−1 Y

k−1 Y k−2 Y

)| =

i=1

this can be rewritten as

(k−1)

|Pk−2 (λi

(k−1)

|λi

(k−2)

− λj

|=

i=1 j=1 k−1 Y

)|.

i=1

(k−1)

|Pk−2 (λi

k−1 Y

(1.154)

(k−1)

|Pk (λi

k−2 Y

(k−2)

|Pk−1 (λj

)|,

(1.155)

j=1

2(k−1)

)| = bk−1

i=1

k−2 Y

(k−2)

|Pk−1 (λj

)|,

j=1

and iteration shows

n−1 Y

(n−1)

|Pn (λi

)| =

i=1

n−1 Y

b2i i .

i=1

Use of (1.155) with k = n + 1 and substitution into (1.153) gives (1.148).

P ROPOSITION 1.9.3 The Jacobian for the change of variables (1.145) to (1.146) can be written as n−1 1 i=1 bi  . (1.156) qn ni=1 qi

Proof. [223] Rewriting (1.147) in the form ((1 − λT)−1 )11 =

n X j=1

qj2 1 − λλj

(1.157)

47

GAUSSIAN MATRIX ENSEMBLES

and equating successive powers of λ on both sides gives 1=

n X

qj2 ,

n X

an =

j=1

qj2 λj ,

∗ + b2n−1 =

j=1

∗ + an−1 b2n−1 =

n X

n X j=1

qj2 λ3j ,

∗ + b2n−2 b2n−1 =

j=1

∗ + an−2 b2n−2 b2n−1 =

qj2 λ2j ,

n X

qj2 λ4j ,

j=1 n X

qj2 λ5j , . . . ,

∗ + a1 b21 · · · b2n−2 b2n−1 =

j=1

n X

qj2 λ2n−1 , j

j=1

where the ∗ denotes terms involving only variables already having appeared on the l.h.s. of preceding equations (thus the variables an , bn−1 , an−1 , bn−2 , . . . occur in a triangular structure). The first of these equations implies qn dqn = −

n−1 X

qj dqj .

(1.158)

j=1

Taking differentials of the remaining equations, substituting for qn dqn , and then taking wedge products of both sides (making use of the triangular structure on the l.h.s.) shows n−1 Y

b4j−1 da ∧ db = qn2 j

n−1 Y

j=1

i h q, qj3 det [λjk − λjn ] j=1,...,2n−1 [jλj−1 ] j=1,...,2n−1 dλ ∧ d k k=1,...,n−1

j=1

where da :=

n ^

daj ,

db :=

j=1

n−1 ^ j=1

dbj ,

dλ :=

n ^ j=1

dλj ,

k=1,...,n

d q :=

n−1 ^

dqj .

j=1

By definition the Jacobian J is positive and such that da ∧ db = ±Jdλ ∧ d q for some sign ±. Making use of the determinant evaluation (1.175) below we thus read off that Qn−1 Qn 2 1 j=1 bj “ j=1 qj ”2 Y Q (λk − λj )4 . J= Qn−1 2j n qn j=1 qj j=1 bj 1≤j 0 [140]. P ROPOSITION 1.9.4 Let β > 0 be fixed. In the notation of Proposition 1.9.1 ensemble as the set of symmetric tridiagonal matrices ⎡ N[0, 1] χ ˜(N −1)β ⎢ χ ˜ N[0, 1] χ ˜(N −2)β (N −1)β ⎢ ⎢ χ ˜ N[0, 1] χ ˜(N −3)β (N −2)β ⎢ Tβ := ⎢ . . .. . . ⎢ . . . ⎢ ⎣ χ ˜2β N[0, 1] χ ˜β χ ˜β N[0, 1]

define the Gaussian β⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(1.159)

The eigenvalues and first component of the eigenvectors (which form the vector q) are independent, with the

48

CHAPTER 1

distribution of the former given by 1

N 

˜ β,N G

l=1

e



−λ2l /2

|λk − λj | dλ,

˜ β,N = (2π)N/2 G

β

N −1  j=0

1≤j 0,

i=1

N

qi2 = 1,

where

cβ,N =

i=1

ΓN (β/2) 2N −1 Γ(βN/2)

.

(1.161)

Proof. Denote the joint distribution of Tβ by P (Tβ ). We have N−1 βl−1 −b2 Y N 2 e l 2N−1 Y bl e−al /2 da ∧ db (2π)N/2 l=1 Γ(βl/2) l=1 QN−1 βl N−1 bl −Tr(T2β )/2  2N−1 Y 1 1 e dλ ∧ d q, = Ql=1 N (2π)N/2 l=1 Γ(βl/2) qN l=1 ql

P (Tβ )(dTβ ) =

where the second equality follows using (1.156). But 2

e−Tr(Tβ )/2 = e−

PN

j=1

λ2 j /2

,

N−1 Y

bβl l =

l=1

N Y l=1

qlβ

Y

|λj − λi |β ,

1≤i<j≤N

q factorizes into the functional forms where the latter formula follows from (1.148), so indeed the dependence on λ and  specified in (1.160) and (1.161). The normalization for (1.161) follows from the Dirichlet integral [541] Z Pn+1 i=1

dρ1 · · · dρn ρi =1, ρi >0

n+1 Y

ρsi i −1 =

i=1

Γ(s1 ) · · · Γ(sn+1 ) Γ(s1 + · · · + sn+1 )

(1.162)

˜ β,N with n = N − 1, si = β/2 and the change of variables ρi = qi2 . With this normalization specified, the value of G follows (an extra factor of N ! is included to effectively remove the ordering on {λi } implicit in the above working; recall the remark below Proposition 1.3.4).

˜ β,N given in (1.160) implies, after a simple change of variables, that We remark that the evaluation of G the normalization constant in (1.28) has the evaluation Gβ,N = β −N/2−N β(N −1)/4 (2π)N/2

N −1  j=0

Γ(1 + (j + 1)β/2) . Γ(1 + β/2)

(1.163)

Another point of interest is that the recurrence (1.154) with ak ∈ N[0, 1],

b2k ∈ Γ[kβ/2, 1]

(1.164)

can be used to generate the characteristic polynomial for a member of the Gaussian β-ensemble, so the p.d.f. (1.160) can be sampled by simply computing the zeros of this polynomial. 1.9.3 Sturm sequences For tridiagonal matrices, the task of computing the cumulative microscopic eigenvalue density N (μ), that is, the number of eigenvalues less than μ, has a number of special features. This in turn follows from special features of the corresponding Sturm sequences [13]. D EFINITION 1.9.5 Let An be a general n × n matrix, and let An−k (k = 1, . . . , n − 1) denote the matrix

49

GAUSSIAN MATRIX ENSEMBLES

obtained by deleting the first k rows and columns. Let di := det Ai (i = 1, . . . , n) and set d0 := 1. The Sturm sequence refers to (d0 , d1 , . . . , dn ). P ROPOSITION 1.9.6 Let An be a real symmetric matrix with no repeated eigenvalues and no zero eigenvalues, and similarly An−k . The number of sign changes in the Sturm sequence (reading from right-to-left, say) is equal to the number of negative eigenvalues of An . Proof. For a given k = 2, . . . , n it is a fundamental result (see Exercises 4.2 q.2(iii) below) that the eigenvalues {ai } of Ak interlace the eigenvalues {αi } of Ak−1 , ak < αk−1 < ak−1 < · · · < α1 < a1 . We know too that the determinant is equal to the product of eigenvalues. Consequently, the number of negative eigenvalues of Ak equals the number of negative eigenvalues of Ak−1 , if dk /dk−1 is positive, while we must add one if dk /dk−1 is negative. Iteratively applying this for k = n, . . . , 1 gives the stated result.

Applying Proposition 1.9.6 to the matrix An − μ1n gives that N (μ) is equal to the number of sign changes in the Sturm sequence for An − μ1n . In the case that An is the tridiagonal matrix (1.143), one has that dk = (−1)k Pk (μ) as specified by (1.154), and using the recurrence (1.154) shows that ri := di /di−1 can be specified by the recursive formula  i = 1, a1 − μ, ri = (1.165) (ai − μ) − b2i−1 /ri−1 , i = 2, . . . , n. As each sign change in the Sturm sequence {di } corresponds to a negative in the ratio sequence {ri }, we see that the number of negative values in {ri } equals N (μ). This latter result can be related to so-called shooting eigenvectors. D EFINITION 1.9.7 The vector x satisfying all but the first of the n linear equations implied by the matrix equation (An − μ1n )x = 0, with x = (xn , . . . , x1 )T and x1 given, is referred to as a shooting eigenvector. (Note that the first equation can only be satisfied as well if and only if μ is an eigenvalue.) For the tridiagonal matrix (1.143), and with xn+1 defined as the first component of (An − μ1n )x, a recurrence for the ratio si = xi /xi−1 , i = 2, . . . , n + 1 is readily obtained, and comparison with (1.165) shows si = −ri−1 /bi−1 (in the case i = n + 1 this requires setting bn := 1). Thus with each bi > 0, the number of positive values in {si } equals N (μ). This can equivalently be stated in terms of {xi }. P ROPOSITION 1.9.8 The number of sign changes in the shooting eigenvector x equals n − N (μ), which is the number of eigenvalues of A greater than μ. 1.9.4 Prufer ¨ phases There is a parametrization, in terms of Pr¨ ufer phases and amplitudes, of the shooting vectors well suited to analysis of the large n limit of the bulk eigenvalues (see Section 13.6). To introduce the parametrization, first observe that the three-term recurrence satisfied by the shooting vector bj xj+1 + aj xj + bj−1 xj−1 = μxj

(j = 1, . . . , n; b0 := 0, bn := −1)

is equivalent to the matrix equation      uj uj+1 (μ − aj )/bj −1/bj = bj 0 vj vj+1 where



uj vj



 =

1 0 0 bj−1



xj xj−1

(j = 1, . . . , n),

(1.166)

(1.167)

 (1.168)

50

CHAPTER 1

(note that the matrix in (1.167) has unit determinant and so as a transformation is volume preserving). Choosing the initial condition u1 = 1, v1 = 0 we see that     1 uj , = Tj 0 vj where Tj := Vj−1 · · · V1 is referred to as a transfer matrix D EFINITION 1.9.9 The Pr¨ufer phases θjμ and amplitudes Rjμ > 0 are such that    μ  Rj cos θjμ uj = , vj Rjμ sin θjμ

(1.169)

μ where −π/2 < θj+1 − θjμ < 3π/2.

Note that it follows from (1.167) and (1.168) that {θjμ } satisfies the first order recurrence μ b2j cot θj+1 = − tan θjμ + (μ − aj ),

θ1μ = 0.

(1.170)

A consequence is an identity which tells us that θjμ is a decreasing function of μ (see also Exercises 1.9 q.5). P ROPOSITION 1.9.10 We have ∂ μ θj = − u2l . ∂μ j−1

(Rjμ )2

(1.171)

l=1

Proof. Differentiating (1.170) with respect to μ and making use of (1.168) and (1.169) gives the recurrence μ )2 (Rj+1

μ ∂θj+1 ∂θjμ = (Rjμ )2 − u2j . ∂μ ∂μ

This together with the initial condition ∂θ1μ /∂μ = 0 implies (1.171).

We are now in a position to relate θnμ to N (μ) for the tridiagonal matrix (1.143) [326] . First note from the recurrence (1.166) that for μ → ∞, xj is positive while xj−1 /xj → 0. Recalling (1.169), this implies limμ→∞ θjμ = 0. But it has just been shown that θjμ is a decreasing function of μ. The facts that xn+1 = μ μ un+1 = Rn+1 cos θn+1 and that xn+1 = 0 if and only if μ is an eigenvalue then imply the kth largest μ λk eigenvalue λk of T is such that θn+1 = (π/2) + π(k − 1), and moreover that θn+1 relates to the number of eigenvalues of T greater that μ, n − N (μ), according to 1  1  μ  (1.172)  θn+1 − (n − N (μ)) ≤ . π 2 E XERCISES 1.9 1. The objective of this exercise is to derive the Vandermonde determinant evaluation

]j,k=1,...,N det[xk−1 j

˛ ˛ ˛ ˛ ˛ := ˛ ˛ ˛ ˛

1 1 .. . 1

x1 x2 .. . xN

x21 x22 .. . x2N

··· ··· .. . ···

xN−1 1 xN−1 2 .. . xN−1 N

˛ ˛ ˛ ˛ Y ˛ (xk − xj ). ˛= ˛ ˛ 1≤j 0. For Qm positive definite (i.e. all eigenvalues positive) one has i n−m i 2m π m(m+1)/2 im (−1)m(m−1)/2  n Im,n (Qm ) = det Qm e 2 μTr(Qm ) 2 j=n−m+1 Γ(j) while Im,n (Qm ) vanishes if Qm has an eigenvalue less than or equal to zero.

95

LAGUERRE AND JACOBI ENSEMBLES

Proof. We proceed in an analogous way to the proof of statement (b) in Proposition 2.5.1. Because the integral (3.24) is

invariant under the transformation Hm → Um Hm U−1 m for Um unitary, Im,n (Qm ) is a function of the eigenvalues of Qm only and so we can take Qm = diag[q1 , . . . , qm ]. Now introduce the decomposition

" Hm =

Hm−1 h†

#

h

,

hmm

where h is a vector of length m − 1 with complex entries, and proceed as in the derivation of (2.59) to show a := μ + h† (Hm−1 − μ1m−1 )−1h.

det(Hm − μ1m ) = det(Hm−1 − μ1m−1 )(hmm − a), This shows

Z Im,n (Qm ) =

i

(dHm−1 ) e 2 Tr(Hm−1 Qm−1 )

Z “ ”−n Z × det(Hm−1 − μ1m−1 ) (dh)

∞ −∞

i

dhmm e 2 qm hmm (hmm − a)−n .

Since by assumption Im(μ) > 0 we have Im(a) > 0, which allows the integral over hmm to be computed by closing the contour in the upper half-plane (for qm > 0) to give Z

∞ −∞

e 2 qm hmm 2πi “ iqm ”n−1 2i qm a dhmm = e . n (hmm − a) Γ(n) 2 i

On the other hand, closing the contour in the lower half-plane shows that for qm < 0 the integral vanishes. The next task then is to evaluate Z i (dh) e 2 qm a . (3.25)  where Um−1 is a unitary matrix such that U−1 Changing variables h = U−1 m−1 w m−1 Hm−1 Um−1 = diag[h1 , . . . , hm−1 ] and applying Proposition 3.2.4 to deduce (dh) = (dw),  separates (3.25) into 2(m − 1) one-dimensional integrals, Z

i i (dh) e 2 qm a = e 2 qm μ

m−1 Y Z ∞ l=1

i

= e 2 qm μ

−∞

m−1 Y “ l=1

Thus Im,n (Qm ) =

−

dwlr e

iqm 2

(wlr )2 (hl −μ)−1

Z

∞ −∞

dwli e

iqm 2

(wli )2 (hl −μ)−1

“ “ 2πi ”m−1 ” i 2π ” det Hm−1 − μ1m−1 . (hl − μ) = e 2 qm μ iqm qm

2π m i(−1)m−1 “ iqm ”n−m 2i qm μ e Im−1,n−1 (Qm−1 ). Γ(n) 2

Iterating this and noting I0,n (Q0 ) := 1 gives the stated result.

To make use of (3.24), note that with X an n × m complex Gaussian matrix, P (X) = and A = X† X we have

 P (A) =

1 π mn

†

e−Tr(X

X)

,

δ(A − X† X)P (X)(dX).

(3.26)

Here δ(A − X† X) is equal to the product of one-dimensional delta functions over the independent real and

96

CHAPTER 3

imaginary parts of A. Writing each of these as a Fourier integral shows  † 1 eiTr(H(A−X X)) (dH), δ(A − X† X) = (2π)m2

(3.27)

where H is an m × m Hermitian matrix. Substituting this in (3.26) and noting that   −n † † e−Tr(X X) e−iTr(HX X) (dX) = π mn det(1 + iH) , which in turn is deduced from the fact that the integral is a function of the eigenvalues of H only and then separating the integration into a product of one-dimensional integrations, gives  π 2mn eiTr(HA) P (A) = (dH). 2 (det(1 + iH))n (2π)m Now the result (3.24) can be applied, thus reclaiming (3.22).  = Xφ|  ψ  and the property of the inner  † ψ 1. (i) Use the definition of the adjoint φ|X †     product φ|φ > 0 for φ = 0 to show that the eigenvalues of X X are non-negative.

E XERCISES 3.2

(ii) Consider the matrix product X† X with X an n × m matrix with n < m. Show that X† X = Y † Y where Y is an m × m matrix obtained from X by the addition of m − n rows of zeros. Hence show that X† X has m − n zero eigenvalues. (iii) By considering the corresponding characteristic polynomials, and making use of (5.26) below, show that in the setting of (ii) the nonzero eigenvalues of X† X and XX† are equal. 2. Here Proposition 3.2.4 will be established.  = [uj + ivj ]j=1,...,N and A = [ajk + ibjk ]j,k=1,...,N , show that the (i) With  z = [xj + iyj ]j=1,...,N , w equation d z = Adw  can be rewritten as the real matrix equation » – » –» – [dxj ]j=1,...,N [ajk ]j,k=1,...,N −[bjk ]j,k=1,...,N [duj ]j=1,...,N = . [dyj ]j=1,...,N [bjk ]j,k=1,...,N [ajk ]j,k=1,...,N [dvj ]j=1,...,N (ii) To evaluate the determinant of the 2N ×2N matrix on the r.h.s. of the above equation, and thus the Jacobian, add i times the blocks in the bottom half to the blocks in the top half. Then subtract i times the blocks in the left half to the blocks in the right half so that the top right block is now the zero matrix. 3.

(i) Show that the problem of calculating the eigenvalue p.d.f. of the Wishart matrices in the case m = 1 is p P 2 2 equivalent to calculating the p.d.f. of βn β/2πe−βxj /2 . j=1 xj where xj has distribution (ii) Calculate the p.d.f. in (i), p(λ) say, according to the formula „ p(λ) =

β 2π

«βn/2 Z

∞

2

dx1 e−βx1 /2 . . .

−∞

Z

∞ −∞

βn “ ” X 2 dxβn e−βxβn /2 δ λ − x2j , j=1

and thus reclaim Proposition 3.2.2 in the case m = 1. 4. [410] The objective of this exercise is to compute the eigenvalue p.d.f. for real Wishart matrices. (i) Let X be a real n × m (n ≥ m) matrix. Suppose the Gram-Schmidt orthogonalization procedure has been used to write X = RT where R is an n × m real matrix such that RT R = 1m and T is an upper triangular m × m real matrix with positive diagonal entries. Use the method of Proposition 3.2.5 to show that (dX) =

m Y j=1

T tn−j jj (dT)(R dR).

97

LAGUERRE AND JACOBI ENSEMBLES

(ii) Let A = TT T where T is as in (i). Use the method of the proof of Proposition 3.2.6 to show that (dA) = 2m

m Y

tm+1−j (dT). jj

j=1

(iii) Use the results of (i) and (ii) and Definition 3.2.1 to show that the p.d.f. of a real Wishart matrix is 1 − 12 Tr(A) (det A)(n−m−1)/2 , e C where C is a normalization constant, and then use (1.11) to express this p.d.f. in terms of the eigenvalues of A. 5. The objective of this exercise is to compute the eigenvalue p.d.f. for quaternion real Wishart matrices. (i) Let X be an n × m (n ≥ m) matrix with real quaternion elements. Suppose the Gram-Schmidt orthogonalization procedure has been used to write X = UT where U is an n × m matrix of real quaternions such that U† U = 1 and T is an upper triangular m × m matrix with diagonal entries positive real multiples of 12 and off-diagonal entries real quaternions. Use the method of the proof of Proposition 3.2.5 to show that (dX) =

m Y

4(n−j)+2

tjj

(dT)(U† dU).

j=1

(ii) Let A = T† T where T as in (i) above. Use the method of the proof of Proposition 3.2.6 to show that (dA) = 2m

m Y

4(m−j)+1

tjj

(dT).

j=1

(iii) Use the results of (i) and (ii) and Definition 3.2.1 to show that the p.d.f. of a quaternion real Wishart matrix is 1 −2Tr(A) (det A)2(n−m+1/2) , e C where C is a normalization constant and the operations Tr and det are not to include repeated eigenvalues, and then use (1.27) for β = 4 to express this p.d.f. in terms of the eigenvalues of A. 6. [433] Let the p.d.f. of the n × m matrix X (with real (β = 1), complex (β = 2), real quaternion (β = 4) elements) be of the form F (X† X). The p.d.f. of the elements of A = X† X is then h(A)F (A) for some h. The objective of this exercise is to determine h. (i) Let A = B† VB, where V is positive definite. Making use of (1.35), show that the p.d.f. of V is then F (B† VB)h(B† VB) det(B† B)(β/2)(m−1+2/β) .

(3.28)

(ii) Let X = YB, where Y is such that V = Y† Y. Use an appropriate generalization of Proposition 3.2.4 to show that the p.d.f. of Y is F (B† Y † YB)(det B† B)βn/2 and hence that of V is F (B† VB)h(B† B) det(B† B)βn/2 h(V).

(3.29)

(iii) Equate (3.28) and (3.29) with V = 1 to deduce h(B† B) = h(1)(det B† B)(β/2)(n−m+1−2/β) , and note that h(1) = c for some constant c to conclude h(A) ∝ (det A)(β/2)(n−m+1−2/β) .

(3.30)

7. Let Y, Z be m × m positive definite Hermitian matrices (denoted Y > 0, Z > 0) with real (β = 1), complex (β = 2) or real quaternion (β = 4) elements. By changing variables X = Z1/2 YZ1/2 , making use of (1.35), show Z 1 e−(β/2)Tr(YZ) (det Y)β(a−(m−1)−2/β)/2 (dY) = (det Z)−βa/2 , (3.31) C Y>0

98

CHAPTER 3

where C is such that both sides equal unity when Z = 1.

3.3 FURTHER EXAMPLES OF THE LAGUERRE ENSEMBLE IN QUANTUM MECHANICS The chiral ensemble, which from the viewpoint of its eigenvalue distribution is equivalent to the Laguerre ensemble, is motivated from a problem in quantum mechanics. Here some further problems in quantum mechanics are studied which lead to the Laguerre ensemble. As part of this four random Hamiltonians are isolated, which together with those forming the Gaussian and chiral ensembles, make up the ten Hermitian random matrix ensembles in correspondence with the ten matrix Lie algebras associated with infinite families of symmetric spaces. 3.3.1 Eigenvalue statistics of Wigner-Smith delay time matrix In this subsection it will be shown, following [102], that for the quantum cavity problem in Section 2.1.1 the distribution of the scaled reciprocal eigenvalues of ∂S −1/2 S (3.32) ∂E is given by the Laguerre ensemble (3.16) with a = N . Here E is the energy of the waves as they enter the cavity (for a long lead and fixed number of channels N the energy will to leading order be constant). The eigenvalues of QE , denoted τ1 , . . . , τN say, are referred to as the proper delay times, and are also the eigenvalues for the Wigner-Smith matrix Q = −iS−1 ∂S/∂E. A precise formulation of the problem requires a Hamiltonian approach to the coupled lead-cavity system [531]. This can be done by introducing a basis of states {|ai }i=1,...,N for the lead, and a basis of states {μj }j=1,...,M for the cavity (typically M  N ). The Hamiltonian is then defined as   ¯ ij μi | , |μj Wji ai | + |ai W |ai Eai | + |μj Hjj νj  | + H= QE := −iS−1/2

i

j,j 

j,i

where the matrix elements Hjj form a random Hermitian M × M matrix H with real (β = 1), complex (β = 2) or quaternion real (β = 4) elements, while the coupling constants form a fixed M × N matrix W. A key point is that for this H the corresponding N × N scattering matrix S can be calculated exactly as [380] S=

1N + iπW† (H − E)−1 W . 1N − iπW† (H − E)−1 W

In the simplest case W can be taken as proportional to the identity, √ ΔM W= 1M×N , π where Δ is the mean level spacing in the cavity. Furthermore it is assumed that the Hermitian matrix H is a Gaussian random matrix with   (3.33) P (H) ∝ exp − βπ 2 TrH2 /4Δ2 M . Writing H − E = UM×M diag[E1 − E, . . . , EM − E]U†M×M , where the columns of U consist of the eigenvectors of H, gives   ΔM 1N + iK , K= UN ×M diag (E1 − E)−1 , . . . , (EM − E)−1 (UN ×M )† . S= 1N − iK π

(3.34)

99

LAGUERRE AND JACOBI ENSEMBLES

Here UN ×M denotes the matrix UM×M with the last M − N rows deleted. Since H has distribution (3.33) and N  M , it follows that the distribution of the submatrix UN ×N is given by  1    P √ U ∝ exp − βTr UU† /2 (3.35) M (cf. Exercises 1.3 q.2). The formula (3.34), which expresses S as a function of E, can be used to compute the eigenvalue distribution of (3.32) [102]. But before undertaking this task we will present a matrix integration formula which is required in the course of the derivation. P ROPOSITION 3.3.1 Let X be a N × N random matrix with real elements (β = 1), complex elements (β = 2) or real quaternion elements (β = 4), and let B be an N × N Hermitian random matrix which is real (β = 1), complex (β = 2) or quaternion real (β = 4). Furthermore suppose the joint distribution of X and B is proportional to †

e−βTr(XX

)/2

δ(X†−1 BX−1 ),

where δ denotes the Dirac delta function. Then the distribution of the eigenvalues a1 , . . . , aN of A = XX† , defined up to a multiplicative constant as the eigenvalue dependent factor in  −βTr(XX† )/2 δ(X†−1 BX−1 )(dB), e (3.36) is proportional to N 

βN/2 −βaj /2

aj

e

j=1



|ak − aj |β .

(3.37)

1≤jx(j+1) >x(j) >x(j+1) . 1

1

j

j

j+1

(iii) Use the characteristic equation for the eigenvalues of a general (n + 1) × (n + 1) Hermitian matrix X, implied by the first displayed equation in (i) to deduce the interlacing property (4.95) with the eigenvalues of a principal minor (assuming the latter are distinct). Deduce the same result from the final displayed relation in (i). 4.

(i) [222] Let A = X† X, where X is an n × N (n > N ) complex Gaussian matrix such that A is a member of the LUE with a = n − N . Suppose b times a row of complex Gaussians is appended to X to form Y, and x†x as considered consider B = Y† Y. By noting that this is equivalent to the rank 1 perturbation B = A+b in (3.72), use (3.73) and Proposition 4.3.4 to conclude that the joint distribution of the eigenvalues of A and B is N PN PN Y Y aal e− j=1 aj e−(1/b) j=1 (bj −aj ) (bj − bk )(aj − ak ), (4.97) l=1

1≤j a 1 > b 2 > a 2 > · · · > b N > a N . (ii) [215] Let X(N) denote an N × N complex Gaussian matrix such that

(4.98) X†(N) X(N)

is a member of the (n)

LUE with a = 0. Let X(n) denote the n × N matrix formed from the first n rows of X(N) . Let xj

156

CHAPTER 4

(j = 1, . . . , n) denote the nonzero eigenvalues of X(n) . Proceed as in (i) to show that the joint p.d.f. of (n) {xj }j=1,...,n for n = 1, . . . , N is equal to N Y l=1

Y

(N )

e−xl

(N)

(xj

(N)

− xk )

N−1 Y

χ(x(p+1) > x(p) ),

(4.99)

p=1

1≤j x(p) ) := χx(p+1) >x(p) >···>x(p+1) >x(p) >x(p+1) >0 . 1

p

1

p

(4.100)

p+1

5. [222] Let A be an n × n member of the GSE with variance such that the eigenvalue p.d.f. of the independent eigenvalues is proportional to n Y Y 2 e−yl (yj − yk )4 . (4.101) l=1

1≤j · · · > yn > xn+1 . Note that (4.102) in the case b = 1 is identical to the eigenvalue p.d.f. for GOE2n+1 , and so conclude from (4.101) that even(GOE2n+1 ) = GSEn , thus realizing (4.27).

4.4 GENERALIZATION OF THE DIXON-ANDERSON INTEGRAL We know from Proposition 4.2.6 that the Dixon-Anderson integral (4.15) implies an inter-relation between the distribution of every second eigenvalue in a Jacobi orthogonal ensemble, and the eigenvalue PDF of a Jacobi symplectic ensemble. This integral can be generalized to give an integration formula which relates the distribution of every (r + 1)th eigenvalue in a certain Jacobi ensemble with β = 2/(r + 1) to the eigenvalue PDF of another Jacobi ensemble with β = 2(r + 1) [199]. First the generalization of (4.15) will be presented.

P ROPOSITION 4.4.1 Let Ar denote the interlaced region aj > λr(j−1)+1 > λr(j−1)+2 > · · · > λrj > aj+1

(j = 1, . . . , n − 1),

(4.103)

and let Cˆ be specified in terms of the Selberg integral by Cˆ =

n−1  l=1

l  1   Sr sp + 2(l − 1)r/(r + 1) − l, sl − 1, 1/(r + 1) . r! p=1

(4.104)

157

THE SELBERG INTEGRAL

One has 1 Cˆ





dλ1 · · · dλr(n−1) Ar

(λj − λk )2/(r+1)

1≤j p + 1 the integrand of Ip is symmetrical in tk and tp+1 . Thus » – » – tp+1 tk = Ip(1) . Ip(1) tp+1 − tk tk − tp+1 Taking the arithmetic mean of both sides gives » – » – tp+1 1 tp+1 1 tk = Ip(1) = Ip(1) . Ip(1) + tp+1 − tk 2 tp+1 − tk tk − tp+1 2 (1)

For k ≤ p the integrand of Ip is symmetrical in tk and tp . Thus » – » – » – tp+1 tp+1 tp tp+1 (1) = Ip(1) = Ip−1 . Ip(1) tp+1 − tk tp+1 − tp tp+1 − tp h i tp tp+1 (1) (1) is antisymmetrical under But the integrand of Ip−1 is symmetrical in tp and tp+1 so the integrand of Ip−1 tp+1 −tp the interchange tp ↔ tp+1 and this quantity therefore vanishes. The summand has thus been simplified for all k, and the stated equation follows.

P ROPOSITION 4.6.2 We have   !  (1) 1 (1) = λ2 Ip(1) + λ1 + λ2 + 2 + (2N − p − 2)λ Ip+1 . λ2 Ip 1 − tp+1 Proof. Setting α = 1 and performing the partial derivative in (4.129) gives » (λ1 + 2)Ip(1) [tp+1 ] − λ2 Ip(1)

– » – N X t2p+1 t2p+1 + 2λ = 0. Ip(1) 1 − tp+1 tp+1 − tk k=1 k=p+1

(1)

(1)

Now Ip [tp+1 ] = Ip+1 and » Ip(1)

t2p+1 1 − tp+1

–

» = Ip(1)

– » – (tp+1 − 1)(tp+1 + 1) + 1 1 (1) = −Ip(1) − Ip+1 + Ip(1) . 1 − tp+1 1 − tp+1

Furthermore, for k > p + 1 » – » – t2p+1 t2p+1 1 1 t2k (1) Ip(1) = Ip(1) = Ip(1) [tp+1 + tk ] = Ip+1 , + tp+1 − tk 2 tp+1 − tk tk − tp+1 2 while for k < p + 1

» Ip(1)

t2p+1 tp+1 − tk

–

» (1)

= Ip+1

tp+1 tp+1 − tk

– =

1 (1) I . 2 p+1

Substitution of these simplifications and rearrangement gives the stated equation.

The recurrence (4.130) results from subtracting the equation in Proposition 4.6.1 from the equation in Proposition 4.6.2. 4.6.3 Solution of the recurrence Iteration of (4.131) M − 1 times to increase λ1 at each step gives SN (λ1 + M, λ2 , λ) = SN (λ1 , λ2 , λ)

N −1  p=0

Γ(λ1 + 1 + M + pλ)Γ(λ1 + λ2 + 2 + (N − 1 + p)λ) . Γ(λ1 + 1 + pλ)Γ(λ1 + λ2 + 2 + M + (N − 1 + p)λ)

168

CHAPTER 4

Taking the limit M → ∞ on both sides using the asymptotic formula of Proposition 4.5.2 on the l.h.s. and the asymptotic formula (4.126) on the r.h.s. determines SN as a function of λ1 , SN (λ1 , λ2 , λ) = AN (λ2 , λ)

N −1  p=0

Γ(λ1 + 1 + pλ) . Γ(λ1 + λ2 + 2 + (N − 1 + p)λ)

But SN is symmetrical in λ1 and λ2 , so SN (λ1 , λ2 , λ) = cN (λ)

N −1  p=0

Γ(λ1 + 1 + pλ)Γ(λ2 + 1 + pλ) . Γ(λ1 + λ2 + 2 + (N − 1 + p)λ)

This is precisely the equation obtained using Selberg’s derivation, so the function cN (λ) is specified as in Section 4.1, and the derivation of (4.3) completed. 4.6.4 A Fuchsian differential equation With multivariable notation t = {t1 , . . . , tN }, the elementary symmetric functions are defined by tp1 · · · tpr . er (t) = er (t1 , . . . , tN ) :=

(4.132)

1≤p1 p + 1, + I , k > p + 1, p+1 − tk p p p+1 2 2 (α)

where here Ip

(α)

(α)

(α)

:= Ip (x) and Ip [g] := Ip (x)[g].

(ii) Use (i) and the method of the proof of Proposition 4.3.4 to show » – “ α ” ´ ` x d (α) 1 (α) Ip − xp + λ Ip−1 . λ2 Ip(α) = λ1 + λ2 + α + λ(N − p − 1) Ip(α) − 1 − tp+1 N − p dx N −p (iii) Use (i) and the method of the proof of Proposition 4.6.2 to show ` ´ (α) −Ep Ip+1 = (λ1 + 2)x + λ2 (x + 1) + 2xλ(N − p − 1) + 2x(α − 1) Ip(α) » – 1 x2 d (α) “ pαx2 ” (α) Ip−1 − λ2 Ip(α) . − Ip − λpx2 + N − p dx N −p 1 − tp+1 (iv) Combine the results of (ii) and (iii) to derive the stated recurrence. 2. [20] In this exercise a q-integral which reduces to the Selberg integral in the limit q → 1 will be evaluated. The q-integral to be considered is Z J(α) := Φ(t) 1 D(t)w, ˜ ξF 

where Φ(t) :=

n Y

α+(j−1)(1−2γ)

tj

j=1

and w ˜ :=

(qtj ; q)∞ (q β tj ; q)∞

Y 1≤i<j≤n

dq t n dq t 1 ··· , t1 tn

(q 1−γ tj /ti ; q)∞ , (q γ tj /ti ; q)∞

Q D(t)

The q-integral over ξF  is defined as Z ξF 

f (t)w ˜ := (1 − q)n

:=

Y

(z; q)∞ :=

∞ Y

(1 − zq j ),

j=0

(ti − Qtj ).

1≤i<j≤n

X

f (t1 , . . . , tn ),

where the sum is over all points such that t1 = q ν1 , t2 /t1 = q ν2 q γ , t3 /t2 = q ν3 q γ , . . . , tn /tn−1 = q νn q γ with each νj ∈ Z≥0 . In the limit q → 1− J(α) tends to the Selberg integral (4.1) with λ1 = α + n − 2 − (n − 1)γ, λ2 = β − 1, λ = γ, and integration domain 1 ≥ t1 ≥ t2 · · · ≥ tn ≥ 0.

170

CHAPTER 4

The strategy is again to seek a recurrence relating J(α + 1) to J(α). Using the fact that the α → ∞ behavior can be computed explicitly (this contrasts with the Selberg integral, for which a multiple integral is obtained), the recurrence uniquely specifies J(α). (i) Show that for α → ∞, the behavior of J(α) is given by the term t1 = 1, t2 = q γ , . . . , tn = q (n−1)γ in its definition as a sum, and thus J(α) ∼ q An where An =

Pn

j=1 (α + (j

n Y Γq (β + (j − 1)γ)Γq (jγ)(1 − q)2(j−1)γ+β , Γq (γ) j=1

− 1)(1 − 2γ) + n − j)((j − 1)γ) and Γq (u) := (1 − q)1−u (q; q)∞ /(q u ; q)∞ .

(ii) Define the q-shift operator of the i-th coordinate of φ(t) by Ti φ(t) = φ(t1 , . . . , qti , . . . , tn ) and define the covariant derivative by ∇i φ(t) := φ(t) −

Ti Φ(t) Ti φ(t). Φ(t)

By noting that Φ(t) vanishes at t1 = q −1 , tj+1 /tj = q −1+γ (j = 1, . . . , n−1), which defines the boundary of ξF , show from the definitions that Z Φ(t)∇i φ(t)w ˜ = 0. ξF 

(iii) Let σ ∈ Sn denote a permutation of indices, σf (t) = f (tσ(1) , . . . , tσ(n) ). Show that σΦ(t) = Uσ (t)Φ(t),

where

Y

Uσ (t) =

1≤i<j≤n σ−1 (i)>σ−1 (j)

“ t ”2γ−1 θ(q γ t /t ; q) j j i , ti θ(q 1−γ tj /ti ; q)

with θ(x; q) := (x; q)∞ (q/x; q)∞ (q; q)∞ , and use the property θ(qx) = −(1/x)θ(x) to show that Ti Uσ (t) = Uσ (t) for every i so that Uσ (t) is a constant on ξF . Use these properties and the result of (ii) to show Z Φ(t)σ∇i φ(t)w ˜=0 ξF 

and hence conclude that Z Φ(t)Asym(∇i φ(t))w ˜ = 0,

where

Asymf :=

ξF 

(iv) Show that

X σ∈Sn

n 1 − q β t1 Y t1 − q −γ tj T1 Φ(t) = qα , Φ(t) 1 − qt1 j=2 t1 − q γ−1 tj

and use this result to show that with φ(t) := (1 − t1 )(t2 · · · tr )

Y 1≤i<j≤n

(ti − q γ tj )

sgn(σ)σf.

(4.135)

171

THE SELBERG INTEGRAL

we have ∇1 φ(t) = a − b − c + d, where Y

a := t2 · · · tr

(ti − q γ tj ),

c := q α+n−1 t2 · · · tr

1≤i<j≤n

b := t1 · · · tr

Y

n Y

k=2

(ti − q γ tj ),

1≤i<j≤n

Y

(t1 − q −γ tk )

k=2

Use (iii) to conclude

(ti − q γ tj ),

2≤i<j≤n

n Y

d := q α+β+n−1 t1 · · · tr

Y

(t1 − q −γ tk )

(ti − q γ tj ).

2≤i<j≤n

Z ξF 

Φ(t)Asym(a − b − c + d)w ˜ = 0.

(v) Kadell’s lemma [336, lemma 4] states that for M ⊂ {1, . . . , n} Asym

“ Y

” (Q; Q)|M | (Q; Q)n−|M | tj Q D(t) = Qa(M ) e|M | (t) 1 D(t), (1 − Q)n j∈M

where a(M ) = |{(i, j) : 1 ≤ i < j ≤ n, i ∈ / M, j ∈ M }|,

(Q; Q)n :=

n Y

(1 − Qj )

j=1

and er (t) denotes the elementary symmetric function (4.132). Noting that by the interchanges t2 ↔ t1 , t3 ↔ t2 , . . . , tn ↔ tn−1 , Asym(c) = q α+n−1 q −(n−1)γ Asym(t1 · · · tr−1 qγ D(t)), Asym(d) = q α+β+n−1 q −(n−1)γ Asym(t1 · · · tr−1 tn qγ D(t)) use Kadell’s lemma to show (q γ ; q γ )r−1 (q γ ; q γ )n−r+1 (1 − q α+n−1−(n+r−2)γ )er−1 (t) 1 D(t), (1 − q γ )n (q γ ; q γ )r (q γ ; q γ )n−r Asym(−b + d) = − (1 − q α+β+n−1+(1−r)γ )er (t) 1 D(t). (1 − q γ )n Asym(a − c) = q γ(r−1)

Substitute in the result of (iv) to deduce the recurrence Z Z 1 − q n−r+1 1 − q α+n−1−(n+r−2)γ Φ(t)er (t) 1 D(t)w ˜ = q 1+(r−1)γ Φ(t)er−1 (t) 1 D(t)w, ˜ 1 − qr 1 − q α+β+n+(1−r)γ ξF  ξF  and iterate this recurrence to obtain the desired recurrence for J(α), J(α + 1) =

n Y r=1

q (r−1)γ

1 − q α+n−1−(n+r−2)γ J(α). 1 − q α+β+n−1+(1−r)γ

(vi) Iterate the recurrence above and make use of the functional equation Γq (x + 1) = [x]q Γq (x),

[x]q :=

1 − qx , 1−q

(4.136)

to get a relationship between J(α + M ) and J(α). Now use the formula Γq (x + p) ∼ ([x]q )p Γq (x) x→∞

(4.137)

172

CHAPTER 4

to take the limit M → ∞, and then use the asymptotic result in (i) to obtain the evaluation J(α) = q An

n Y Γq (β + (j − 1)γ)Γq (α + n − 1 − (n + j − 2)γ)Γq (jγ) , Γq (γ)Γq (α + β + n − 1 − (n − j)γ) j=1

where An is specified in (i). 3.

(i) In the case γ ∈ Z>0 show that the integrand in the q-integral defining J(α) in q.2 above is symmetric and thus deduce that in this case Z 1 Z 1 1 J(α) = ··· Φ(t)1 D(t)w ˜ n! 0 0 R1 P d t j where 0 f (t) qt := (1 − q) ∞ j=0 f (q ). (ii) From (i) and Kadell’s lemma (q.2(v) above) with M = ∅, show that in the case γ ∈ Z>0 Z 1 Z 1 1 J(α) = ··· Φ(t) qγ D(t)w. ˜ Γqγ (n + 1) 0 0 (iii) [36] Assuming the identities Z 1

tx+p−1

0

“q

(qt; q)∞ (q x ; q)p Γq (x)Γq (y) , dq t = x+y y (q t; q)∞ (q ; q)p Γq (x + y)

”

(q −b ; q)p (q; q)a+b ; q (q −(b+1) t)p = a+1 , t (q ; q)p (q; q)a (q; q)b b deduce that for any Laurent polynomial f CT (t; q)a

„

Γq (x + y) Γq (x)Γq (y) „ =

«N Z

1

Z

0

(q; q)a (q; q)b (q; q)a+b

1

dq t 1 · · ·

dq t N 0

«N CT

N Y

(tj ; q)a

N Y

tx−1 j

j=1

“q

j=1

tj

” ;q

b

(x; q)a :=

(x; q)∞ , (xq a ; q)∞

(qtj ; q)∞ f (t1 , . . . , tN ) (q y tj ; q)∞

f (q −(b+1) t1 , . . . , q −(b+1) tN ),

where x = −b and y = a + b + 1. (iv) Use (ii), (iii) and the evaluation of J(α) given in q.2 (vi) to deduce the q-Morris identity [552] CT

N Y

(xj ; q)a

j=1

“q ” ;q xj b

Y 1≤jxN/2 >yN/2

N  j=1



e−V (wj )



(wj − wk )

1≤j0 ]j,k=1,...,N/2 ,

(6.115)

assuming the inequalities (5.173). On the l.h.s. of this identity is the Boltzmann factor for the ordered log-gas at β = 1. The r.h.s. naturally separates the even and odd labeled coordinates, and has the further significant feature of being a symmetric function of {xi } and {yj } separately. The identity (6.115) can be used to compute ρ(n1 ,n2 ) . P ROPOSITION 6.5.1 Let ρ(n1 ,n2 ) (x1 , . . . , xn1 ; y1 , . . . , yn2 ) denote the (n1 , n2 )-point correlation function corresponding to (6.115) for n1 particles with odd numbered w-coordinates and n2 particles with even numbered w-coordinates. With x, x odd numbered w-coordinates and y, y  even numbered w-coordinates let 1 hoo (x, x ) = hee (y, y  ) = 0, hoe (x, y) = −heo (y, x) = χx−y>0 ,  ∞  ∞2 o −V (y) e Φk (x) = heo (y, x)Rk (y)e dy, Φk (y) = hoe (x, y)Rk (x)e−V (x) dx, −∞

−∞

and let {Rk (t)} denote the monic skew orthogonal polynomials of Definition 6.3.1, with corresponding normalizations {rj }. Substitute these quantities in Proposition 6.3.2 and denote the corresponding expressions ˜ respectively. Then we have for S1 , D1 and I˜1 by the symbols S, D and I, ρ(k1 ,k2 ) (x1 , . . . , xk1 ; y1 , . . . , yk2 )  [foo (xj , xl )]j,l=1,...,k1 = qdet [feo (yj , xl )] j=1,...,k2 l=1,...,k1

[foe (xj , yl )] j=1,...,k1 l=1,...,k2

[fee (yj , yl )]j,l=1,...,k2

,

(6.116)

270

CHAPTER 6

where with p, p ∈ {e, o}

 fpp (w, z) =

Spp (w, z) I˜pp (w, z) Dpp (w, z) Spp (z, w)

 .

Proof. This result can be deduced from Proposition 6.3.2. It is necessary first to replace each integration R

(6.117) R

dw over the appropriate w-variables in (6.115) by a Riemann sum approximation dμ(w). In the latter the lattice points which make up the domain of integration are chosen to be the set X if w is odd numbered (an x variable), and the set Y if w is even numbered (a y variable). These lattice points must interlace so that for coordinates xj restricted to X and yj restricted to Y (j = 1, . . . , N/2) it’s always possible to have xj > yj . Now consider N Y

e−V (wj ) det[Rj−1 (wk )]j,k=1,...,N Pf[−h(wj , wk )]j,k=1,...,N .

(6.118)

j=1

It follows from the definition of h(w, w ) = h·,· that (6.118) vanishes unless exactly half the wk variables are on X and the other half are on Y . In the latter case (6.118) equals 2−N/2 times (6.115). The expression (6.118) has a structure identical to the l.h.s. of (6.65), although the variables are confined to lattice points. We can therefore use Propositions 6.3.2 and 6.3.3 to write down the (n1 , n2 )-point correlation function, where n1 coordinates are on X (odd numbered) and n2 coordinates on Y (even numbered). The polynomials Rk (z) must be skew orthogonal with respect to the inner product Z Z dμ(w) dμ(z) e−V (w)−V (z) f (w)g(z)h(w, z) f, g = X∪Y X∪Y Z Z “ ” = dμ(x) dμ(y) e−V (x)−V (y) f (x)g(y)χx−y>0 − f (y)g(x)χy−x>0 , X

Y

which in the continuum limit is identical to (6.61).

Propositions 6.5.1 and 6.3.2 together give  x  x  e−V (x )  R2k+1 (x ) e−V (t) R2k+1 (t) dt , e−V (t) R2k (t) dt − R2k (x ) 2rk −∞ −∞ k=0  x ∂ 1 I˜oo (x, x ) = − Soo (x, t) dt, Doo (x, x ) = 2 Soo (x, x ), ∂x 2 x  ∞  ∞ N/2−1 −V (y  )   e   −V (t)  See (y, y ) = − R2k+1 (y ) e R2k (t) dt − R2k (y ) e−V (t) R2k+1 (t) dt , 2rk y y k=0  y ∂ 1 I˜ee (y, y  ) = − See (y, t) dt. (6.119) Dee (y, y  ) = 2 See (y, y  ), ∂y 2 y N/2−1

Soo (x, x ) =

Next we will show that in the classical cases See (y, y  ) is closely related to S4 (y, y  ). First we note, by comparing the definition of See in (6.119) and the definition of S1 in Proposition 6.3.2, with h, Φ therein given by (6.67), (6.69), that in general  1 See (y, y  ) = S1 (y, y  ) − S1 (∞, y  ) (6.120) 2 ˜

(in the Jacobi case S1 (∞, y  ) means S1 (1, y  )). With e−V → e−V1 (recall (6.100)), let the corresponding summations in (6.120) be denoted See → S˜ee , S1 → S˜1 . Now S˜1 is summed in all the classical cases by (6.107). Furthermore, taking x → ∞ in (6.107) we see  ∞ ˜1 (y  ) ˜  −V  ˜ S1 (∞, y ) = γN −2 e pN −1 (y ) pN −2 (t)e−V1 (t) dt. −∞

271

CORRELATION FUNCTIONS AT β = 1 AND 4

Thus we have 

˜

˜



2S˜ee (y, y  ) = e−(V (y)−V1 (y)) eV (y )−V1 (y ) KN −1 (y, y  )  ∞  ˜ ˜ pN −2 (t)e−V1 (t) dt. −γN −2 e−V1 (y ) pN −1 (y  ) We remark that, after replacing (y, y  ) by (x, x ) and

∞

(6.121)

y

x

, the same formula applies for S˜oo (x, x ). The expression (6.121) has the same structure as the summation formula (6.56) for S˜4 (x, y) (recall here that the tilde denotes the modified weight functions (6.53)) in the classical cases. Noting that in general ˜ ˜ e−2V (t)+V4 (t) = e−V1 (t) we thus observe     = S˜ee (y, y  ) . (6.122) S˜4 (y, y  ) N →N/2

y

by −

−∞

N →N +1

We will see in the next subsection that the r.h.s. of (6.122) does indeed correspond to S˜ee (y, y  ) for N odd, or equivalently that (6.121) remains valid for N odd. Recalling the analogous structure of (6.14) and the formula (6.116) for ρ(0,n) (which too remains valid for N odd), we therefore have the following result [220]. P ROPOSITION 6.5.2 Let OEN (e−V (x) ), SEN (e−4V (x) ) and the operation even be as in Section 4.2.3. We have ˜

˜

ρ(n) (y1 , . . . , yn ; SEN/2 (e−2V4 (x) )) = ρ˜(n) (y1 , y2 , . . . , yn ; even(OEN +1 (e−V1 (x) ))).

(6.123)

The identity (6.123) with n = N/2 says that integrating the p.d.f. at β = 1, with weight function (6.100) and N + 1 particles, over every odd numbered coordinate gives the p.d.f. at β = 4 with weight function (6.53). Equivalently, in terms of the distributions, ˜

˜

even(OEN +1 (e−V1 (x) )) = SEN/2 (e−2V4 (x) ).

(6.124)

In the case N → 2n, this summarizes in one formula the results of Section 4.2.3 relating to eigenvalue p.d.f.’s on the real line. In fact a simple direct derivation of (6.124) based on (5.64) can be given (see Exercises 6.5 q.1). The result (6.124) raises the question as to the relationship of ρ(0,n) to a β = 4 correlation in the cases when the weight function is not classical. It was proved in [220] that the only continuous weight functions for which (6.124) holds are the classical weight functions (or weight functions related to the classical weight functions by a fractional linear transformation). Another question of interest is the relationship of the distribution of every even labeled coordinate for OEN , N even, to a β = 4 distribution. The following result was proved in [220]. P ROPOSITION 6.5.3 Let N be even. We have even(OEN (f )) = SEN/2 ((g/f )2 ) for

 (f, g) =

x > 0, (e−x/2 , e−x ), ((1 − x)(a−1)/2 , (1 − x)a ), 0 < x < 1,

and up to linear fractional transformations these pairs of weights are unique. It remains to consider the N odd case, and to verify that indeed (6.121) still holds true.

(6.125)

(6.126)

272

CHAPTER 6

6.5.2 N odd The computation of ρ(n1 ,n2 ) for N odd yields to a similar strategy as does the N even case. First one notes that in the limit yn → −∞, (5.174) reads   det [χxj >yk ] j=1,...,N [1]j=1,...,N = χx1 >y1 >···>yn−1 >xn , k=1,...,N −1

valid for x1 > · · · > xn , Furthermore, with the latter ordering   det [χxj >yk ] j=1,...,N [1]j=1,...,N k=1,...,N −1 ⎡ 0 χx1 >y1 0 ⎢ −χx1 >y1 0 −χ x2 >y1 ⎢ ⎢ 0 χx2 >y1 0 = Pf ⎢ ⎢ −χx1 >y2 0 −χ x2 >y2 ⎣ .. .

χx1 >y2 0 χx2 >y2 0

y1 > · · · > yn−1 .

0

−χx3 >y1 0 −χx3 >y2

··· ··· ··· ···

χx1 >yN −1 0 χx2 >yN −1 0

0

−χxN >y1 0 −χxN >y2

1 1 1 1 .. .

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

and this extends to the setting (6.118), thereby allowing Proposition 6.3.7 to be used in an analogous way to the use made of Proposition 6.3.2 in the derivation of Proposition 6.5.1. P ROPOSITION 6.5.4 Let x, x be odd numbered w-coordinates and y, y  be even numbered w-coordinates as in Proposition 6.5.1. Let 1 1 hoo (x, x ) = hee (y, y  ) = 0, hoe (x, y) = −heo (y, x) = χx−y>0 , F (x) = F (y) = , 2  ∞  2∞ 1 1 o −V (y) e −V (x) ˆ k (x) = ˆ k (y)e ˆ k (y) = ˆ k (x)e Φ h(y, x)R dy, Φ h(x, y)R dx, 2 −∞ 2 −∞ ˆ k (t)} be specified in terms of the monic skew orthogonal polynomials of Definition 6.3.1 as in Propolet {R sition 6.3.8, and similarly specify {ˆ rj }. Substitute these quantities in Proposition 6.3.7 and denote the corresponding expressions for S1odd , D1odd and I˜1odd by the symbols S odd , Dodd and I˜odd , respectively. Then the formulas (6.116) and (6.117) again apply, where in the latter the superscript “odd” is to be attached to the matrix elements. Comparing the above specification of S odd with the formula for S1odd in Proposition 6.3.7 we see that  1  odd odd S1 (y, y  ) − S1odd (∞, y  ) . (y, y  ) = See 2 For the modified weight functions (6.100), indicated by the use of a tilde, we have the summation formula (6.107) of S˜1 . Substituting this in the above formula gives  ˜  e−V1 (y )  odd (y, y  ) = 2S˜ee (y, y  ) + γN −3 s˜N −3 2S˜ee s˜N −1 N →N −1  ∞  ∞   ˜ ˜ × pN −2 (y  ) e−V1 (t) pN −1 (t) dt − pN −1 (y  ) e−V1 (t) pN −2 (t) dt , (6.127) y

y

where S˜ee (y, y  ) is specified by (6.121). The formula (6.127) can be further simplified. The first step is ˆ N −3 (x)/Φ ˆ N −1 (x) = to note from (6.103), (6.105) and (6.98) that taking the limit x → ∞ in the ratio Φ

273

CORRELATION FUNCTIONS AT β = 1 AND 4

φ˜N −3 (x)/φ˜N −1 (x) implies γN −3

s˜N −3 = γN −2 . s˜N −1

(6.128)

Substituting this in (6.127) and recalling (6.121) we see that odd (y, y  ) = S˜ee (y, y  ), S˜ee

(6.129)

provided ˜ γN −3 φ˜N −3 (x) − γN −2 φ˜N −1 (x) = e−2V (x)+V1 (x)

pN −2 (x) (pN −2 , pN −2 )2

(6.130)

for N odd. This latter identity is verified by checking that both sides agree for x → ∞ and that it reduces to (6.131) below upon differentiation. E XERCISES 6.5

1.

(i) Note from (5.60) and (5.64) that ” ˜ d “ −V˜4 (x) cl cl−1 pl (x) = − pl+1 (x) + pl−1 (x) eV1 (x) e dx (pl+1 , pl+1 )2 (pl−1 , pl−1 )2

(6.131)

for l = 0, 1, . . . , where c−1 := 0. (ii) For N odd let I(x2 , x4 , . . . , xN−1 ) :=

1“ C

(N+1)/2 Z x Y 2l l=1

x2l−2

dx2l−1

N ”Y j=1

˜

e−V1 (xj )

Y

|xk − xj |,

1≤j 2π, − |k| ⎩ 1 + (β − 2) 12 log |k|−2π + 4π log 1 − k2 up to terms O((β − 2)2 ). 7.1.3 Soft edge scaling limit

√ The edges of the spectrum are predicted by Proposition 1.4.4 to occur at ± 2N . Since the eigenvalues are not confined by a wall at these points but have a nonzero √ density on either side, each edge is referred to as a soft edge. For x in the neighborhood of the right edge ( 2N ) we have the large N asymptotic formula [508]   (7.9) exp(−x2 /2)HN (x) = π 1/4 2N/2+1/4 (N !)1/2 N −1/12 πAi(t) + O(N −2/3 ) , where x = (2N )1/2 + 2−1/2 N −1/6 t and with Ai(x) denoting the Airy function. The Airy function in turn can be specified by the integral representation  3 dv Ai(x) = e−xv+v /3 , (7.10) 2πi A where the contour starts at e−πi/3 ∞ and finishes at eπi/3 ∞, following the corresponding rays asymptotically, staying in the sector −π/3 < arg z < π/3. The expansion (7.9) suggests that in order to evaluate the √ Christoffel-Darboux sum in the neighborhood of x = 2N we should make the change of variables x = (2N )1/2 +

X , 21/2 N 1/6

y = (2N )1/2 +

Y 21/2 N 1/6

(7.11)

(the factors of 1/21/2 are chosen for later convenience). Using the above asymptotic formula and Stirling’s formula, the Christoffel-Darboux sum is then readily evaluated [189]. P ROPOSITION 7.1.2 We have

  X Y 1 (G) 1/2 1/2 (2N ) K + , (2N ) + N →∞ 21/2 N 1/6 N 21/2 N 1/6 21/2 N 1/6   Ai(X)Ai (Y ) − Ai(Y )Ai (X) , = X −Y

K soft (X, Y ) := lim

(7.12)

287

SCALED LIMITS AT β = 1, 2 AND 4

(G)

(G)

where KN (x, y) is given by Proposition 5.1.3 with pN (x) = pN (x). We see from the formula of Proposition 5.1.2 that to specify ρ(n) we must also calculate K soft (X, X). This can be deduced from (7.12) by taking the limit Y → X and simplifying using the fact that Ai(x) satisfies the differential equation y  (x) = xy(x). This gives K soft (X, X) = −X(Ai(X))2 + (Ai (X))2 .

(7.13)

7.1.4 Soft edge scaling of the perturbed Hermite kernel The perturbed Hermite kernel is given by (5.172), and thus consists of the unperturbed kernel, which has the soft edge scaling (7.12), plus a sum of r correction terms. The latter depend on parameters a1 , . . . , ar , which to give a well-defined soft edge limit must be replaced by the parameters s1 , . . . , sr according to the scaling [442], [132] √ √ ak = − 2N + 2N 1/6 sk . To specify the limiting functional form, we introduce a class of incomplete multiple Airy functions  3 (j) e−xv+v /3 dv ˜ , Ai (x) := j 2πi A{s1 ,...,sj } k=1 (v − sk )  j−1  3 dv Ai(j) (x) := (−1)j . e−xv+v /3 (v + sk ) 2πi A

(7.14)

k=1

Here A{s1 ,...,si } is a contour which starts at e−πi/3 ∞ and finishes at eπi/3 ∞, following the corresponding rays asymptotically, staying in the sector arg z > π/3, arg z < −π/3, and crossing the real axis to the left of {sk }. The contour A is defined as in (7.10). P ROPOSITION 7.1.3 For large N  2 ˜ (j) (x) √ e−x /2 Γ = (−1)N +r+1 x= 2N +X/21/2 N 1/6  2  ex /2 Γ(j) (x)

ak =

x=

√

2N (−1+sk /N 1/3 )

√ 2N +X/21/2 N 1/6 √ 2N (−1+sk /N 1/3 )

ak =

  N (j−1)/3 ˜ (j) (x) 1 + O(N −1/3 ) , Ai (2N )(N +j−r−1)/2  (2N )(N +j−r)/2 (j)  −1/3 = (−1)N +r+1 Ai (x) 1 + O(N ) . (7.15) N j/3

Proof. Consider the first formula. In the definition (5.171) make the change of variables z → This shows e−x

2

/2 ˜ (j)

Γ

˛ ˛ (x)˛

x=

√

2N (−1 + w/N 1/3 ).

√ 2N +X/21/2 N 1/6 √ 2N (−1+sk /N 1/3 )

ak =

N (j−1)/3 = (−1)N−r √ ( 2N )N−r+j−1

Z

2/3

C

{N 1/3 ,s1 ,...,sr }

2

1/3

dw e−wX e−N w e−w N /2 . Qj 1/3 N−r 2πi (1 − w/N ) k=1 (w − sk )

But for large N 2/3

2

1/3

3 e−N w e−w N /2 ∼ ew /3 (1 + O(N −1/3 )), (1 − w/N 1/3 )N−r and the method of steepest descent says we must deform the contour to rays such that the exponent is minimized. This occurs along the rays arg w = ±π/3, giving the first formula in (7.15). The second asymptotic formula is derived similarly.

It follows immediately from these asymptotic formulas, and (7.12), that with the scaled variables X, Y ,

288

CHAPTER 7

{sk } specified by √ x = 2N + X/21/2 N 1/6 ,

y=

√ 2N + Y /21/2 N 1/6 ,

ak =

√ 2N (−1 + sk /N 1/3 ),

(7.16)

the perturbed Hermite kernel (5.172) has the soft edge scaled form r 2 2 1 ˜ (j) (X)Ai(j) (Y ). e−x /2+y /2 KN (x, y) = K soft (X, Y ) + Ai lim √ N →∞ 2N 1/6 j=1

We see from (7.10) and (7.14) that ˜ (1) (x) = Ai

Ai(1) (x) = Ai(x),



(7.17)

x

es1 (x−t) Ai(t) dt, −∞

so in the case r = 1 this simplifies to read



K soft (X, Y ) + Ai(Y )

X

−∞

es1 (X−t) Ai(t) dt.

(7.18)

7.1.5 Soft edge of GUE minor process The correlations for the GUE minor process are given by (5.198), with all quantities referring to the Gaussian case. They permit a soft edge scaling, in which the species are separated by O(N 2/3 ) [215]. The limiting correlation is the so-called dynamical extension of the Airy kernel, specified by soft ρsoft ((τj , yj ), (τk , yk ))]j,k=1,...,n , (n) ((τ1 , y1 ), . . . , (τn , yn )) = det[K

where K

soft

⎧  ⎪ ⎪ ⎨ ((τx , x), (τy , y)) =

∞

0

⎪ ⎪ ⎩ −

e−(τy −τx )u Ai(x + u)Ai(y + u) du, 0

−∞

(7.19)

τy ≥ τx ,

e−(τy −τx )u Ai(x + u)Ai(y + u) du, τy < τx

(see Section 11.7 below). P ROPOSITION 7.1.4 In the Gaussian case of (5.198), introduce the scalings si = N − 2ci N 2/3 , One has

 lim

N →∞

Yi yi = (2si )1/2 + √ 1/6 . 2si

r 1 √ ρ(r) ({(sj , yj )}j=1,...,r ) = det[K soft ((cj , Xj ), (ck , Xk ))]j,k=1,...,r . 2N 1/6

(7.20)

Proof. Substituting in (5.198) the appropriate Gaussian quantities we obtain, for cl ≥ cj , 2 sl 1 e−yj X Hs −k (yj )Hsl −k (yl ), K(sj , yj ; sl , yl ) = √ π k=1 2sl −k (sl − k)! j

(7.21)

while for cl < cj the r.h.s. is to be modified by multiplying by −1 and changing the summation to k ∈ Z≤0 . Only the former case will be considered explicitly, as the latter is essentially the same. For the analysis of the sum (7.21) we

289

SCALED LIMITS AT β = 1, 2 AND 4

substitute (7.9) to obtain K(sj , yj ; sl , yl ) ∼ e−N ×

1/3

(Yj −Yl ) −(cj −cl )N 2/3 21/2 N −1/6

N „ X k=1

2

(N − 2cj N 2/3 − k)! (N − 2cl N 2/3 − k)!

«1/2 Ai(Yj + 2k/(2N )1/3 )Ai(Yl + 2k/(2N )1/3 ).

(7.22)

Noting that the leading order contribution to the summation comes from k of order N 1/3 , then using Stirling’s formula to simplify the ratio of factorials in this regime, we can recognize the sum as the Riemann sum approximation to the first integral in (7.20).

7.1.6 Global limit of density The scaled global density (1.53) was computed in Exercises 1.6 q.1. Here, following [257], we will show how this same result can be derived from the formula ρ(1) (x) = KN (x, x), where KN (x, x) is given by (5.13) √ with the quantities therein specified in the Gaussian case of Section 5.4.1, but with x scaled x → 2Nx, so 2 the weight becomes e−2N x . With this scaling the relevant monic orthogonal polynomials are √ −3n/2 −n/2 N Hn ( 2Nx). p˜(G) n (x) = 2 Making use of the differentiation formula for the Hermite polynomials we have that d (G) (G) p˜ (x) = n˜ pn−1 (x), dx n and thus e−2N x

(G)

KN (x, x) =

2

  (G) (G) (G) N (˜ pN −1 (x))2 − (N − 1)˜ pN −2 (x)˜ pN (x) .

(7.23) (G) (G) (˜ pN −1 , p˜N −1 )2 √ The asymptotic form of Hn ( 2N x) for n near N is given by the Plancheral-Rotach formula [508]. It tells us that for x ∈ (−1, 1)     1  2 2−m (G) (G) (G) N x2 , p˜N +m (x) = (˜ p , p ˜ ) e cos f (x) − m arccos x 1 + O 2 N N N π (1 − x2 )1/4 N   π 1 fN (x) := N x 1 − x2 − N + arccos x + . (7.24) 2 4 Substituting this in (7.23), with N → N + 1 therein, and making use of the trigonometric identity cos2 a − cos(a − b) cos(a + b) = 1 − cos2 b, we reclaim (1.54). Asymptotic analysis can also be carried out on integral representations of (7.23) [341], [258]. This allows correction terms to the Wigner semicircle law to be obtained. In particular, with |x| < 1 fixed, one finds  1   √ 2 cos(2N πP (x)) 1 +O , (7.25) 2/N ρ(1) ( 2Nx) = ρW (x) − 3 W 2 π (ρ (x)) N N2 where ρW (x) =

2 1 − x2 , π

P (x) = 1 +

x W 1 ρ (x) − arccos x. 2 π

(7.26)

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CHAPTER 7

The analysis of [341] also applies at the soft edge, where it gives 1 X ρ(1) ((2N )1/2 + 1/2 1/6 ) 21/2 N 1/6 2 N  1 1 1 2   2 2 (7.27) = ρsoft,2 + O (1) (X) − 20 3X (Ai(X)) − 2X(Ai (X)) − 3Ai(X)Ai (X) N N 2/3 with ρsoft,2 (1) (X) given by (7.13) (the 2 in the superscript soft,2 indicates β = 2; below we will encounter ρsoft,β (1) (X) for other values of β). E XERCISES 7.1

1. Show from (7.8) that ∞ “ |k| ”2j ˜ β) ˛˛ ∂ S(k; 1 |k| X =− , + ˛ ∂β 4π j=1 2j(2j − 1) 2π β=2

|k| < 2π.

(7.28)

7.2 SCALED LIMITS AT β = 2 — LAGUERRE AND JACOBI ENSEMBLES 7.2.1 Laguerre ensemble — hard edge The global density for the Laguerre ensemble is predicted by Proposition 3.2.3. In particular the density of the large eigenvalues is identical to the density of the large eigenvalues in the Gaussian ensemble. In contrast, the small eigenvalues in the Laguerre ensemble bunch together as the origin is approached, giving rise to the x−1/2 divergence. Due to the hard wall at x = 0 in the Laguerre ensemble this is referred to as the hard edge of the spectrum, whereas the boundary of the support of the large eigenvalues is a soft edge, as there is no wall in the statistical mechanics interpretation. To compute the correlation functions in the neighborhood of the origin, we first change scales so that the average inter-particle (inter-eigenvalue) spacing is O(1) in the N → ∞ limit. From the large N asymptotic formula [508, p.199] Γ(N + a + 1) Ja (2(M x)1/2 ) + O(N a/2−3/4 ), N! ∼ N a/2 Ja (2(N x)1/2 ),

e−x/2 xa/2 LaN (x) ∼ M −a/2

a > −1, (7.29)

where M = N + (a + 1)/2 and Ja (z) denotes the Bessel function, we see that the appropriate choice of scale is provided by the change of variable x=

X 4N

(7.30)

(the factor of 14 is chosen for convenience). With this scale, by using the above asymptotic formula and Stirling’s formula, the Christoffel-Darboux summation is readily evaluated in the N → ∞ limit [189]. P ROPOSITION 7.2.1 We have

1 (L)  X Y  KN , N →∞ 4N 4N 4N Ja (X 1/2 )Y 1/2 Ja (Y 1/2 ) − X 1/2 Ja (X 1/2 )Ja (Y 1/2 ) = , 2(X − Y )

Kahard (X, Y ) := lim

(L)

(7.31)

(L)

where KN (x, y) is given by Proposition 5.1.3 with pN (x) = pN (x). By taking the limit Y → X in the above formula and using the Bessel function identities α uJα (u) = αJα (u) − uJα+1 (u), Jα (u) = Jα−1 (u) − Jα (u), u

(7.32)

291

SCALED LIMITS AT β = 1, 2 AND 4

we see from (7.31) that

 1 (Ja (X 1/2 ))2 − Ja+1 (X 1/2 )Ja−1 (X 1/2 ) . 4 The Bessel function identities also show that (7.31) can be rewritten Kahard (X, X) =

Kahard (X, Y ) =

(7.33)

X 1/2 Ja+1 (X 1/2 )Ja (Y 1/2 ) − Y 1/2 Ja+1 (Y 1/2 )Ja (X 1/2 ) . 2(X − Y )

(7.34)

7.2.2 Laguerre ensemble — soft edge It was remarked above that at the soft edge the global density in the Laguerre ensemble is the same as the global density at the spectrum edge in the Gaussian ensemble. In fact, use of the asymptotic expansion [508, p.201]   e−x/2 Lan (x) = (−1)n 2−a−1/3 n−1/3 Ai(t) + O(n−2/3 ) , (7.35) where x = 4n + 2a + 2 + 2(2n)1/3 t, shows that (L)

lim 2(2N )1/3 KN (4N + 2(2N )1/3 X, 4N + 2(2N )1/3 Y ) = K soft (X, Y ).

N →∞

(7.36)

Thus, with this scaling of the coordinates, all correlation functions are those of the Gaussian ensemble at the soft edge. As for the Gaussian kernel, correction terms to the l.h.s. of (7.36) regarded as a function of N can (L) be computed using integral representations of KN . For the corresponding density, one finds [341], [258]  2 a (L) Ai(X) (X) + + O(N −2/3 ). (7.37) 2(2N )1/3 ρ(1) (4N + 2(2N )1/3 X) ∼ ρsoft,2 (1) (2N )1/3

7.2.3 Laguerre ensemble — global density (L)

The result (3.57) for the global density can be derived by computing the large N limit of KN (4N y, 4N y). One approach is to make use of the differentiation formula for the Laguerre polynomials, as given in (7.130) (L) below, to write the formula for KN in terms of {Lα n } for certain n and α, and then to make use of the Laguerre analogue of the Plancheral-Rotach formula (see, e.g., [202])   1   (L) xa/2 e−x/2 Lan+m (x)|x=4nX = (−1)n+m (2π X(1 − X))−1/2 na/2−1/2 gm,n , (X) + O    n √ √ (L) gm,n (X) = sin 2n( X(1 − X) − arccos X) − (2m + a + 1)arccos X + 3π/4 . (7.38) (L)

An alternative approach is to make use of a contour integral form of KN . This has the advantage of allowing corrections to (3.57) to be computed, giving the expansion up to terms O(N −2 ) [258]  cos((2N + a)πP (X) − aπ(1 + XρMP (X)))  1 a MP 4ρ(1) (4N X) ∼ ρMP (X) − , −  4πX(1 − X) 2π X(1 − X) N (7.39) where  √ 2 1−x 2 MP , PMP (x) := 1 + xρMP (x) − arccos x, ρ (x) := π x π valid for 0 < X < 1.

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CHAPTER 7

7.2.4 Hard edge scaling of the perturbed Laguerre kernel Analogous to the situation with the perturbed Hermite kernel, its Laguerre counterpart (5.169) admits a scaling at both the soft and hard edges [29], [132]. At the soft edge the eigenvalues {xj }j=1,...,k and parameters {aj }j=1,...,r are to be scaled according to xj = 4N + 2(2N )1/3 Xj ,

sj 1 aj = − + , 2 2(2N )1/3

(7.40)

with the result r   ˜ (l) (Xα )Ai(l) (Xβ ) Ai lim (2(2N )1/3 )k ρ(k) (x1 , . . . , xk ) = det K soft (Xα , Xβ ) +

N →∞

l=1

α,β=1,...,k

, (7.41)

as found in (7.17). We will concentrate on the hard edge scaling. The limiting functional form involves incomplete multiple Bessel functions    exw−1/4w p−1 e−xz+1/4z z a+r dz (p) (p) k=1 (w − sk ) dw ˜ p , J (x) = . J (x) = a+r w 2πi C{0,s1 ,...,sp } C{0} k=1 (z − sk ) 2πi Explicitly, the following result holds [132]. P ROPOSITION 7.2.2 For large N , with ak = 4N sk (k = 1, . . . , r)   X    1 ˜ (i) X = J˜(p) (X) + O 1 , (4N )a+r−p Λ(i) = J (p) (X) + O . (7.42) (4N )p−a−r−1 Λ 4N N 4N N Consequently r  Y (a+r)/2 1 a  X Y  hard KN , = K (X, Y ) + (7.43) J˜(p) (X)J (p) (Y ). lim  a+r N →∞ 4N 4N 4N ak =4N sk X p=1 Proof. The first term in (7.43) follows immediately from (5.169), (5.167) and (7.31), while the sum in (7.43) follows from (5.169) and (7.42). To derive (7.42), substitute ak = 4N sk in (5.168), change variables z → 4N z, w → 4N w, scale the contours and make use of the elementary limit (1 + u/N )N → eu .

From the integral representation of the Bessel function  −1 ex(z−z )/2 dz , Jα (x) = z α+1 2πi C{0} together with the formula J−α (x) = (−1)α Jα (x), both valid for α ∈ Z, we can check from the definitions that √ J (1) (x) = (4x)(a+r−1)/2 Ja+r−1 ( x),  x √ t−(a+r+1)/2 es1 (t−x) Ja+r+1 ( t) dt J˜(1) (x) = −2−(a+r+1) −∞  x √ √ −(a+r)/2 Ja+r ( x) − s1 2−(a+r) t−(a+r)/2 es1 (t−x) t−(a+r)/2 Ja+r ( t) dt. = (4x) −∞

Here the final line follows on integrating by parts, making use of the identity 1 d −α (u Jα (u)). u du Furthermore, making use of the form (7.34), together with the Bessel function three-term recurrence u−α−1 Jα+1 (u) = −

tJα+2 (t) = 2(α + 1)Jα+1 (t) − tJα (t),

(7.44)

293

SCALED LIMITS AT β = 1, 2 AND 4

it is straightforward to verify that  Y 1/2 1 hard Ka+1 (X, Y ) = Kahard (X, Y ) − X −1/2 Ja+1 (X 1/2 )Ja (Y 1/2 ). X 2 As a consequence of these facts, in the case r = 1 (7.43) can be written in terms of Bessel functions according to  X  Y a/2   √ √ s1 Kahard (X, Y ) − Ja ( Y )X a/2 es1 (t−X) t−(a+1)/2 Ja+1 ( t) dt . X 2 −∞ 7.2.5 Jacobi ensemble For the Jacobi ensemble, Proposition 3.6.3 predicts the same global eigenvalue density in the neighborhood of the edges x = ±1 as for the Laguerre ensemble in the neighborhood of the edge x = 0. From the Jacobi polynomial large N asymptotic formula [508, p.197]  x   x1/2 −a (a,b) PN 1− ∼ Ja (x1/2 ) (7.45) 2N 2 2N X we see that by making the shift of origin and change of scale x = 1 − 2N 2 the scaled Christoffel-Darboux summation can be evaluated in the N → ∞ limit to give the same expression as in Proposition 7.2.1. Thus the scaled correlations in the neighborhood of the edges x = 0 and x = 1 of the Laguerre and Jacobi ensembles, respectively, are the same. This is to be expected as in the neighborhood of these points the corresponding Boltzmann factors are proportional.

7.2.6 Circular Jacobi ensemble — spectrum singularity Consider the n-point correlation function (5.55). Our interest is in the neighborhood of the point z = −1, which is analyzed by writing xj → xj + L/2 or equivalently zj → −zj . In preparation for taking the thermodynamic limit, we make use of the formula  2  Γ(β + 1 + 2n + α)  x − 1 n , Pn(α,β) (x) = 2 F1 − n, −n − α; −2n − α − β; − n!Γ(β + 1 + n + α) 2 x−1 where 2 F1 denotes the Gauss hypergeometric function (5.83), to write  1 + z  2 N (CJ) = i−N − i pN 2 F1 (−N, a; 2a; 1 − z), 1−z 1−z  1 + z  2 N −1 (CJ) = i−(N −1) − pN −1 i 2 F1 (−(N − 1), a + 1; 2a + 2; 1 − z). 1−z 1−z To take the thermodynamic limit we use the formulas lim 2 F1 (−n, b; c; t/n) = 1 F1 (b; c; −t),

n→∞

1 F1 (b; c; −t) :=

∞ (b)n (−t)n , (c) n! n n=0

 1  x −(a−1/2) ix F (a; 2a; 2ix) = Γ a + e Ja−1/2 (x), 1 1 2 2 which give

 1 + z   (−1)N 1 j  (πρx/2)−(a−1/2) Ja−1/2 (πρx), i ∼ Γ a+  N 1 − zj α=N +a (sin πx/L) 2  1 + z   (−1)N −1 3 j  (CJ) (πρx/2)−(a+1/2) Ja+1/2 (πρx). ∼ Γ a + pN −1 i  1 − zj α=N +a (sin πx/L)N −1 2 (CJ)

pN

(7.46)

(7.47)

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CHAPTER 7

Using (5.48) to evaluate the normalization in (5.55) and using the duplication formula (4.180) we find that in the thermodynamic limit [418]   s.s. ρ(n) (x1 , . . . , xn ) = det K s.s. (xj , xk ) , j,k=1,...,n   Ja+1/2 (πρx)Ja−1/2 (πρy) − Ja+1/2 (πρy)Ja−1/2 (πρx) , K s.s. (x, y) := (πρx)1/2 (πρy)1/2 2(x − y) (7.48) where ρ := N/L, which is valid for all values of x1 , . . . , xn in the case a a non-negative integer, while it is valid for x1 , . . . , xn all positive otherwise. In the limit y → x we have  (πρ)2 x  2a (Ja−1/2 (πρx))2 + (Ja+1/2 (πρx))2 − Ja−1/2 (πρx)Ja+1/2 (πρx) . (7.49) K s.s. (x, x) = 2 πρx Using the Bessel function formulas  2 1/2  2 1/2 sin x, J−1/2 (x) = cos x, J1/2 (x) = πx πx together with the addition formula for the sine function, we see that  sin πρ(x − y)  =: K bulk (x, y). = K s.s. (x, y) π(x − y) a=0

(7.50)

This is expected because with a = 0 the circular Jacobi ensemble corresponds to Dyson’s circular ensemble, and the latter is locally identical to the bulk of the Gaussian ensemble. Another relationship between (7.48) and the bulk correlation of Proposition 7.1.1 is given in Exercises 7.2 q.3.

7.2.7 The classical groups There are three possible scalings of k-point correlations for the classical groups specified by Proposition 5.5.3 and (5.87). One is the bulk scaling with unit density. For the CUE this is achieved by θ → 2πx/N , while for the cases of Proposition 5.5.3 one requires θ → π/2 + πx/N . A simple calculation gives ρbulk (k) as specified by the result of Proposition 7.1.1 with ρ = 1 as the scaled k-point correlation. In addition to the bulk scaling, there are two possible edge scalings—when there is an eigenvalue at the endpoint, and when there is not. Typical of the former case is O+ (2N + 1) in the neighborhood of θ = 0. A straightforward calculation shows  π k + O (2N +1) ρ+ ρ(k) (πx1 /N, . . . , πxk /N ) (k) (x1 , . . . , xk ) := Nlim →∞ N  sin π(x − x ) sin π(x + x )  j l j l − . (7.51) = det π(xj − xl ) π(xj + xl ) j,l=1,...,k Typical of the latter is O− (2N + 1) in the neighborhood of θ = 0. It follows from Proposition 5.5.3 that  π k − O (2N +1) (x , . . . , x ) := lim ρ(k) (πx1 /N, . . . , πxk /N ) ρ− 1 k (k) N →∞ N  sin π(x − x ) sin π(x + x )  j l j l + . (7.52) = det π(xj − xl ) π(xj + xl ) j,l=1,...,k The two edge scalings can be related to the hard edge result (7.31). Thus we see that with Kach (X, Y ) := 2π 2 (XY )1/2 Kahard ((πX)2 , (πY )2 )

(7.53)

295

SCALED LIMITS AT β = 1, 2 AND 4

determining the edge correlations in the ensemble (3.4) with β = 2 and the bulk density unity, ch (X, Y ) = K±1/2

sin π(X − Y ) sin π(X + Y ) ± , π(X − Y ) π(X + Y )

(7.54)

where use has been made of (7.50). E XERCISES 7.2

1. Use the asymptotic formulas (7.9) and (7.29) to deduce from (5.70) and (5.71) that Z ∞ Z √ √ 1 1 K soft (X, Y ) = Ai(X + t)Ai(Y + t) dt, Kahard (X, Y ) = Ja ( Xt)Ja ( Y t) dt. 4 0 0

Note too the integral formula K bulk (x, y) =

ρ 2

Z

1

eπiρ(x−y)t dt.

−1

2. The objective of this exercise is to show the connection between the quantities K soft (X, Y ), Kahard (X, Y ), K s.s. (X, Y ) and K bulk (X, Y ). (i) Use the asymptotic expansions (7.69) and (7.74) below to deduce that Z

√ c+πρx/ c

lim

c→∞

Z ρsoft (1) (−X) dX = ρx,

√ c+2πρx c

lim

c→∞

c

ρhard (1) (X) dX = ρx

c

so the densities are asymptotically constant in the variable x. (ii) Use the asymptotic expansions (7.68) and (7.73) to show √ √ πρ lim √ K soft (−(c + πρx/ c), −(c + πρy/ c)) c √ √ √ = lim 2πρ cKahard (c + 2πρx c, c + 2πρy c) = K bulk (x, y).

c→∞

c→∞

(iii) Use the asymptotic expansion [539] Ja (x) ∼

“ 2 ”1/3 a

“ 21/3 (a − x) ” Ai , x1/3

(7.55)

valid for a and x large such that the argument of the Airy function is order one, to show that lim 2a(a/2)1/3 Kahard (a2 − 2a(a/2)1/3 x, a2 − 2a(a/2)1/3 y) = K soft (x, y).

a→∞

p (iv) With φ(x) := x/2Ja+1/2 (x), show by using the Bessel function identities (7.32) that for ρ = 1/π (7.48) can be rewritten φ(x)yφ(y) − φ(y)xφ (x) K s.s. (x, y) = . x−y (v) Use the asymptotic expansion (7.55) in the result of (iv) to show that ” “ lim (a/2)1/3 K s.s. a − (a/2)1/3 x, a − (a/2)1/3 y = K soft (x, y). a→∞

pj (x)} (vi) Consider the chiral ensemble (3.4) with β = 2 and α = a + 12 (note that this is defined on R+ ). Let {˜ 2

be orthonormal polynomials with respect to the weight function w2 (x) = |x|2a+1 e−x , x ∈ R. Show that KN (x, y) = (w2 (x)w2 (y))1/2

N−1 X

p˜2ν (x)˜ p2ν (y)

ν=0

= (w2 (x)w2 (y))1/2

2N−1 X “ ν=0

” pν (y) + p˜ν (x)˜ pν (−y) , p˜ν (x)˜

296

CHAPTER 7

where the second line follows from the parity of the p˜ν (x). Conclude from this that “ ”˛ ˛ 2(XY )1/2 K hard (X 2 , Y 2 ) = K s.s. (X, Y ) + K s.s. (X, −Y ) ˛ a →a+1/2 . ρ=1/π

p2μ−1 (x) = p˜2μ (x)|a→a−1 and use this to show (vii) With {˜ pj (x)} as in (vi), note that x˜ ˛ ˛ ˛ “ ˛ ˛ ˛ K s.s. (X, Y )˛ = (XY )1/2 K hard (X 2 , Y 2 )˛ + K hard (X 2 , Y 2 )˛ a→a−1/2

ρ=1/π

3. Argue that

˛ ˛ s.s. ρbulk (n+1) (x1 , . . . , xn , 0)/ρ = ρ(n) (x1 , . . . , xn )˛

a=1

” a→a+1/2

.

,

(7.56)

s.s. where ρbulk (n+1) refers to the correlation function in Proposition 7.1.1 while ρ(n) refers to the correlation function (7.48) for the spectrum singularity. Use the trigonometric formula for J1/2 (x) in (7.50), an analogous formula for J3/2 (x) deducible from (7.50) and the three term recurrence (7.44) together with elementary row operations in the determinant formula for the l.h.s. to check this directly.

4. [215] In this exercise the soft edge limit of the Laguerre case of (5.198) will be analyzed. Explicitly, it will be shown that with the scalings si = N − s˜i , s˜i := 2ci (2N )2/3 , one has

“ lim

N→∞

2(2N )2/3 )

”r

yi = 4si + 2(a + N − si ) + 2(2N )1/3 Yi ,

ρ(r) ({(sj , yj )}j=1,...,r ) = det[K soft ((cj , Yj ), (ck , Yk ))]j,k=1,...,r ,

(7.57)

where K soft is specified by (7.19).

(i) Substitute in (5.198) the appropriate Laguerre quantities to obtain, for s˜j ≤ s˜l , a+˜ sj −yj

K(N − s˜j , yj ; N − s˜l , yl ) = yj

e

X Γ(N − s˜j − k + 1) (a+˜s ) (a+˜ s ) j (yj )LN−˜sll−k (yl ), (7.58) L Γ(N − k + a + 1) N−˜sj −k

N−˜ sl

k=1

and note that for s˜j > s˜l the r.h.s. is to be modified by multiplying by −1 and changing the summation to k ∈ Z≤0 . (ii) Use the generalization of (7.35) applicable for a = o(n) [334], p xa/2 e−x/2 Lan−k (x) = (−1)n−k (2n)−1/3 (n − k + a)!/(n − k)! ( „ “ 1/3 2k ” O(e−k/n ), −2/3 ) + O(n × Ai X + 1/3 (2n) O(1),

k≥0 k 1,

on the critical line Re(s) = 12 . Indeed, it has been noted [62] that there are formal similarities between a certain representation of the density of zeros of ζ(s) on the critical line (Proposition 7.5.4 below), and the Gutzwiller trace formula for the density of states in a chaotic quantum system without time reversal symmetry, so from this viewpoint the occurrence of the GUE distributions is not unexpected. On the other hand we know that in the large N bulk scaling the GUE distributions coincide with those for the CUE or equivalently U (N ). In fact, as will be seen subsequently, it is U (N ) eigenvalues which more accurately models the Riemann zeros. First we recall that ζ(s) is closely related to the prime numbers, as is seen from a product formula due to Euler. P ROPOSITION 7.5.1 For Re(s) > 1, ζ(s) =

 (1 − p−s )−1 , p

where the product is over all primes p (= 2, 3, 5,. . . ).

302

CHAPTER 7

Proof. Subtracting 2−s ζ(s) from the definition of ζ(s) gives ∞ X

ζ(s)(1 − 2−s ) =

n=1 n=2m,m∈Z+

1 . ns

Similarly, subtracting from this series 3−s times the series gives ζ(s)(1 − 2−s )(1 − 3−s ) =

∞ X ∗ 1 = 1−s + 5−s + 7−s + · · · , s n n=1

where the * indicates all terms n which are multiples of 2 or 3 are to be omitted. Subtracting all the primes to the power of −s in this fashion gives Y ζ(s) (1 − p−s ) = 1−s = 1 p

as required.

Our interest is in the statistical properties of the zeros of ζ(s) in the complex plane. The famous Riemann hypothesis asserts that all the zeros of ζ(s) with nonzero imaginary part lie on the so-called critical line Re(s) = 21 . As noted above such zeros are called the Riemann zeros. The Riemann hypothesis dates back to 1859. Over a century later, beginning with a conjecture of Montgomery [405] for the pair correlation function ∞ d(E)d(E + ε), d(E) := j=1 δ(E − Ej ) (Ej denotes the (positive) imaginary part of the jth zero), the statistical properties of the Riemann zeros with large imaginary part have been the subject of study. The Montgomery conjecture states that 1 + S(ˆ ε) := lim

E→∞ ε ˆ fixed

d(E)d(E + ε) ε sin2 πˆ = δ(ˆ ε) + 1 − , 2 d(E)d(E + ε) (πˆ ε)

(7.80)

¯ where the averages · are over a region [E, E + ΔE] such that 1  ΔE  E, and εˆ = ε/d(E) with ¯ d(E) denoting the mean spacing between zeros at point E on the critical line. Since in general S(x) = δ(x) + ρT(2) (x, 0)/ρ2 , we see from (7.65) that (7.80) is identical to S(x) for the GUE in the bulk limit. Indeed the so called Montgomery-Odlyzko law [352] asserts that after scaling the large Riemann zeros have the same distribution as the bulk eigenvalues for large GUE matrices. This conjecture has supporting evidence from large scale numerical computations [426], involving the accurate evaluation of the zero number of order 1020 along the critical line, and over 107 of its neighbors. Moreover for test functions f (x) with Fourier transform fˆ(k) supported on |k| < 2π, Montgomery rigorously proved that  2π  2π ˆ fˆ(k) dk = 1 |k|fˆ(k) dk, (7.81) S(k) 2π −2π −2π ˆ where Sˆ denotes the Fourier transform of S, which is consistent with (7.80) since S(k) is given by (7.4) (note ˆ ˜ that S(k) = S(k) for ρ = 1). This rigorous result has been extended to higher order correlations in [472]. By abandoning rigor, the general n-point correlation function for the limiting zeros along the critical line can be calculated exactly [73], and the result of Proposition 7.1.1 for the GUE obtained. Here we will give the latter argument in the simplest case (n = 2), and so derive (7.80). 7.5.2 Rigorous theory Since Proposition 7.5.1 gives that there are no zeros of ζ(s) for Re(s) > 1, to study the zeros it is necessary to analytically continue ζ(s). For this purpose a contour integral representation can be used. Furthermore, this representation can be used to establish a functional equation satisfied by ζ(s).

303

SCALED LIMITS AT β = 1, 2 AND 4

P ROPOSITION 7.5.2 We have ζ(s) = −

Γ(1 − s) 2πi

 C

(−z)s−1 dz ez − 1

where C is any simple contour starting at z = ∞ + iμ (μ > 0), looping around z = 0, and finishing at z = ∞ − iμ without crossing the real axis, and |arg(−z)| < π. Furthermore ζ(s) = 2s π s−1 sin(πs/2)Γ(1 − s)ζ(1 − s). Proof. For Re(s) > 1, the first result follows by writing on the portion of C with Re(z) > 0 X −kz 1 e , = ez − 1 k=1 ∞

noting that the contribution from other portions of C can be made arbitrarily small, then changing variables kz → z and integrating term by term using the formula [541] Z 1 1 =− (−z)s−1 e−z dz Γ(1 − s) 2πi C (cf. (5.156)). For Re(s) ≤ 1, the integral representation is the analytic continuation. The second result follows by supposing Re(s) < 0 and expanding C to a circle of infinite radius about the origin (the resulting contour integral vanishes), and calculating the original contour integral as 2πi times the sum of the residues of the poles at z = ±2πi, ±4πi, . . . .

It follows from Proposition 7.5.2 that the only singularity of ζ(s) is a simple pole at s = 1, that there are zeros (trivial zeros) at s = −2, −4, . . . and that the only other zeros of ζ(s) must occur in the critical strip 0 ≤ s ≤ 1. To investigate other zeros, Riemann introduced the entire function 1 1 s(s − 1)ζ(s)Γ(s/2)π −s/2 , (7.82) s := − it. 2 2 The Riemann hypothesis is equivalent to the statement that all zeros of ξ(t) are on the real t-axis. Use of the functional equation for ζ(s), the functional equation (4.5) for the gamma function and the duplication formula (1.111) shows that ξ(t) :=

ξ(t) = ξ(−t),

¯ = ξ(t). and thus ξ(t)

This last equation implies ξ(t) is real for real t. Using ξ(t), the number of zeros N (E) of ζ(s) in the critical strip with 0 ≤ Im(s) < E can be computed (see, e.g., [516]). P ROPOSITION 7.5.3 We have ¯ (E) + Nosc (E), N (E) = N where ¯ (E) := 1 − 1 E log π − 1 Im log Γ N 2π π



1 iE − 4 2

 ,

1 Nosc (E) := − Im log ζ π



 1 − iE . 2

Proof. In terms of ξ(t), N (E) is equal to the number of zeros of ξ(t) in the region − 21 ≤ Im(t) ≤ 12 , 0 ≤ Re(t) < E. According to the residue theorem, this is given by N (E) = 

1 1 2 2πi

Z C

1 ξ  (t) dt = Im ξ(t) 4π

Z C

ξ  (t) dt, ξ(t)

where C is the rectangular contour with sides parallel to the real and imaginary t-axes, which passes through the points ±E and ±i( 12 + α), with α > 0. It is assumed that E is not a zero of ξ(t). The factor of 1/2 comes from the fact that ¯ ξ(t) is even. Since ξ(t) is even and ξ(t) = ξ(t), the integral over C  can be replaced by 4 times the integral over C1 ,

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where C1 is the contour over the portion of C  in the upper right quadrant of the complex t plane. This gives N (E) =

1 Δ arg ξ(t), π

where Δ argξ(t) refers to the change in argument of ξ(t) in going from t = E to t = i( /2 + α) along C1 . But at t = i( 21 + α), ξ(t) > 0 and thus N (E) = −

1 1 arg ξ(E) = − Im log ξ(E). π π

The result now follows by substituting (7.82).

7.5.3 Heuristic argument The above theory is entirely rigorous. However, to achieve our goal of establishing (7.80), it is necessary to proceed heuristically. We begin with a formula for the density of zeros, and assume the validity of the Riemann hypothesis, which gives that all zeros of ξ(t) are real. P ROPOSITION 7.5.4 Let E1 , E2 , . . . denote the positive zeros of ξ(E). We have d(E) :=

∞

δ(E − Ej ) =

j=1

d ¯ N (E) = d(E) + dosc (E), dE

where E 1 ¯ log , d(E) ∼ 2π 2π

dosc (E) = −

∞ 1 log p cos(Ek log p). π p pk/2 k=1

¯ ¯ (E) Derivation. To derive the formula for d(E), which is rigorous, we simply use Stirling’s formula in the formula for N in Proposition 7.5.3 and differentiate. To derive the formula for dosc (E), which is heuristic, we substitute the Euler product for ζ( 12 − iE) in the formula for Nosc (E) in Proposition 7.5.3 (note this is not justified since the product only converges for Re(s) > 1). Expanding log(1 − p−s ) and differentiating gives the stated result.

In an obvious notation, this result gives Nosc (E) = −

∞ 1 exp(− k2 log p) sin(Ek log p). π p k

(7.83)

k=1

It should be remarked [63], [64] that in the study of classically chaotic quantum systems, the so-called trace formula gives a formula of the same structure for the oscillating part of the quantum spectrum, as calculated from the classical data. Thus, after a simplifying approximation, one has for  → 0 ∞ k 1 exp(− k2 λp˜Tp˜) πk  Nosc (E) ∼ sin Sp˜(E) − μp˜ . (7.84) π k  2 p˜ k=1

Here p˜ labels primitive periodic orbits (i.e., orbits traversed once), k labels their repetitions, Sp˜(E) is the action of the primitive orbit with the property that the period of the latter is given by Tp˜ = ∂Sp˜/∂E, λp˜ is the instability exponent and μp˜ is the Maslov phase. The formulas (7.83) and (7.84) coincide (apart from an overall minus sign; see [69] for a discussion of the possible significance of this) with the primes as primitive periodic orbits of period log p, while λp˜ = 1 and μp˜ = 0. Also, because time reversal symmetry would require all non-self-retracing orbits to have a partner, the fact that the prefactor in (7.83) is 1/π rather than 2/π implies the analogous quantum system does not possess time reversal symmetry. Note that if we average d(E) over some region [E, E + ΔE] with 1  ΔE  E, Proposition 7.5.4 gives ¯ d(E) = d(E). Use of Proposition 7.5.4 also allows d(E)d(E + ε), where · denotes this same average,

305

SCALED LIMITS AT β = 1, 2 AND 4

to be reduced into a form suitable for further analysis. P ROPOSITION 7.5.5 For large E we have ¯ d(E ¯ + ε) + dosc (E)dosc (E + ε) d(E)d(E + ε) ∼ d(E) ¯ d(E ¯ + ε) + dosc (E)dosc (E + ε)diag + dosc (E)dosc (E + ε)off , = d(E) where log2 p 1 e−iε log p , Re 2π 2 p p / 0 log2 p −iε log p 1 e dosc (E)dosc (E + ε)off ∼ 2 Re cos Eh/p , π p h p

dosc (E)dosc (E + ε)diag ∼

hp

p denoting a prime, and where h is such that p + h is prime. Derivation. The first asymptotic equality follows by substituting the formula of Proposition 7.5.4 for d(E) and d(E + ε), ¯ ¯ ¯ and noting that due to the averaging d(E)d osc (E + ε) ∼ 0. The averaging does not affect d(E)d(E + ε) so the first term of the equality follows. This leaves the final asymptotic expressions. Now + * ∞ 1 X X log p log p   dosc (E)dosc (E + ε) = 2 cos(Ek log p) cos((E + ε)k log p ) . π pk/2 pk /2   p,p k,k =1

Due to the averaging, we can replace the cosine terms by 1 cos(Ek log p − (E + ε)k log p ) 2 (the term involving (1/2) cos(Ek log p + (E + ε)k log p ) will vanish). Furthermore, again due to the averaging, only terms with k log p ∼ k log p are relevant in the limit E → ∞. Now for either k > 1 or k > 1 the sum over these contributions converges, so we expect the sum to be dominated by the terms with k = k = 1, which formally diverges. Thus + * X log p log p iE(log p−log p )−iε log p 1 . Re e dosc (E)dosc (E + ε) ∼ 2π 2 p1/2 p1/2 p,p P P P Now split the double sum into two parts p,p = p=p + p =p . The first sum gives the first of the last two equations in the proposition. To obtain the last equation, make the ordering p < p (and thus multiply the sum by 2) and notice that the leading order contribution comes from primes p ∼ p. Writing p = p + h, expanding the logarithm to first order in h/p, and noting that for h  p, e−iEh/p ∼ cos Eh/p, gives the final stated equation.

A straightforward (heuristic) analysis gives the limiting value of the term ·diag in Proposition 7.5.5 (a rigorous analysis can also be given; in fact the result (7.81) was obtained in such a way). This requires use of the prime number theorem, which asserts that to leading order the number of prime numbers less than X is given by X/ log X. P ROPOSITION 7.5.6 We have lim

E,ε→∞ ¯ ε/d(E)= ε ˜

1 1 dosc (E)dosc (E + ε)diag ∼ − . 2 ¯ 2(πˆ ε)2 (d(E))

Derivation. The basic idea is to replace the sum in the formula ·diag given in Proposition 7.5.5 by an integral. According to the prime number theorem, to leading order the density of primes is 1/ log X, and so Z ∞ X log2 p −iε log p Z ∞ log X −iε log X ∼ dX = (2π)2 τ e−2πiτ ε dτ, e e p X 1 0 p

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CHAPTER 7

where the last equation follows from the change of variables log X = 2πτ . Finally note that for Re(z) > 0 Z ∞ Z ∞ 1 1 τ e−τ z dτ = 2 τ e−τ dτ = 2 , z z 0 0

(7.85)

and using this as an analytic continuation to evaluate the integral.

To mimic the strategy used to derive Proposition 7.5.6 requires an asymptotic formula not for the number of primes less than X, but for the more complicated quantity πm (X), the number of primes p not exceeding X and such that (p − m) is a prime. For this one uses the Hardy-Littlewood conjecture [290], which asserts πm (X) ∼ α(m)

X , log2 X

where α(m) (which was given explicitly by Hardy and Littlewood) has after averaging the large m behavior [353] α(m) ∼ 1 −

1 . 2m

(7.86)

P ROPOSITION 7.5.7 We have lim

E,ε→∞ ¯ ε/d(E)= ε ˆ

1 cos 2πˆ ε dosc (E)dosc (E + ε)off ∼ . 2 2 ¯ 2(πˆ ε) (d(E))

Derivation. The first step is to use the Hardy-Littlewood conjecture to replace the sums over p and h in the expression of Proposition 7.5.5 for ·off by the corresponding density α(k)/ log2 p to give Z ∞ Z dp −iε log p ∞ 1 e dh α(h) cos Eh/p. dosc (E)dosc (E + ε)off ∼ 2 Re π p 1 l(p) Here l = l(p) = O(log p), and the upper terminal of integration in the second integral has been set equal to infinity as only the region h  p contributes to the leading order. ¯ ¯ Now make two changes of variables: w = log p/(2π d(E)) and y = (2π)w E 1−w h. Since 2π d(E) ∼ log(E/2π) we w have p ∼ (E/2π) and y ∼ Eh/p. Thus the change of variables gives Z ∞ ¯ 2d(E) E w−1 ¯ dw exp(−2πid(E)εw) dosc (E)dosc (E + ε)off ∼ Re π (2π)w 0 fi „ w−1 «fl Z ∞ yE cos y. × dy α (2π)w w 1−w l(2π) E Consider the case w < 1. When E → ∞, E 1−w → ∞ so the second integral vanishes. When w > 1, E 1−w → 0 and E w−1 → ∞ so we can use (7.86) to conclude fi „ w−1 «fl (2π)w yE ∼1− α . w (2π) 2yE w−1 Multiplying by cos y and integrating, we take the first term to integrate to zero, while the second term is to leading order given by the small y behavior Z ∞ cos y ¯ dy ∼ (w − 1) log E ∼ (w − 1)2π d(E). y l(2π)w E 1−w Thus

Z ∞ ¯ d(E) ¯ ¯ e−2πid(E)εw (w − 1)(2π d(E)) dw. Re π 1 Evaluating the integral by changing variables w − 1 → w and using (7.85) establishes the result. dosc (E)dosc (E + ε) ∼ −

307

SCALED LIMITS AT β = 1, 2 AND 4

Combining the results of Propositions 7.5.5–7.5.7 gives the statement (7.80) of the Montgomery conjecture for εˆ = 0 (the delta function at εˆ = 0 follows from the definition). The above heuristic analysis can also be carried out without taking the limit E → ∞, although E is still assumed to be large. Then it is found that [64], [81] ∞   1 − m im log pε 2  1 ∂2  dosc (E)dosc (E + ε)diag = − 2 2 log |ζ(1 + iε)|2  exp e  , 4π ∂ε m2 p m p m=1   1 (1 − piε )2  ¯ dosc (E)dosc (E + ε)off = 2 |ζ(1 + iε)|2 Re e2πid(E)ε ) . (1 − 2π p−1 p Expanding as a function of ε gives [72] 1 (γ 2 + 2γ1 + c0 ) − 0 + O(ε2 ), 2 2 2π ε 2π 2   1  1 ¯ 2 2 2πid(E)ε , dosc (E)dosc (E + ε)off = 2 Re + (γ + 2γ + c ) + iQ + O(ε ) e 1 0 0 2π ε2 where γ0 , γ1 are specified by dosc (E)dosc (E + ε)diag = −

∞

ζ(1 + x) =

1 (−1)n + γn xn x n=0 n!

while c0 :=

p

(log p)2

∞ (n − 1) , pn n=1

Q :=

log3 p . (p − 1)2 p

¯ With εˆ := ε/d(E) this implies that for E → ∞   ε 1 sin2 πˆ Λ 1 2 , d (E)d (E + ε) = 1 − − sin (πˆ ε ) + O osc osc 2 2 3 ¯ ¯ ¯ (πˆ ε)2 (d(E)) (π d(E)) (d(E))

(7.87)

where Λ := γ02 + 2γ1 + c0 = 1.57314... On the other hand (5.87) and (5.86) give that for matrices from U (N ), with lengths scaled so that ρ = 1,  1  sin2 πx 1 2 ρ(2) (x, 0) = 1 − . (7.88) − sin πx + O (πx)2 3N 2 N4 Comparison of (7.87) and (7.88) gives agreement of the first three terms provided one makes the identification E ¯ 1 π d(E) . (7.89) =√ log N= √ 2π 3Λ 12Λ It should be noted that the U (N ) type correction term in (7.87) can be seen in Odlyzko’s numerical data [72]. 7.5.4 L functions The Riemann zeta function is an explicit example of a general class of functions referred to as L-functions. Such functions are Dirichlet series ∞ an L(s) = (7.90) ns n=1 with the an such that the series converges for Re(s) > 1, and with three additional properties. Briefly, these relate to the analytic continuation, functional equation and Euler product, which are all required to be similar

308

CHAPTER 7

to that for ζ(s). The generalized Riemann hypothesis asserts that all complex zeros of an L-function lie on the critical line Re(s) = 12 . Furthermore there is analytic [472] and numerical [471] evidence to suggest that the statistics of these zeros for large imaginary part agrees with the statistics of eigenvalues from large U (N ) matrices. In addition to this, Katz and Sarnak [352], [351] have related the statistical properties of the zeros on the critical line nearest the real axis for families of L-functions to the eigenvalues of large random matrices from the classical groups near θ = 0. One example of a family of L-functions are those associated with the quadratic Dirichlet characters, an = χd (n) in (7.90), for all allowed d. Here d is the fundamental discriminant, which can be either positive or negative. It has the further property of being square free and congruent to 1 mod 4 if it is odd, and of being 4 times a square free integer congruent to 2 or 3 mod 4 if it is even. The so called character χd (n) only takes on values ±1, 0 and has the periodicity property χd (n + |d|) = χd (n), with |d| being referred to as the conductor. (−2) (−1) Consider now the zeros on the critical line closest to the real axis, 1/2 + iγ (n) with · · · γd < γd < (1) (2) (j) 0 < γd < γd < · · · . Define the scaled variable αd log(|d|/2π), and define an ensemble by allowing d to vary and averaging. Analytic [352], [351] and numerical [471] evidence suggests that the distribution of the resulting zeros is that of the eigenvalues of matrices from the classical group Sp(2N ) for N large, about θ = 0. Moreover, an analogous result is expected to hold true for general families of L-functions, but with the correspondence to L-functions now relating to one of the classical groups U (N ), O+ (2N ), O− (2N + 1) or Sp(2N ). We refer to [403], [115] as starting points for studies into this topic.

7.6 SCALED LIMITS AT β = 4 — GAUSSIAN ENSEMBLE 7.6.1 The bulk ˜

In the Gaussian ensemble e−4V (x) = e−2V4 (

√

2x)

, which implies √ √ √ S4 (x, y) = 2S˜4 ( 2x, 2y).

(7.91)

In the bulk we must scale x and y as in Proposition 7.1.1. It turns out that then only the Christoffel-Darboux term in (6.59) contributes to the leading order asymptotics. P ROPOSITION 7.6.1 For the Gaussian ensemble with β = 4  πρ n  πρx πρxn  1 ρ(n) √ ,..., √ lim √ N →∞ 2N 2N 2N ⎡  2πρ(xj −xk ) sin 2πρ(xj − xk ) sin x 1 dx ⎢ 2πρ(x − x ) 2πρ x j k ⎢ 0 = ρn qdet ⎢ ⎣ ∂ sin 2πρ(xj − xk ) sin 2πρ(xj − xk ) ∂xj

2πρ(xj − xk )

⎤ ⎥ ⎥ ⎥. ⎦

2πρ(xj − xk )

Proof. In view of the above remarks, the only point which requires checking is that the second term in the expression (6.59) for S˜4 (x, y) does not contribute to

√ √ √ lim (πρ/ 2N )S4 (πρx/ 2N , πρy/ 2N ).

N→∞

This is a consequence of the N → ∞ asymptotic estimate [420] with (7.1).

R∞ x

e−t

2

/2

H2N−1 (t)dt = O(22N (N − 1)!) together

309

SCALED LIMITS AT β = 1, 2 AND 4

7.6.2 Properties of the two-point function With ρbulk (2) (x, y) now denoting the scaled two-point correlation function in the bulk, Proposition 7.6.1 gives 2   1 ∂ sin 2πρ(x − y)   2πρ(x−y) sin t T bulk 2 sin 2πρ(x − y) 2 dt. (7.92) +ρ ρ(2) (x, y) = −ρ 2πρ(x − y) 2πρ ∂x 2πρ(x − y) t 0 It follows from this that for small x 2 ρbulk (2) (x, 0) = ρ

while for large x,

 (2πρx)4 135

−

 2(2πρx)6 2(2πρx)8 + + O((ρx)10 ) , 4725 165375

(7.93)



  1  1 3 π cos 2πρx − 1 + O + 2 2πρx (2πρx)2 2(2πρx)4 (ρx)2      1 1  cos 4πρx π sin 2πρx 1+O +O , + − 2 (2πρx)2 2(2πρx)4 (ρx)2 (ρx)5

ρT(2)bulk (x, 0) = ρ2

(7.94)

where the terms O(1/x2 ) do not contain any oscillatory factors. Also, making use of the 2 × 2 determinant form of ρT(2) implied by Proposition 7.6.1, the dimensionless Fourier transform (7.3) can be computed as (see Exercises 7.6 q.1) - |k| |k| |k| 4π − 8π log |1 − 2π |, |k| ≤ 4π, . S(k) = (7.95) 1, |k| ≥ 4π. Note in particular the small |k| behavior |k| k2 . S(k) ∼ + + O(|k|3 ). 4π 16π 2

(7.96)

7.6.3 Perturbation about β = 4 in the bulk The expansion (7.5) tells us the first order correction in (β − β0 ) to the two-point correlation function at coupling β0 in terms of higher order correlations at β0 . As was the case at β0 = 2, these higher order correlations are known for β0 = 4. Also, according to (6.4) the expansion of a quaternion determinant is formally the same as that of an ordinary determinant allowing (7.5) in the case of β0 = 4 to be written in a form analogous to (7.6) and computed exactly. Moreover the Fourier transform of this first order correction can be computed exactly in terms of elementary functions together with the dilogarithm  x log t dilog(x) = dt. 1 1−t It has a different analytic form for each of the regions |k| < 2π, 2π < |k| < 4π, |k| > 4π. For |k| < 2π, the exact expression (which is rather lengthy) implies the expansion [207]  . β)  ∂ S(k; |k|3 |k| 5k 4 3|k|5 27k 6  + = − + + + ∂β β=4 16π 256π 3 3072π 4 4096π 5 81920π 6 37|k|7 1273k 8 887|k|9 4423k 10 + + + 245760π 7 18350080π 8 27525120π 9 293601280π 10 1949|k|11 + + ··· . 275251200π 11 +

(7.97)

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CHAPTER 7

7.6.4 Global limit of the density The global density ρ˜(1) (x) for general Gaussian β-ensembles with eigenvalue p.d.f. (1.28) is defined by (1.53). We read off from (6.55) and (7.91) that for β = 4  √ √ √ √ 2/Nρ(1) ( 2N x) = (2/ N )S˜4 (2 N x, 2 N x). To compute the large N form of this expression, we see from (6.59) that in addition the global limit of  ∞to −t 2 K2N , which is already known from (7.25), we must also compute the global limit of x e /2 H2N −1 (t) dt. For this, use of the integral evaluation implied by (7.146) below gives   ∞  x 2 π (2N − 1)! −t2 /2 − e H2N −1 (t) dt = e−t /2 H2N −1 (t) dt. 2 (N − 1/2)! x 0 The global limit of the integral on the r.h.s. can now be analyzed using the Plancheral-Rotach formula (7.24), and the following result obtained [341], [202]. √ P ROPOSITION 7.6.2 Let −1 < x < 1, set AN (x) := cos(2N x 1 − x2 + (2N + 12 )arcsin x − N π) and let ρW (x) denote the Wigner semicircle density. One has  1  √ 1 1 (−1)N  cos AN (x) √ 2/Nρ(1) ( 2N x) = ρW (x) − √ + + O(N −3/2 ). (7.98) + 2 1/4 2πN (1 − x ) 4πN 1 − x2 2πN 7.6.5 Soft edge Here we want to study (7.91) in the scaled limit with x and y given by (7.11), and thus to compute   1 X Y S4soft (X, Y ) := lim 1/2 1/6 S4 (2N )1/2 + 1/2 1/6 , (2N )1/2 + 1/2 1/6 . N →∞ 2 N 2 N 2 N Now, use of the asymptotic formula (7.68) shows that   1 X Y (G) lim 1/2 1/6 K2N 21/2 ((2N )1/2 + 1/2 1/6 ), 21/2 ((2N )1/2 + 1/2 1/6 ) N →∞ 2 N 2 N 2 N −1/3 soft 2/3 2/3 K (2 X, 2 Y ), =2 where K soft is given by (7.12). Writing out the explicit form of the second term in (6.59), substituting in (7.91) and introducing the scaled variables gives  ∞  √ √ 2 2−2N −1/2 −y2  −√ e−t H2N −1 ( 2t) dt x →(2N )1/2 +X/21/2 N 1/6 =: A4 (X, Y ). e H2N ( 2y) N Γ(2N ) x y →(2N )1/2 +Y /21/2 N 1/6 Use of the asymptotic expansion (7.9) shows 1 1 A4 (X, Y ) = − 2/3 Ai(22/3 Y ) N →∞ 21/2 N 1/6 2



∞

lim

Ai(22/3 v) dv.

X

Hence S4soft (X, Y ) =

1

2

K soft (22/3 X, 22/3 Y ) − 1/3

1

2

Ai(22/3 Y ) 2/3



∞

Ai(22/3 v) dv.

(7.99)

X

The density is obtained by setting X = Y in (7.99). According to (7.68) the second term decays as X → −∞. This implies √ 1 X soft 2/3 2/3 ρsoft , (7.100) (−X) ∼ K (−2 X, −2 X) ∼ (1) X→∞ 21/3 X→∞ π which is the same leading order behavior as that exhibited at β = 2 (recall (7.69)). Another feature of interest

311

SCALED LIMITS AT β = 1, 2 AND 4

relating to the density is the correction terms to the soft edge scaled form. According to (7.27) the leading correction to the first term in (6.59) is O(N −2/3 ). The correction to the second term is found by making use of (7.9). It is O(N −1/3 ), and gives the expansion [202]    2 1 X 1 soft,4 1/2 2/3 (2N ) ∼ ρ Ai(2 ρ + (X) + X) + O(N −2/3 ). (7.101) (1) (1) 21/2 N 1/6 21/2 N 1/6 24/3 (2N )1/3 From Proposition 6.1.7 we have that the truncated two-particle correlation is given by  ∂  Y T soft soft soft soft S (X, Y ) ρ(2) (X, Y ) = −S4 (X, Y )S4 (Y, X) − S4soft (X, Y  ) dY  . ∂X 4 X

(7.102)

We can check from (7.68) that only K soft in the formula (7.99) for S4soft contributes to the leading nonoscillatory behavior of ρT(2) (X, Y ) for X, Y → −∞. One finds ρT(2)soft (−X, −Y )

∼

X,Y →∞

−

8π 2

X +Y 1 √ , XY (X − Y )2

(7.103)

which is exactly one half of the asymptotic expression (7.70) found for the same quantity at β = 2. E XERCISES 7.6

1.

(i) Let

Z FT f = fˆ(k) :=

∞

f (x)eikx dx.

−∞

Verify that

Z

∞ −∞

f (y − x)g(x)eikx dx =

(ii) Let

1 2π

2

sin 2πx 6 2πx f4 (x) := 4 ∂ sin 2πx ∂x 2πx ( 1 , sin 2πx 2 FT = 2πx 0,

Use the formula

to check that

Z

2

1 6 2 FT f4 (x) = 4 ik − 2

3 i 2k 7 (|k| < 2π), 1 5 2

∞ −∞

Z

e−ily fˆ(l − k)ˆ g (l) dl.

(7.104)

3 sin 2πt dt 7 2πt 5. sin 2πx 2πx

x 0

|k| < 2π, |k| > 2π » FT f4 (x) =

0 0

0 0

–

(iii) Use the result (7.104) with y = 0, together with the result of (ii) to show “ ” FT f4 (x)f4 (−x) = 8 2 3 ˛ ˛ |k| |k| ˛ |k| ˛ > > 1− + log ˛1 − 0 ˛ > 7 > 6 4π 8π 4π > < 4 ˛ ˛ 5, |k| ˛ |k| |k| ˛ + log ˛1 − 0 1− ˛ > 4π 8π 2π » – > > > 0 0 > : , |k| > 4π 0 0

(|k| > 2π).

|k| < 4π, (7.105)

and thus deduce (7.95). 2. [221] Use the integral form of the kernels given in Exercises 7.1 q.1 to show that for scaled = bulk or soft Z ∞ 1 1 ∂ S˜4scaled (X, Y ) = K scaled (X, Y ) + K scaled (t, Y ) dt, (7.106) 2 2 ∂Y X

312

CHAPTER 7

where S˜4bulk (x, y) = S4bulk (x/2, y/2), S˜4soft (x, y) = 21/3 S soft (x/22/3 , y/22/3 ) and thus 1“ ∂ ∂ ˜scaled ∂ ” scaled (X, Y ) = (X, Y ), S4 − K ∂X 2 ∂X ∂Y Z Y Z ∞ Z ∞ ” “ 1 S˜4 (X, u) du = − K scaled (X, t) dt − K scaled (Y, t) dt . 2 Y X X

(7.107)

7.7 SCALED LIMITS AT β = 4 — LAGUERRE AND JACOBI ENSEMBLES 7.7.1 Hard edge In the Laguerre case

 ˜  e−4V (x) ∝ e−2V4 (2x) 

a →2a−1

and so

  S4 (x, y) = 2S˜4 (2x, 2y)

a →2a−1

.

(7.108)

Furthermore, the explicit form of S˜4 is, from (6.59),  ∞ 1  x 1/2 (L) (2N )! y (a−1)/2 e−y/2 La2N (y) K2N (x, y) + t(a−1)/2 e−t/2 La2N −1 (t) dt. S˜4 (x, y) = 2 y 4Γ(a + 2N ) x (7.109) The scaling at the hard edge is as specified in Proposition 7.2.1, so we seek to compute X Y  1 S4hard (X, Y ) := lim S4 , . N →∞ 4N 4N 4N Now, according to Proposition 7.2.1 1 (L)  X Y  K2N , = 2K hard(4X, 4Y ), lim N →∞ 4N 2N 2N while the asymptotic form of the second term in (7.109) is deduced from (7.29), giving in total    X 1/2 J2a−1 (2Y 1/2 ) ∞  S4hard (X, Y ) = 2 K hard (4X, 4Y ) + J2a−1 (2t) dt. (7.110) Y a →2a−1 2Y 1/2 X 1/2 Bessel function identities applied to this give the alternative form S4hard (X, Y

) = 2K

hard

  (4X, 4Y )

J2a−1 (2Y 1/2 ) − a →2a 2Y 1/2



X 1/2

J2a+1 (2t) dt

(7.111)

0

(see also Exercises 7.7 q.3). Note that in the case a = 0 the integral in (7.111) can be evaluated, and thus we have the simplified result     1   1/2 1/2 S4hard (X, Y ) = 2K hard (4X, 4Y ) − J (2Y ) J (2X ) − 1 . (7.112) 1 0 a=0 a=0 4Y 1/2 In the case of the Jacobi ensemble a similar calculation shows 1 ˜(J)  1 ˜(L)  X Y  X Y  , , = lim ,1 − S4 1 − lim S4 2 2 2 N →∞ 2N N →∞ 4N 2N 2N 4N 4N which is to be expected in accordance with the remarks following (7.45).

313

SCALED LIMITS AT β = 1, 2 AND 4

7.7.2 Asymptotics of the one- and two-point functions From Proposition 6.1.7 the density in the scaled limit near the hard edge is given by hard (X, X) ρhard (1) (X) = S4   1 (J2a+1 (2X 1/2 ))2 − J2a+2 (2X 1/2 )J2a−2 (2X 1/2 ) = 2  ∞ 1 1/2 J2a−1 (2X ) J2a−1 (2s) ds, + 2X 1/2 X 1/2

(7.113)

where the second equality follows from (7.111) and (7.33). For large X, use of (7.73) shows that the leading order decay of S4hard (X, X) is given by the leading order behavior of 2K hard (4X, 4X). Use of (7.74) then gives ρhard (1) (X)

∼

X→∞

1 , 2πX 1/2

(7.114)

which is identical to the behavior exhibited at β = 2. The truncated two-particle correlation is given by (7.102) with S4soft replaced by S4hard . For the leading asymptotics, we again see from (7.73) that the leading behavior of S4hard (X, Y ) is given by the leading behavior of 2K hard (4X, 4Y ). Substituting this in the modified form of (7.102) shows that ρT(2)hard (X, Y )

∼

X,Y →∞

−

8π 2

X +Y 1 √ XY (X − Y )2

(7.115)

independent of a. This is the same behavior as found in (7.103) for the soft edge at β = 4.

7.7.3 Laguerre ensemble — global and soft edge densities For finite N the density is given by (7.108) with x = y. The global density is then four times this expression with x → 4N X (recall (3.57)). The first term in (7.109) is the Christoffel-Darboux kernel, which has been analyzed in this limit to give (7.39). For the second term, use of (7.131) below allows the integral over (x, ∞) to be replaced by minus the same integral over (0, x). This latter integral can be analyzed using the Plancheral-Rotach type asymptotic expansion (7.38), with the final result being [202] (L)

4ρ(1) (4N X) ∼ ρ

MP

g0,2N (X)|a →2a−1 1 a  (X) − + + o(N −1 ), 2(πN )1/2 X 3/4 (1 − X)1/4 2πN X(1 − X)

(7.116)

(L)

where g0,2N (X) is specified by (7.38), and ρMP (X) is as in (7.39). At the soft edge, the expansion of the first term in (7.109) is given by (7.37), while that of the second term is determined by making use of (7.35). With ρsoft (1) (X) as in (7.101), one finds [202] 2(2N )1/3 ρ(1) (4N + 2(2N )1/3 X)  ∞  a   2/3 2/3 2 (Ai(2 ∼ ρsoft,4 (X) + X)) + Ai (2 X) Ai(t) dt + O(N −2/3 ). (7.117) (1) 2N 1/3 2/3 X 2

7.7.4 Circular Jacobi ensemble — spectrum singularity We recall from (3.123) that to study the log-gas on a circle with weight (5.50), it is sufficient to study the log-gas on a line with Cauchy weight as specified by (5.56). According to (3.123) the parameter α in (5.56) is dependent on N , β as well as the parameter a in (5.50). For purposes of using the general formula (6.59),

314

CHAPTER 7

it is convenient to define α independent of β, by setting e−2V (x) =

1 , (1 + x2 )α

α = N + a,

(7.118)

which according to (3.123) is indeed the correct choice at β = 2. Then (6.53) gives ˜

e−2V4 (x) =

1 , (1 + x2 )α−1

(7.119)

and comparison with (3.123) shows that the correct weight function for the Cauchy ensemble equivalent to ˜ the circular Jacobi ensemble at β = 4 is e−2V4 (x) | α →2α . This means that the quantity S˜4 (x, y)| α →2α gives α=N +a α=N +a the sought n-point correlation at β = 4. Now, with the weight function (7.118), use of (7.119) and (5.65) in (6.59) shows 1  1 + x2 1/2 (C) S˜4 (x, y) = K2N (x, y) 2 1 + y2  ∞ (1 + y 2 )−(α+1)/2 p2N −1 (t) (α − 2N )p2N (y) dt, (7.120) − 2(p2N −1 , p2N −1 )2 (1 + t2 )(α+1)/2 x where (C)

(CJ) K2N (x, y)

(C)

(C)

(C)

p2N (x)p2N −1 (y) − p2N (y)p2N −1 (x) 1 . := x−y (1 + x2 )α/2 (1 + y 2 )α/2

As in the analysis leading to (7.48), our task is to compute the thermodynamic limit of S˜4 with 1+z 1+w , z := e2πix/L , w = e2πiy/L . y → i (7.121) 1−z 1−w Analogous to (7.46) we have  1 + z   2 2N  (CJ) p2N i  α →2α = i−2N − 2 F1 (−2N, 2a; 4a; 1 − z), 1 − z α=N +a 1−z  1 + z   2 2N −1  (C) p2N −1 i  α →2α = i−(2N −1) − 2 F1 (−(2N − 1), 2a + 1; 4a + 2; 1 − z), 1 − z α=N +a 1−z (7.122) x → i

(C)

while (p2N −1 , p2N −1 )2

is specified by (5.54). Making use of the asymptotic formula (7.47) shows   1 + z 1 + w  2πρ|xy| s.s.   (CJ) ,i K (2x, 2y) , (7.123) K2N i  α →2α ∼ 1 − z 1 − w α=N +a N a →2a

where K s.s. is specified by (7.48). Regarding the second line in (7.120), making the substitutions (7.121) and (7.122), a similar analysis which leads to (7.123) shows that the leading asymptotic form is  πρx a − (πρy)3/2 J2a−1/2 (2πρy) s−1/2 J2a+1/2 (2s) ds. N 0 Hence

 1 + z 1 + w  πρy 2 s.s.  ,i S (x, y), S˜4 i  α →2α ∼ 1 − z 1 − w α=N +a N 4   S4s.s. (x, y) := K s.s. (2x, 2y)

a →2a

− aπρ

J2a−1/2 (2πρy) (πρy)1/2

 0

πρx

s−1/2 J2a+1/2 (2s) ds.

(7.124)

(7.125)

315

SCALED LIMITS AT β = 1, 2 AND 4

˜ 4 as specified by (6.15). One The result (7.124) can be used to deduce the asymptotic behavior of I˜4 and D finds  y   2 ∂  1 + z 1 + w  ˜ 4 i 1 + z , i 1 + w ∼ πρxy I˜4 i ,i ∼− S s.s. (x, y). S4s.s. (x, u) du, D 1−z 1−w 1−z 1−w N ∂x 4 x We substitute these asymptotic forms in the expression of (6.14) for f4 , form qdet according to Proposition 6.1.7, and extract common factors from the odd numbered columns and even numbered rows which give an overall contribution n  πρ 2 x . (7.126) N j j=1 This procedure gives the leading asymptotic form in the thermodynamic limit of ρ(n) as specified by the first equality in (5.55). The factor (7.126) cancels with the prefactors on the r.h.s. of the first equality in (5.55), leaving the final expression [213] x  − xjk S4s.s. (xj , u) du S4s.s. (xj , xk ) s.s. ρ(n) (x1 , . . . , xn ) = qdet . (7.127) ∂ s.s. S4s.s. (xk , xj ) ∂xj S4 (xj , xk ) j,k=1,...,n

When a = 0 the circular Jacobi ensemble reduces to the circular ensemble of Dyson. In this case (7.127) should agree with the scaled n-point correlation given in Proposition 7.6.1 for the Gaussian ensemble in the bulk. Indeed, we see from (7.125), (7.48) and (7.50) that  sin 2πρ(x − y)  S4s.s. (x, y) , =ρ 2πρ(x − y) a=0 which when substituted in (7.127) gives the result of Proposition 7.6.1. Another case of interest is a = 1. Then the formula (7.56) must apply, with the l.h.s. given by Proposition 7.6.1, and the r.h.s. by (7.127). In particular, taking n = 1 we must have   sin 2πρx 2  1 ∂ sin 2πρx   2πρx sin t  S4s.s. (x, 0) dt. (7.128) =ρ−ρ +ρ 2πρx 2πρ ∂x 2πρx t a=1 0 This can be checked using (7.125) and (7.49). E XERCISES 7.7

1. Analogous to Exercises 7.1 q.2, verify that √ √ πρ lim √ S4soft (−(α + πρx/ α), −(α + πρy/ α)) α √ √ √ = lim 2πρ aS4hard (a + 2πρx a, a + 2πρy a) = S4bulk (x, y),

α→∞

a→∞

lim 2a(a/2)1/3 S4hard (a2 − 2a(a/2)1/3 x, a2 − 2a(a/2)1/3 y) = S4soft (x, y),

a→∞

and with ρ = 1/π lim (a/2)1/3 S4s.s. (a − (a/2)1/3 x, a − (a/2)1/3 y) = S4soft (x, y).

a→∞

2. [3] The objective of this exercise is to verify that (7.109) can be rewritten as [544], [216] ˛ ˛ (L) (L) 2S˜4 (x, y) = K2N (x, y)˛ a→a+1 (a+1)/2 −y/2 La+1 (y) − La+1 (y) Z x e d (2N )!y 2N 2N−1 + t(a+1)/2 e−t/2 La+1 (t) dt. (7.129) 2Γ(2N + a + 1) y dt 2N 0 (y), k = 0, . . . , 2N in (7.109) and (7.129). The general strategy is to compare coefficients of y (a−1)/2 e−y/2 La+1 k

316

CHAPTER 7

This requires using the Laguerre polynomial identities La−1 (x) = Lan (x) − Lan−1 (x), n d a Ln (x) = −La+1 n−1 (x), dx a+1 a+1 xLa+1 (x) + (2n + a)La+1 n−1 (x) = −nLn n−1 (x) − (n + a)Ln−2 (x),

(7.130)

and the definite integral (see (6.113) and (6.114)) Z

(

∞

x

a/2 −x/2

e

La+1 (x) dx n

0

=

Γ((n+3)/2))Γ(a+n+2) , 2a/2−1 Γ(n+2)Γ((n+a+3)/2)

n even,

0,

n odd.

(7.131)

(i) Use the first and third identities in (7.130) to show that the coefficient of y (a−1)/2 e−y/2 La2N (y) in (7.129) is Z x 2 a+1 ” “d (2N )! 1 (a+1)/2 −x/2 ((2N − 1)!) L2N−1 (x) e t(a+1)/2 e−t/2 x (−2N )+ La+1 2N (t) dt, 2 Γ(2N )Γ(a + 1 + 2N ) 4Γ(2N + a + 1) 0 dt and read off that the same coefficient in (7.109) is Z ∞ (2N )! t(a+1)/2−1 e−t/2 La2N−1 (t) dt. 4Γ(a + 2N ) x Use the definite integral (7.131) to show that both these expressions agree when x = 0. Then use the identities (7.130) to show that the two expressions have the same derivative. (ii) Proceed as in (i) to show that for k < 2N , the coefficient of y (a−1)/2 e−y/2 Lak (y) in (7.129) is ” “ k! a+1 (x) x(a+1)/2 e−x/2 La+1 k+1 (x) − Lk 2Γ(a + 1 + k) while in (7.109) the coefficient is k! x(a+1)/2 e−x/2 Lak (x), 2Γ(a + 1 + k) and note from the first identity in (7.130) that these expressions are equal. (iii) Use (7.129) to show that an alternative expression for (7.110) is (7.111).

7.8 SCALED LIMITS AT β = 1 — GAUSSIAN ENSEMBLE 7.8.1 The bulk In the Gaussian ensemble V˜1 = V and so S˜1 (x, y) = S1 (x, y). Thus from (6.107) we read off that for N even  2 2 e−y /2 HN −1 (y) ∞ (G) (G) ˜ S1 (x, y) = KN −1 (x, y) + N 1/2 sgn(x − t)e−t /2 HN −2 (t) dt. (7.132) 2 π (N − 2)! −∞ Proceeding as in the proof of Proposition 7.6.1 we can obtain from this formula the scaling limit of the n-point correlation function in the bulk.

317

SCALED LIMITS AT β = 1, 2 AND 4

P ROPOSITION 7.8.1 For the Gaussian ensemble at β = 1 we have  πρ n  πρx πρxn  1 ρ(n) √ , . . . , √ lim √ N →∞ N N N ⎡  πρ(xj −xk ) 1 sin πρ(xj − xk ) sin t 1 dt − sgn(xj − xk ) ⎢ πρ(x − x ) πρ t 2ρ n j k 0 = ρ qdet ⎢ ⎣ ∂ sin πρ(xj − xk ) sin πρ(xj − xk ) ∂xj πρ(xj − xk ) πρ(xj − xk )

⎤ ⎥ ⎥ ⎦

.

j,k=1,...,n

7.8.2 Properties of the two-point function The truncated two-particle correlation, according to Proposition 7.8.1, is given by    πρx sin t  sin2 πρx 1  d sin πρx  π T 2 ρ(2) (x, 0) = ρ − − sgn x + dt . + (πρx)2 πρ dx πρx 2 t 0 This implies the small x expansion ρ(2) (x, 0) = ρ2

π 6

|πρx| −

 π 1 |πρx|3 + (πρx)4 − · · · 60 135

and the large x asymptotic expansion    1  cos 2πρx   1  1 3 1 + O + 1 + O , + ρT(2) (x, 0) = ρ2 − (πρx)2 2(πρx)4 x2 2(πρx)4 x2

(7.133)

(7.134)

(7.135)

where the terms O(1/x2 ) do not contain any oscillatory factors. Analogous to (7.95), the corresponding dimensionless Fourier transform can be computed in terms of elementary functions [395], ⎧   ⎨ |k| − |k| log 1 + |k| , |k| ≤ 2π, π 2π π . S(k) = (7.136) ⎩ 2 − |k| log |k|/π+1 , |k| ≥ 2π, 2π |k|/π−1 which exhibits the small |k| expansion |k| k2 . S(k) ∼ − 2 + O(k 3 ). π 2π

(7.137)

7.8.3 Perturbation about β = 1 in the bulk As in the cases of β0 = 2 and β0 = 4, the explicit form of the n-point correlation (n = 2, 3, 4) can be used to compute the first order perturbation of the dimensionless two-point correlation about β0 = 1, or equivalently the first order perturbation of the dimensionless structure function. However, as far as the |k| < π form of the ˜ β) in Section 13.7.4 below shows that latter goes this is superfluous, as an integral formula obtained for S(k; if we set πβ f (k; β) := S(k; β), 0 < k < min (2π, πβ) (7.138) |k| and define f for k < 0 by analytic continuation, then

 2k 4  ; . f (k; β) = f − β β

(7.139)

318

CHAPTER 7

Thus if we write ∞  |k| j πβ ˜ Aj (β/2) , S(k; β) = 1 + |k| πβ j=1

|k| < min(2π, πβ)

(7.140)

the coefficients Aj (β/2) must satisfy the functional relation Aj (1/x) = (−x)−j Aj (x).

(7.141)

˜ β)/∂β|β=4 we can use this to deduce the corresponding In particular, knowing the expansion (7.97) of ∂ S(k; ˜ β)/∂β|β=1 . expansion of ∂ S(k; 7.8.4 Global limit of the density Suppose for definiteness that N is even, and write ΦN (x) = 2

−N



x

e−t

2

/2

HN (t) dt.

(7.142)

0

Using this in (7.132) gives, after simple manipulation, 2−(N −1) e−x e−x /2 (G) (HN −1 (x))2 + √ HN −1 (x)ΦN −2 (x). ρ(1) (x) = KN (x, x) − √ π(N − 1)! 2 π 2

2

(7.143)

(G)

The global asymptotic expansion of KN (x, x) is given by (7.25), while that of the other terms can be deduced by making use of (7.24). These together show [341]  √ 1 √ 2/N ρ(1) ( 2N X) = ρW (X) − + O(N −2 ). (7.144) 2πN 1 − X 2 Inspection of (7.25), (7.98) and (7.144) shows that with oscillatory terms ignored they can be combined into the single formula  √ 1 1 1 1 √ − 2/Nρ(1) ( 2N X) ∼ ρW (X) − , −1 < X < 1. (7.145) πN β 2 1 − X2 Macroscopic arguments will be used in Section 14.2 below to derive this formula for general β, and to extend it to all real X = 1. 7.8.5 Soft edge The task here is to evaluate the scaled limit of (7.132) with coordinates (7.11). For the first term the limit is equal to K soft (X, Y ) as specified by (7.12). To compute the scaled limit of the second term we make use of (6.113) to rewrite (7.142) as   ∞ 2 π N! −N ΦN (x) = − 2 e−t /2 HN (t) dt, (7.146) N 2 2 (N/2)! x and then use the asymptotic expansion (7.9). This gives  ∞    1 1  √ −y 2 /2 1 √ HN −1 (y)ΦN −2 (x) x → 2N +X/21/2 N 1/6 = Ai(Y ) 1 − Ai(t) dt , lim 1/2 1/6 e √ N →∞ 2 2 π 2 N X y → 2N +Y /21/2 N 1/6 and thus, in an obvious notation S1soft (X, Y

)=K

soft

 ∞   1 (X, Y ) + Ai(Y ) 1 − Ai(t) dt . 2 X

(7.147)

319

SCALED LIMITS AT β = 1, 2 AND 4

The density is given by (7.147) with X = Y . Since then the second term decreases as X → −∞, we see that √ X soft , (7.148) ρ(1) (−X) ∼ K (−X, −X) ∼ X→∞ X→∞ π which is identical to the leading asymptotic behavior of the soft edge density at β = 2 and β = 4. The asymptotic expansion of the density at the soft edge can be computed starting from (7.143). We know from (7.27) that the corrections to the first term therein are O(N −2/3 ). Using (7.146) and (7.9) shows the remaining terms have corrections O(N −1/3 ), giving the result [202]  ∞     X 1 1 soft,1  1/2 (2N ) ∼ ρ ρ + (X) + Ai (X) 1 − Ai(t) dt + O(N −2/3 ). (1) (1) 21/2 N 1/6 21/2 N 1/6 2N 1/3 X (7.149) From Propositions 6.3.2 and 6.3.3 the truncated two-particle correlation is given by 1 ∂ soft S (X, Y ) ρT(2) (X, Y ) = −S1soft (X, Y )S1soft (Y, X) + 2 ∂X 1  ∞  × S1soft (X, z)sgn(z − Y ) dz − sgn(X − Y ) .

(7.150)

−∞

Use of (7.68) shows that only K soft (X, Y ) in the formula (7.147) contributes to the leading non-oscillatory behavior of (7.150), which one calculates as X +Y . (7.151) (X − Y )2 XY Note that this is the same asymptotic behavior found in (7.70) and (7.103) provided we replace the factor of two in the denominator by 2β. ρT(2) (−X, −Y )

E XERCISES 7.8

∼

X,Y →∞

−

2π 2

1 √

1. From the definition (7.138) read off from (7.136) and (7.95) that “ “ 1 1 k” k ” , f (k; 4) = 1 − log 1 − f (k; 1) = 1 − log 1 + 2 π 2 2π

(7.152)

and thus illustrate (7.139) for β = 1. 2. [221] Use the integral form of the kernel given in Exercises 7.1 q.1 to show that for scaled = bulk or soft Z X 1 1 ∂ K scaled (t, Y ) dt (7.153) S1scaled (X, Y ) = K scaled (X, Y ) − 2 2 ∂Y −∞ (cf. (7.106)) and thus the equations (7.107) hold with S˜4scaled replaced by S1scaled .

7.9 SCALED LIMITS AT β = 1 — LAGUERRE AND JACOBI ENSEMBLES 7.9.1 Hard edge

  In the Laguerre case at β = 1 we have V (x) = V˜1 (x)

a →a+1

  S1 (x, y) = S˜1 (x, y)

and so

a →a+1

.

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CHAPTER 7

Making use of (6.107) we thus have that for N even   x 1/2  (L) KN −1 (x, y) S1 (x, y) = y a →a+1  ∞ (N − 1)! a/2 −y/2 a+1 a/2 −u/2 y e − LN −1 (y) sgn(x − u)La+1 e du N −2 (u)u 4Γ(a + N ) 0   x 1/2 (N − 1)! a/2 −y/2 a+1  (L) y e = KN −1 (x, y) LN −1 (y) − y 4Γ(a + N ) a →a+1    x Γ((N + 1)/2)Γ(a + N ) a/2 −u/2 , × 2 La+1 (u)u e du − N −2 2a/2−1 Γ(N )Γ((N + 1 + a)/2) 0

(7.154)

where the second equality follows from (7.131). Making use of (7.29) then shows X Y  1 S1 , S1hard (X, Y ) := lim N →∞ 4N 4N 4N  X 1/2   X 1/2  Ja+1 (Y 1/2 )   hard = 1 − K (X, Y ) + J (v) dv , (7.155) a+1 Y a →a+1 4Y 1/2 0 where K hard is given by (7.31). Bessel function identities applied to this expression give the alternative form [216]  X 1/2  Ja+1 (Y 1/2 )  hard hard S1 (X, Y ) = K 1 − (X, Y ) + J (v) dv (7.156) a−1 4Y 1/2 0 (see also Exercises 7.9 q.1). Both (7.155) and (7.156) have the feature that in the case a = 0 the integrals therein can be evaluated explicitly. Doing this in (7.156) gives [418]   √ √ 1   S1hard (X, Y ) = K hard (X, Y ) − √ J0 ( X)J1 ( Y ). (7.157) a=0 a=0 4 Y As in the β = 4 theory, for the Jacobi ensemble it can be shown that [415] 1 (J)  1 (L)  X Y  X Y  S1 , . = lim S1 1 − ,1 − lim 2 2 2 N →∞ 2N N →∞ 4N 2N 2N 4N 4N 7.9.2 Asymptotics of the one- and two-point functions From Propositions 6.3.3, 6.3.2 and (7.33), (7.155), we have that for general a > −1 the scaled density at a point X from the hard edge is given by  1 1/2 2 1/2 1/2 ρhard (J (X) = (X )) − J (X )J (X ) a+1 a+2 a (1) 4 √  √X  Ja+1 ( X)  √ Ja+1 (u) du − 1 . (7.158) − 4 X 0 The asymptotic expansion (7.73) allows the leading large X non-oscillatory behavior of (7.158) to be computed, analogous to the determination of the asymptotics of (7.113). Again the result (7.74) is obtained. This suggests a universality property: in general the large X behavior of ρ(1) (X) near the hard edge will be given by (7.74), independent of the value of β and a. The truncated two-particle correlation function is given by the formula (7.150) with S1soft replaced by hard S1 and the lower terminal of integration −∞ replaced by 0. From the fact that the large X, Y asymptotics of S1hard come from K hard (X, Y ), the (modified) formula (7.150) shows that the asymptotics (7.151) persists independent of a.

321

SCALED LIMITS AT β = 1, 2 AND 4

7.9.3 Laguerre ensemble — global and soft edge densities For finite N (assumed even) the density is given by setting x = y in (7.154). Following the strategy detailed in Section 7.7.3 for the β = 4 Laguerre ensemble density, this expression can be analyzed globally by setting x = 4N X, and at the soft edge by setting x = 4N + 2(2N )1/3 X. The final results are [341], [202] 4ρ(1) (4N X) ∼ ρMP (X) +

a 1  + o(N −1 ), 2πN X(1 − X)

(7.159)

2(2N )1/3 ρ(1) (4N + 2(2N )1/3 X)  ∞   a soft,1  2 Ai + O(N −2/3 ). (7.160) ∼ ρ(1) (X) − (X)(1 − Ai(s) ds) − (Ai(X)) 2(2N )1/3 X Comparison of these formulas with their Laguerre counterparts at β = 2 and 4 reveals some common features. Consider first the global density. Ignoring oscillatory terms, inspection of (7.39), (7.116) and (7.159) shows that they all satisfy a  4ρ(1) (4N X) ∼ ρMP (X) + 0 < X < 1. (7.161) + o(N −1 ), 2πN X(1 − X) As discussed in Section 14.2 below, this is closely related to the expansion (7.145). In relation to the expansion of the density at the soft edge, we see from (7.37), (7.117) and (7.160), together with the explicit functional forms of ρsoft,β that they all satisfy (1) 2(2N )1/3 ρ(1) (4N + 2(2N )1/3 X) ∼ ρsoft,β (1) (X) −

d soft,β a ρ (X). 1/3 (2N ) dX (1)

(7.162)

This is in distinction to the analogous expansions for the Gaussian ensembles at the soft edge, which reveal no such structure.

7.9.4 Circular Jacobi ensemble — spectrum singularity Considerations analogous to those of Section 7.7.4 show that the correct weight function for the Cauchy ensemble equivalent to the circular Jacobi ensemble is ˜

e−V1 (x) =

1 , (1 + x2 )(α+1)/2

α = N + a.

(7.163)

Thus the n-point correlation can be computed from the quantity S˜1 (x, y)|N +α=a with e−2V (x) given by (7.118). Use of (7.118), (7.163) and (5.65) in (6.107) shows this is given by   1 + x2 1/2  (C) S˜1 (x, y) = KN −1 (x, y) 1 + y2 N +a=α  ∞ (C) (C) sgn(x − t)pN −2 (t) (a + 1)pN −1 (y) dt. (7.164) + (C) 2 (α+1)/2 (pN −2 , pN −2 )2 (1 + y 2 )(α+1)/2 −∞ (1 + t ) Regarding the thermodynamic limit, proceeding as in the derivation of (7.123) we readily find   1 + z 1 + w  πρxy   (C) ,i Ks.s. (x, y) KN −1 i ∼ .  1 − z 1 − w α=N +a N a →a+1 The asymptotic analysis of the second line in (7.164) is complicated by the region of integration being the whole real line. Analogous to the manipulations in (7.146) and (7.154), with x → −i(1 + z)/(1 − z) =

322

CHAPTER 7

− cot πx/L we write  ∞  ∞ (C) (C) sgn(x − t)pN −2 (t)  sgn(x − t)pN −2 (t)  dt =− dt 2 (N +a+1)/2 2 (N +a+1)/2 x →− cot πx/L x →cot πx/L −∞ (1 + t ) −∞ (1 + t )  =−

∞

−∞

(C)

pN −2 (t) dt + 2 (1 + t2 )(N +a+1)/2



(C)

∞

cot πx/L

pN −2 (t) dt. (1 + t2 )(N +a+1)/2

(7.165)

The definite integral is evaluated according to the general formula (6.113), which after recalling (6.97) and (5.65) gives  N/2−2  ∞  ∞ (C)  N + a − 1 − 2j (p2j+1 , p2j+1 )(C) pN −2 (t) dt 2 dt = 2 (N +a+1)/2 2 (N +a+1)/2 (C) N + a − 2j −∞ (1 + t ) −∞ (1 + t ) (p2j , p2j )2 j=0 = 2a+2 N −a−2

Γ(a/2 + 1)Γ(a + 5/2) , Γ(a/2 + 3/2)

(7.166)

where to obtain the first equality use has been made of (5.54), while the duplication formula (1.111) has been used in obtaining the final line. The second integral in (7.165), along with the terms outside the integral in (7.164), are analyzed as in Section 7.2.6. Combining results, and substituting in (7.164) we find the asymptotic behavior  1 + z 1 + w  πρy 2 s.s.  S˜1 i ,i S (x, y), ∼ (7.167)  1 − z 1 − w α=N +a N 1   S1s.s. (x, y) = Ks.s. (x, y)

πρ(a + 1)Γ(a/2 + 1) (πρy)−1/2 Ja+1/2 (πρy) 23/2 Γ(a/2 + 3/2)  πρx   1/2 Γ(a/2 + 3/2) × 1−2 s−1/2 Ja+3/2 (s) ds . Γ(a/2 + 1) 0

a →a+1

+

(7.168)

Substituting (7.167) in the formulas of Proposition 6.3.2 for D1 and I˜1 and substituting this n-point function in (5.55), after cancellation of the prefactor in the first equality of (5.55), we see that in the thermodynamic limit the n-point correlation about the spectrum singularity at β = 1 is given by [213] x  − xjk S1s.s. (xj , u) du − 12 sgn(xj − xk ) S1s.s. (xj , xk ) ρ(n) (x1 , . . . , xn ) = qdet . ∂ s.s. S1s.s. (xk , xj ) ∂xj S1 (xj , xk ) j,k=1,...,n

(7.169) Analogous to the corresponding correlation at β = 4 (7.127), we can check from (7.168) that with a = 0 the result of Proposition 7.8.1 is obtained, while with a = 1 the analogue of (7.128) holds.

7.9.5 Superimposed β = 1 systems The formula (6.152) can be used to compute the scaled limit of (6.151) at the soft edge of the Gaussian ensemble. Proceeding as in Sections 7.6.5 and 7.8.5 one finds [195] odd(OEsoft)2

ρ(n)

(x1 , . . . , xn ) n   √ 1 x1 xn odd(GOEN )2 √ := lim ρ(n) 2N + 1/2 1/6 , . . . , 2N + 1/2 1/6 1/2 1/6 N →∞ 2 N 2 N 2 N  ∞   Ai(xk − v) dv . = det K soft (xj , xk ) + Ai(xj ) 

0

j,k=1,...,n

(7.170)

323

SCALED LIMITS AT β = 1, 2 AND 4

Note that this corresponds to the kernel (7.18) with s1 = 0. Consideration of the distribution (6.146) only requires minor modification to the above working. In particular we see the formula (6.149) applies but with  ∞ ˜ ˜4 (x) V f (x) = e e−V1 (t) dt x

in (6.150). This means that formula (6.151) formally applies for even(OEN )2

ρ(n)

(x1 , . . . , xn ),

(7.171)

the only difference being that the first summation in (6.152) must be multiplied by a minus sign, which in turn follows from the fact that  ∞ ˜ e−V1 (t) dt = −φ˜0 (x) + c˜0 x

(cf. (6.153)). This modified form of (6.152) can then be used to compute the scaled limit of (7.171) at the hard edge of the superimposed Laguerre orthogonal ensembles. Proceeding as in Sections 7.7.1 and 7.9 one finds [195]  1 n  xn  even(LOEN |a →a−1 )2 x1 odd(OEhard)2 ρ(n) (x1 , . . . , xn ) := lim ρ(n) ,..., N →∞ 4N 4N 4N  ∞ √   J ( x ) a j = det K hard (xj , xk ) + √ J (t) dt (7.172) a √ 2 xk j,k=1,...,n xk (the notation odd(OE)2 on the l.h.s. refers to the labeling x1 < x2 < · · · which is natural at the hard edge in the scaled limit). E XERCISES 7.9

1. Using Exercises 7.7 q.2 as a guide, verify that (7.154) can be rewritten as [544], [216] N! (L) (L) y a/2 e−y/2 S1 (x, y) = KN (x, y) + 4Γ(N + a) Z ∞ ×La+1 (y) sgn(x − u)ua/2−1 e−u/2 La−1 N−1 N (u) du. 0

Deduce from this (7.156).

7.10 TWO-COMPONENT LOG-GAS WITH CHARGE RATIO 1:2 7.10.1 A local limit theorem and the evaluation of the free energy To compute the free energy per particle from the closed form expression (6.162) for the partition function requires the asymptotic behavior of the coefficient of ζ N1 /2 in the polynomial 

N1 /2+N2

gN1 /2+N2 (ζ) :=

l=1

 1  ζ + (l − )2 . 2

(7.173)

This can be deduced as a special case of a local limit theorem of Bender [55], which relates to the asymptotic behavior of the coefficients of general polynomials given in a factorized form. P ROPOSITION 7.10.1 [55, Th. 2] Let Pn (x) =

n k=0

an (k)xk = an (n)

n  j=1

(x + rn (j))

324

CHAPTER 7

be a polynomial in x whose roots are all real and nonpositive. Associate with Pn (x) the normalized double sequence pn (k) :=

an (k) . Pn (1)

Then with the mean and variance given by μn =

n j=1

1 , 1 + rn (j)

σn2 =

n j=1

rn (j) , (1 + rn (j))2

the an (k) satisfy a local limit theorem

  2 1   σn pn ([σn x + μn ]) − √ e−x /2  = 0 n→∞ x∈(−∞,∞) 2π sup

lim

(7.174)

provided σn → ∞ as n → ∞. (The square brackets in [σn x + μn ] denote the integer part.) When x = 0, (7.174) gives an ([μn ]) ∼

Pn (1) √ , σn 2π

(7.175)

provided σn → ∞ as n → ∞. This formula can be used to deduce the asymptotics of [ζ N1 /2 ]gN1 /2+N2 (ζ). P ROPOSITION 7.10.2 We have [ζ N1 /2 ]gN1 /2+N2 (ζ) ∼

|Γ(iα + N1 /2 + N2 + 12 )|2 cosh πα √ παN1 2πσ 2

(7.176)

where α=

N2 + N1 /2 , ν

(7.177)

with ν specified as the solution of the equation N1 arctan ν , = N1 + 2N2 ν and σ2 ∼

(7.178)

1  N2 + N1 /2  arctan ν − . 2 ν 1 + ν2

Proof. To use (7.175) to compute the coefficient of ζ N1 /2 in gN1 /2+N2 (ζ) requires that we scale ζ by a suitable function

of N1 /2 + N2 so that μN1 /2+N2 = N1 /2. Thus we choose Pn (x) = α−N1 gn (α2 x) in Proposition 7.10.1, where α is to be so determined. From the formula for μn we can check that for large n = N1 /2 + N2 , with α given by (7.177) μN1 /2+N2 ∼ α arctan ν which implies μN1 /2+N2 = N1 /2 provided (7.178) is satisfied. The specification of σ 2 follows similarly to the formula for μn , and we note that indeed σ 2 → ∞ as N1 /2 + N2 → ∞, which is required for the validity of (7.175). The stated asymptotic formula now follows from (7.175) after noting from (7.173) that √ √ 1 cosh πα x gN1 /2+N2 (α2 x) = |Γ(iα x + N2 + N1 /2 + )|2 , 2 π where use has been make of (4.5).

Use of Stirling’s formula in (7.176), together with the definitions (7.177) and (7.178) allows (7.176) to be

325

SCALED LIMITS AT β = 1, 2 AND 4

simplified to read

 [ζ N1 /2 ]gN1 /2+N2 (ζ) ∼ exp N2 log(1 + 1/ν 2 ) + 2N2 log(N2 + N1 /2)  +(N1 /2) log(1 + ν 2 ) − 2N2 + O(1) .

(7.179)

Substituting this result in (6.162), and making a further use of Stirling’s formula specifies the asymptotics of ZN1 ,N2 . Substituting the resulting expression in the general formula (4.160) gives the free energy per particle [179]. P ROPOSITION 7.10.3 Let x1 :=

lim

N1 ,N2 →∞

N1 , N1 + N2

x2 :=

lim

N1 ,N2 →∞

N2 , N1 + N2

ρb =

lim

N1 ,N2 ,L→∞

N1 + 2N2 L

and specify ν by (7.178). Then − log ZN1 ,N2 ∼ (N1 + N2 )βf + O(1), where βf =

  1  x1 ρb 1 2 x1 log − x + x2 . 1 + − log ρ (2π) 2 b 2 4(1 + ν 2 ) ν2 2

(7.180)

The quantities x1 and x2 are referred to as concentrations. Note that in the limit x1 → 1, so that ν → 0, ρb → ρ and x2 → 0, ρ 1 1 log − , 2 4 2 which agrees with the β = 1 case of (4.166) for the free energy per particle for the one-component log-gas at β = 1. Similarly, in the limit x2 → 1, so that ν → ∞, ρb → 2ρ and x1 → 0,   βf ∼ − log 2ρ(2π)2 + 1, βf ∼

which agrees with the β = 4 case of (4.166) for the free energy per particle of the one-component log-gas. 7.10.2 The correlation functions The local limit theorem Proposition 7.10.1 can also be used to compute the thermodynamic limit of the correlation function (6.163). According to (6.163) and (6.168) it suffices to compute the asymptotics of 

N1 /2+N2

[ζ

N1 /2−q

]

l=1 l=l1 ,...,lp

gN1 /2+N2 (ζ) 1 (ζ + (l − )2 ) =: [ζ N1 /2−q ] p 2 2 i=1 (ζ + (li − 1/2) )

for p and q fixed. Application of Proposition 7.10.1 gives 

N1 /2+N2

[ζ N1 /2−q ]

l=1 l=l1 ,...,lp

1 α2q N1 /2 (ζ + (l − )2 ) ∼ p ] gN1 /2+N2 (ζ), 1 2 [ζ 2 2 (α + (l − ) ) i i=1 2

where α is given by (7.177). As a result, with  1 j t sin πρb xt sj (x) := dt, t2 + 1/ν 2 0

 cj (x) := 0

(7.181)

1 j

t cos πρb xt dt, t2 + 1/ν 2

(7.182)

326

CHAPTER 7

g(x)

1

0.8

0.6

0.4

0.2

0.5

1

1.5

2

2.5

3

3.5

x

Figure 7.2 Plot of g(x) = ρ+1,+1 (x, 0)/(ρ+1 )2 (first dashed curve), ρ+1,+2 (x, 0)/ρ+1 ρ+2 (second dashed curve) and ρ+2,+2 (x, 0)/(ρ+2 )2 in the case ρ+1 = ρ+2 = 1/3, ρb = 1. Note the different position of the first maximum in each case.

we obtain from (6.163) and (6.168) the limiting formula ρ+1n1 ,+2n2 (x1 , . . . , xn1 ; y1 , . . . , yn2 )  [A(xj − xk )]j,k=1,...,n1 [B(xj − yk )] j=1,...,n1 k=1,...,n2 = qdet [C(yj − xk )] j=1,...,n2 [D(yj − yk )]j,k=1,...,n2

,

(7.183)

k=1,...,n1

where ρb A(x) = 2 ν



   ρb − π2 ν 2 sgn(x) + s−1 (x) c0 (x) s1 (x) , B(x) = √ , c0 (x) ν 2 −s1 (x) c2 (x)     ρb ρb c2 (x) s3 (x) c2 (x) s1 (x) , D(x) = . C(x) = √ −s −s (x) c (x) 2 1 0 1 (x) c2 (x) ν 2

c0 (x) −s1 (x)

Note that in the limit ν → 0 this expression reduces to the result of Proposition 7.8.1 for the correlations at β = 1, while it in the limit ν → ∞ reduces to the result of Proposition 7.6.1 for the correlations at β = 4. The determinant form of (7.183) allows the general truncated correlation to be written down. Thus we have (0)  ρ+1n1 ,+2n2 (x1 , . . . , xn1 ; y1 , . . . , yn2 ) = (−1)n1 +n2 −1 f (ri , ri ) , (7.184) cycles length n1 +n2

(i,i )

where f (ri , ri ) := A(xi −xi ), i, i ∈ [1, n1 ]; f (ri , ri ) := B(xi −yi −n1 ), i ∈ [1, n1 ], i ∈ [n1 +1, n1 +n2 ]; f (ri , ri ) := C(yi−n1 − xi ), i ∈ [n1 + 1, n1 + n2 ], i ∈ [1, n1 ]; f (ri , ri ) := D(yi−n1 − yi −n1 ), i, i ∈ [n1 + 1, n1 + n2 ]. This of course agrees with the results for the truncated correlations for the one-component log-gas at β = 1 and β = 4, as given by Propositions 7.8.1 and Propositions 7.6.1, respectively, in the appropriate limits.

327

SCALED LIMITS AT β = 1, 2 AND 4

For the two-particle correlation functions (7.183) and (7.184) give [179]  πρ2 ρ2  ρT+1,+1 (x) = − b4 (c0 (x))2 + s1 (x)s−1 (x) + 2b s1 (x), ν 2ν  2  ρ ρT+1,+2 (x) = − b2 (s1 (x))2 + c0 (x)c2 (x) , 2ν  2 ρ (7.185) ρT+2,+2 (x) = − b (c2 (x))2 + s1 (x)s3 (x) . 4 Plots of these quantities for a particular value of the parameter ν are given in Figure 7.2. Note that the value of the first maximum is characteristic of local charge neutrality. E XERCISES 7.10

1. Show that to leading order the large x expansions of (7.185) are ρT+1,+1 (x) ∼ − x→∞

1 ρ2b , (1 + ν 2 )2 (πxρb )2

ρ2b ν 4 ρT+2,+2 (x) ∼ − x→∞ 4(1 + ν 2 )2

1 . (πxρb )2

ρT+1,+2 (x) ∼ − x→∞

1 ρ2b ν 2 , 2(1 + ν 2 )2 (πxρb )2 (7.186)

Chapter Eight Eigenvalue probabilities — Painleve´ systems approach The generating function for the probability that there are exactly n eigenvalues in an interval J of a classical matrix ensemble with unitary symmetry can, for certain J, be identified with the τ -function of a Painlev´e system. In particular, this is possible whenever J is a single interval containing an endpoint of the support of the density. This allows the distribution of certain eigenvalue probabilities relating to the largest and smallest eigenvalue, and the bulk spacing, to be characterized in terms of the solution of nonlinear equations of Painlev´e type. A practical consequence is the rapid computation of the power series expansions of the spacing distribution, and the high precision numerical tabulations which follow from these. To obtain the Painlev´e nonlinear equations we give a self-contained development of the Hamiltonian formulation of the Painlev´e theory, which makes essential use of B¨acklund transformations and the Toda lattice equation. In the case of J = [−a, a] and the weight being classical and even, the gap probability for β = 1 can be related to a β = 2 gap probability. This allows for a Painlev´e characterization of the gap probability of the COE and its bulk scaling limit. Inter-relationships between the COE, CUE and CSE known from earlier chapters then imply the analogous result for the CSE. In the last section, the theory of orthogonal polynomial systems on the unit circle is used to characterize some random matrix averages over U (N ) in terms of discrete Painlev´e equations.

8.1 DEFINITIONS From experimental data of energy spectra, it is a simple matter to construct the histogram corresponding to the p.d.f. for the spacing between consecutive energy levels. For the spectra of heavy nuclei, this was first done in the late 1950s [447]. In accordance with our discussion in Chapter 1, the theoretical p.d.f. should be identical to that of the bulk eigenvalues in the GOE, appropriately scaled. Here we will discuss the mathematical theory of Painlev´e systems as it relates to the calculation of such eigenvalue spacing distributions. In general, for a continuous, one-dimensional statistical mechanical system the p.d.f. for the spacing distribution is simply related to the probability of a particle-free interval (sometimes called the gap, or hole, probability).

D EFINITION 8.1.1 We denote by EN,β (n; J) the probability that there are exactly n particles in the interval J of a continuous, one-dimensional statistical mechanical system at inverse temperature β with N particles. With J = (a− , a+ ) we denote by pN,β (n; J) the p.d.f. for the event that given there is a particle at a− , there is a particle at a+ , with exactly n particles in between. Similarly, we use the notation pN,β (n; a± ) to denote the p.d.f. for the event that there is a particle at a± and exactly n particles in total to the right (left).

329

EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH

P ROPOSITION 8.1.2 Define the generating functions  ∞  ∞ N  1 (l) EN,β (J; ξ) := dx1 · · · dxN (1 − ξχJ ) e−βU(x1 ,...,xN ) , ˆ ZN −∞ −∞ l=1  ∞  ∞ N  N (N − 1) (l) pN,β (J; ξ) := dx3 · · · dxN (1 − ξχJ )e−βU(a− ,a+ ,x3 ,...,xN ) , ρ(1) (a1 )ZˆN −∞ −∞ pN,β (a± ; ξ) := (l) χJ



N ZˆN

∞

−∞

 dx2 · · ·

l=3

∞

−∞

dxN

N 

(1 − ξχJ )e−βU(a± ,x2 ,...,xN ) , (l)

l=2

(l) = 1 if xl ∈ J and χJ = 0 otherwise, in = (a+ , ∞) or (−∞, a− ). For a continuous

where formula J particles, we have

the second formula J = (a− , a+ ), and in the third one-dimensional statistical mechanical system of N

 (−1)n ∂ n  E (J; ξ)  , N,β n! ∂ξ n ξ=1  n n (−1) ∂  pN,β (n; J) = pN,β (J; ξ) n! ∂ξ n ξ=1 2 ∂ 1 =− EN,β (n; J) + 2pN,β (n − 1; J) − pN,β (n − 2; J), ρ(1) (a− ) ∂a− ∂a+  (−1)n ∂ n d  p (a ; ξ) = ± EN,β (n; J) + pN,β (n − 1; J), pN,β (n; a± ) =  N,β ± n! ∂ξ n da ξ=1 EN,β (n; J) =

(8.1)

(8.2) (8.3)

where pN,β (k; J) = 0 for k < 0 and in (8.2) J = (a− , a+ ), while in (8.3) J = (a+ , ∞) or (−∞, a− ).

Proof. The first equalities in (8.1)–(8.3) follow from the definitions of the generating functions and the defining equations Z Z Z Z 1 “N ” dx1 · · · dxn dxn+1 · · · dxN e−βU (x1 ,...,xN ) , ¯ ZˆN n J J J¯ Z ZJ Z Z 1 1 N! pN,β (n; J)= dx3 · · · dxn+2 dxn+3 · · · dxN ρ(1) (a− ) ZˆN n!(N − n − 2)! J J J¯ J¯

EN,β (n; J)=

×e−βU (a−,a+ ,x3 ,...,xN ) , Z Z Z Z N! 1 pN,β (n; a± )= dx2 · · · dxn+1 dxn+2 · · · dxN e−βU (a±,x2 ,...,xN ) , ZˆN n!(N − n − 1)! J J J¯ J¯ where J¯ := (−∞, ∞) − J. For the second equalities in (8.2) and (8.3), we note that it follows from the first equalities that EN,β (J; ξ) =

∞ X

(1 − ξ)n EN,β (n; J),

n=0 ∞ X

pN,β (a± ; ξ) =

pN,β (J; ξ) =

∞ X

(1 − ξ)n pN,β (n; J),

n=0 n

(1 − ξ) pN,β (n; a± ),

(8.4)

n=0

while inspection of the definitions of the generating functions shows pN,β (J; ξ) = −

∂2 EN,β (J; ξ), (1) (a− ) ∂a− ∂a+ 1

ξ2ρ

pN,β (a± ; ξ) = ±

Substituting (8.4) in (8.5) and equating powers of (1 − ξ) gives the second equalities.

1 d EN,β (J; ξ). ξ da±

(8.5)

330

CHAPTER 8

In the case of a matrix ensemble with unitary symmetry specified by the eigenvalue p.d.f. N 1  g(xj ) C j=1



(xk − xj )2

(8.6)

1≤j x1 . Define the averaged spacing p.d.f. by Z ∞ (s) := p(x, x + s) dx pW β with the value of c in (8.19) chosen so that

R∞ 0

−∞

spW β (s) ds

= 1 (mean spacing equals unity).

(i) For general c show that pW β (s) =

1 β −cβs2 /4 s e , ˜ G2

˜2 = G

“ βc ”1/2 π

G2 .

(ii) Use the constraint that the mean spacing be unity to show “ Γ( β + 1) ”2 cβ 2 = 4 Γ( β2 + 12 ) and thus in particular pW 1 (s) =

πs −πs2 /4 , e 2

pW 2 (s) =

32s2 −4s2 /π e , π2

pW 4 (s) =

218 s4 −64s2 /9π e . 36 π 3

(8.20)

W (iii) [65] Show from (8.16) in the case n = 0 that the gap probability associated with pW 1 is E1 (0; s) = R ∞ −πt2 /4 e dt. Use this and your answer to q.2 to show that for a system of unit density, consisting of a s fraction c1 of particles which are a perfect gas (PG), and fraction 1 − c1 of particles obeying the β = 1 Wigner surmise, the probability that the interval (0, s) is free of particles is given by Z ∞ 2 e−πt /4 dt. E1PG (0; c1 s)E1W (0; (1 − c1 )s) = e−c1 s (1−c1 )s

4. [330] Suppose that in a general continuous one-dimensional statistical mechanical system of N particles, the right most particle is labeled x1 , the second rightmost x2 and so on. Set a0 = ∞, suppose a1 > a2 > · · · and put Ij = (aj , aj−1 ). Denote by EN,β ({(nr , Ir )}r=1,...,l ) the probability that exactly nr of the particles are in Ir

333

EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH

(r = 1, . . . , l). Show that X

Pr(x1 < a1 , . . . , xl < al ) =

EN,β ({(nr , Ir )}r=1,...,l ),

(n1 ,...,nl )∈Ll

where Ll = {(n1 , . . . , nl ) ∈ Zl≥0 :

r X

nj ≤ r − 1 (r = 1, . . . , l)}.

j=1

5. Show that with the change of variables t → (is + 1)/2, σV I (t) → i˜ σV I (s)/2, the σPVI equation in (8.15) reads 4 “ “ ”2 ”2 Y σV I + 4 σ ˜V I (˜ σV I − s˜ σV I ) − iv1 v2 v3 v4 + 4 (˜ σV I + vk2 ) = 0. σ ˜V I (1 + s2 )˜

(8.21)

k=1

(This will be referred to as the σ ˜ PVI equation.)

6. [120] In the general theory of Painlev´e equations it is known that any differential equation of the form (y  )2 = −

4 n c1 (ty  − y)3 + c2 y  (ty  − y)2 + c3 (y  )2 (ty  − y) g 2 (t) + c4 (y  )3 + c5 (ty  − y)2 + c6 y  (ty  − y) + c7 (y  )2 + c8 (ty  − y) + c9 y  + c10

o (8.22)

where g(t) := c1 t3 + c2 t2 + c3 t + c4 , referred to as the master Painlev´ e equation, is integrable in terms of Painlev´e transcendents. Show that only four of the ten parameters in (8.22) are essential because the equation retains its form under the gauge transformations at + b t¯ = , ct + d

y¯ =

hy + kt + m . ct + d

8.2 HAMILTONIAN FORMULATION OF THE PAINLEVE´ THEORY 8.2.1 The auxiliary Hamiltonian In the Hamiltonian approach to the Painlev´e equations PII–PVI, one presents a Hamiltonian H = H(p, q, t; v), where the components of v are parameters, such that after eliminating p in the Hamilton equations q =

∂H , ∂p

p = −

∂H , ∂q

(8.23)

q  and p denoting derivatives with respect to t, the equation in q is the appropriate Painlev´e equation. This was first achieved by Malmquist in 1922 [383]. However the consequences of the Hamiltonian formulation to be presented below, in particular the B¨acklund transformations (transformations of p and q which conserve the Hamiltonian structure) and the associated sequences of special solutions, were not explored until the work of Okamoto in the 1980s [428].

334

CHAPTER 8

The forms of the Hamiltonians given in the work of Okamoto, after some renaming of the parameters, are 1 v1 − v2 q, HII = − (2q 2 − p + t)p − 2 2 1 tHIII  = q 2 p2 − (q 2 + v1 q − t)p + (v1 + v2 )q, 2 HIV = (2p − q − 2t)pq − 2(v1 − v2 )p + (v3 − v2 )q, tHV = q(q − 1)2 p2 − {(v1 − v2 )(q − 1)2 − 2(v1 + v2 )q(q − 1) + tq}p t(t − 1)HV I

+ (v3 − v2 )(v4 − v2 )(q − 1),  = q(q − 1)(q − t)p2 − (v3 + v4 )(q − 1)(q − t) + (v3 − v4 )q(q − t)  − (v1 + v2 )q(q − 1) p + (v3 − v1 )(v3 − v2 )(q − t),

(8.24)

where the parameters herein relate to those in (8.9) according to PII

v1 + v2 = 0,

PIII

α = −4v2 ,

1 α = v1 − , 2 β = 4(v1 + 1),

γ = 4,

δ = −4,

α = 1 + 2v3 − v1 − v2 , β = −2α21 , 1 1 1 PV v1 + v2 + v3 + v4 = 0, α = (v3 − v4 )2 , β = − (v1 − v2 )2 , γ = 2v1 + 2v2 − 1, δ = − , 2 2 2 1 1 1 1 2 2 2 2 PVI α = (v1 − v2 ) , β = − (v3 + v4 ) , γ = (v3 − v4 ) , δ = (1 − (1 − v1 − v2 ) ). 2 2 2 2 These Hamiltonians can be systematically derived from the isomonodromy deformation theory associated with the Painlev´e equations [323], [427] (see also Section 9.9.2), although the details are somewhat complicated and will not be presented here. In the Hamiltonian HIII  , q satisfies the differential equation PIV

v1 + v2 + v3 = 0,

y  =

1  2 1  αy 3 1 δ (y ) − y + 2 + 2 (βy 2 + γt) + y t 4t 4t 4y

(8.25)

for suitable parameters α, . . . , δ. This equation, referred to as PIII , reduces to PIII in (8.9) upon making the replacements t → t2 , y → ty. A feature of the Hamiltonians in (8.24) is that they involve parameters v = (v1 , v2 , . . . ) instead of the parameters α, β, . . . in the original Painlev´e equations (8.9). An advantage of introducing the parameters v is that the Hamiltonians display certain symmetries in these variables. These symmetries are revealed by the second order second degree differential equation satisfied by simple modifications of the Hamiltonians.

P ROPOSITION 8.2.1 Define the auxiliary Hamiltonians hII (t) = HII , 1 1 hIII  (t) = tHIII  + v12 − t, 4 2 hIV (t) = HIV − 2v2 t, hV (t) = tHV + (v3 − v2 )(v4 − v2 ) − v2 t − 2v22 , 1 hV I (t) = t(t − 1)HV I + e2 [−v1 , −v2 , v3 ]t, − e2 [−v1 , −v2 , v3 , v4 ], 2  where ep [a1 , . . . , as ] := 1≤j1 t2i ≥ 0 and each (t2i , t2i+1 ) disjoint so that J{t2i ,t2i+1 } consists of p disjoint intervals on the positive half line and these same intervals reflected through the origin onto the negative half line. Suppose also that g(x) is even. Then we have (0)

(0)

+ EN,2 (J{t2i ,t2i+1 } ; ξ; g(x)) = E[(N +1)/2],2 (J{t 2

2 2i ,t2i+1 }

(0)

+ ×E[N/2],2 (J{t 2

2 2i ,t2i+1 }

+ where J{t 2

2 2i ,t2i+1 }

; ξ; x−1/2 g(x1/2 )χx>0 )

; ξ; x1/2 g(x1/2 )χx>0 ),

(8.123)

2 2 := ∪p−1 i=0 (t2i , t2i+1 ).

Proof. By definition (0)

EN,2 (J{t2i ,t2i+1 } ; ξ; g(x)) =

1“ C ×

Z

∞

−∞

N Y

Z

g(xl )

l=1

”

−ξ

dx1 · · ·

“Z

−∞

J{t

2i ,t2i+1 }

Y

∞

Z

”

−ξ

dxN J{t

2i ,t2i+1 }

|xk − xj |2 ,

(8.124)

1≤j0 ((−θ, θ); ξ) = E[(N +1)/2],2 0, tan2 EN,2 2   θ (0) ×E[N/2],2 0, tan2 ); ξ; x1/2 (1 + x)−N χx>0 2   θ    2 θ J J ;ξ  ); ξ) = E[(N 0, sin2 E ((0, sin  +1)/2],2 [N/2],2 2 2 a=−1/2,b=b∗ a=1/2,b=−b∗ = EO ∗

1 2

∗

−

(2[(N +1)/2]+1)

((0, θ); ξ)E O

+

(2[N/2]+1)

((0, θ); ξ),

(8.127)

− 21

(N odd), and the second equality follows from the first by a change of where b = (N even), b = variables y = x/(1 + x), and the third equality from (8.80). For definiteness consider the case N → 2N even. Choosing the second and third equalities (8.127) reads   θ   θ   CUE J J E2N,2 ; ξ  a=−1/2 EN,2 ; ξ  a=1/2 0, sin2 0, sin2 ((−θ, θ); ξ) = EN,2 2 2 b=1/2 b=−1/2 O− (2N +1)

= E2

O+ (2N +1)

((0, θ); ξ)E2

((0, θ); ξ).

(8.128)

(An alternative way to derive this formula is to use the identity (5.95).) We remark too that one can check that this is consistent with the power series expansions (8.79), (8.81). In relation to the first equality in (8.128), we J J know from Proposition 8.3.1 that EN,2 ((0, s); ξ) for general a, b is characterized by the quantity fN (8.75), and furthermore with a = −b the parameters are v1 = v3 = N , v2 = 0, v4 = b. With these parameters we see J that the differential equation (8.76) determining fN is invariant with respect to the mapping b → −b. Thus J the fN characterizing the two terms on the r.h.s. of the first equality in (8.128) satisfies the same differential equation, differing only in the boundary condition. The generating function for the bulk gap probabilities can be obtained from (8.127) by scaling θ → 2πs/N and taking N → ∞. Recalling from Section 7.2 that X → X/2N 2 gives the hard edge scaling in the Jacobi ensemble we see     E2bulk ((−s, s); ξ) = E2hard ((0, π 2 s2 ); ξ) E2hard ((0, π 2 s2 ); ξ) a=−1/2

a=1/2

− + = E2O ((0, s); ξ)E2O ((0, s); ξ),

(8.129)

where on the l.h.s. the bulk density has been set to unity.

8.4.2 Gap probabilities with orthogonal symmetry and an evenness symmetry Analogous to the definition (8.7), let us define EN,1 (J; ξ; f ) :=

N   (l) (1 − ξχJ ) l=1

OEN (f )

.

(8.130)

For f a classical weight (6.100) and furthermore even (this then excludes the Laguerre case, and restricts the Jacobi case to a = b), and with J = (−t, t), it is possible to relate (8.130) to an average over a matrix ensemble with unitary symmetry [197]. ˜

P ROPOSITION 8.4.2 Let f = e−V1 (x) be as specified above, let e−2V (x) correspond to the different cases ˜ of e−V1 (x) as specified in (5.56) and suppose N is even. We have   1/2 ˜   (0) = EN/2,2 ((0, t2 ); ξ; y −1/2 e−2V (y ) χy>0 ) . (8.131) EN,1 ((−t, t); ξ; e−V1 (x) ) ξ=1

ξ=1

366

CHAPTER 8

Proof. According to the method of integration over alternate variables (recall Section 6.3.2) EN,1 ((−t, t); ξ; f ) = where Aj,k =

1“ 2

Z

∞ −∞

Z −ξ

t

N! 1/2 det[Aj,k ]j,k=1....,N , C

“Z ” dx f (x)Rj−1 (x)

−t

∞ −∞

Z −ξ

”

t −t

(8.132)

dy f (y)Rk−1 (y)sgn(y − x)

(8.133)

and C is independent of t. Here {Rj (x)}j=0,1,... is an arbitrary set of monic polynomials. Let us choose Rj (x) even (odd) for j even (odd). Then A2j,2k = A2j−1,2k−1 = 0

for j, k = 1, . . . , N/2

so every alternate element in the matrix [Aj,k ] is zero. Interchanging rows and columns so that the zero elements are all in the top left and bottom right block and noting A2k,2j−1 = −A2j−1,2k shows EN,1 ((−t, t); ξ; f ) =

N! det[A2j−1,2k ]j,k=1,...,N/2 . C

(8.134)

˜

We now make use of the assumption that f is an even classical weight, f = e−V1 (x) . Let {pj (x)}j=0,1,... denote the monic orthogonal polynomials corresponding to the even weight functions e−2V (x) in (5.56). Then pj (x) is even (odd) for j even (odd). Furthermore, from (6.98) and (6.99), ” 1 ˜ d “ −V˜4 (x) ˜ 2j (x) = p2j (x), ˜ 2j+1 (x) = − R R eV1 (x) p2j (x) , (8.135) e γ2j (p2j , p2j )2 dx ˜

where e−V4 (x) is specified by (6.53), have the skew orthogonality property Z ∞ Z 1 ∞ ˜ ˜ ˜ 2j−2 (x) ˜ 2k−1 (y) sgn(y − x) = 1 δj,k . dx e−V1 (x) R dy e−V1 (y) R 2 −∞ γ2k−2 −∞

(8.136)

˜ j (x), j = 0, 1, . . . in (8.133), using (8.136) and integrating by parts shows Choosing Rj (x) = R Z t 1 (2ξ − ξ 2 ) A2j−1,2k = δj,k − e−V2 (x) p2j−2 (x)p2k−2 (x) dx γ2k−2 γ2k−2 (p2k−2 , p2k−2 )2 −t ˜

+

(ξ − ξ 2 )e−V4 (t) p2k−2 (t) γ2k−2 (p2k−2 , p2k−2 )2

Z

t

˜

e−V1 (x) p2j−2 (x)dx.

−t

It follows from this that ˜

EN,1 ((−t, t); 1; e−V1 (x) ) =

h“ 1 det C

Z

∞ −∞

Z −

t −t

”

e−V2 (x) p2j−2 (x)p2k−2 (x) dx

i j,k=1,...,N/2

(8.137)

for some C independent of t. Repeating the workings of the proof of Proposition 5.2.1, but in the reverse order, shows that the r.h.s. of (8.137) is equal to Z t ” Z “Z ∞ Z t ” Y (N/2)! “ ∞ − − (x2k − x2j )2 . dx1 e−V2 (x1 ) · · · dxN/2 e−V2 (xN/2 ) C −∞ −t −∞ −t 1≤j0  ((−θ, θ); ξ) = EN,2 0, tan2 E2N,1 2 ξ=1 ξ=1    − θ  O (2N +1) J 0, 0, sin2 = EN,2 (0; (0, θ)),  a=−1/2 = E2 2 b=1/2

(8.138)

where the second and third equalities follow as in (8.127). Thus the probability of there being no eigenvalues in the interval (−θ, θ) for the COE — a β = 1 quantity — is equal to the probability of there being no eigenvalues in (0, θ) for matrices from O− (2N + 1) — a β = 2 quantity. The evaluation of the latter in terms of a σPVI transcendent has been discussed in the last subsection of Section 8.3.1. As with going from (8.127) to (8.129), we can take the bulk limit (bulk density set equal to unity) in (8.138) to obtain  −  E1bulk (0; (−s, s)) = E2hard (0; (0, π 2 s2 )) = E2O (0; s). (8.139) a=−1/2

This characterization and the results of Section 8.3.4 give the expansion π 4 (ρs)5 π 4 (ρs)6 π 6 (ρs)7 π 6 (ρs)8 π 2 (ρs)3 − + + − + O(s9 ). (8.140) 36 1200 8100 70560 264600 Note that this could have been obtained from the first series expansion in (8.81), by replacing N → N/2, x → ρπs/N , setting ξ = 1 and taking N → ∞, in agreement with (8.138). It follows from (8.140) that E1bulk (0; (0, s)) = 1 − ρs +

1 bulk 1 d2 p1 (0; s) = 2 2 E1bulk (0; (0, s)) ρ ρ ds π 4 (ρs)4 π 6 (ρs)5 π 6 (ρs)6 π 2 (ρs) π 4 (ρs)3 − + + − + O(s7 ). = (8.141) 6 60 270 1680 4725 Substituting the Painlev´e evaluation (8.102) of (8.139) in the first equality of (8.141) gives a Painlev´e evaluation of pbulk (0; s). This can be simplified if we first note from (8.93) and (8.95) that 1    (πt)2   ds    (πt)2  d 1 3  ds   exp − = − exp − , σIII  (s) μ=2 + v s; − ; ξ = 1 dt 2 s 2 s 0 0 a=−1/2 and the following result deduced [229]. P ROPOSITION 8.4.3 We have pbulk (0; s) 1

 2˜ u((πs/2)2 ) exp − = s

 0

(πs/2)2

u ˜(t)  dt , t

where u ˜ satisfies the nonlinear equation 9  2 3  1 u) − u ˜ + s2 (˜ u )2 = (4(˜ u )2 − u ˜ )(s˜ u − u ˜) + (˜ 4 2 4 subject to the boundary condition s s2 8s5/2 − + . 45 135π s→0 3 Although this result is of theoretical interest as an exact form of the Wigner surmise (8.20), for purposes − of numerical computations we can make use of the fact that the moments of pO 2 (s; ξ)|ξ=1 are known from u ˜(s) ∼ +

368

CHAPTER 8

p.d.f.

mean

variance

skewness

kurtosis

pbulk (0; s) 1 pbulk (1; s) 1 pbulk (2; s) 1 pbulk (3; s) 1 pbulk (4; s) 1 pbulk (5; s) 1 pbulk (6; s) 1 pbulk (7; s) 1 pbulk (8; s) 1 bulk p1 (9; s)

1 2 3 4 5 6 7 8 9 10

0.28553 06557 0.41639 36889 0.49745 52604 0.55564 24180 0.60091 83521 0.63794 46245 0.66925 53948 0.69637 60657 0.72029 45046 0.74168 65573

0.68718 99889 0.34939 68438 0.22741 44134 0.16645 68639 0.13042 07251 0.10679 47124 0.09018 32871 0.07790 15490 0.06847 07897 0.06101 25387

0.37123 80638 0.02858 27332 −0.01329 56588 −0.01994 68028 −0.02007 29233 −0.01884 07449 −0.01743 19487 −0.01613 54800 −0.01500 75200 −0.01404 07984

Table 8.14 Statistical properties of pbulk (n; s) for various values of n. In the cases n ≥ 4 these are from [79]. 1

Section 8.3.4. Thus we see from (8.139) and the first equality in (8.141) that for p ≥ 1  ∞  ∞ − p−1 sp pbulk (0; s) ds = p2 sp−1 pO 1 2 (0; s) ds, 0

(8.142)

0

where we have set ρ = 1. The results of Table 8.12 then give the results listed in Table 8.14 for the case n = 0. The result of Proposition 8.4.2 leading to (8.138) applies only in the case ξ = 1. Thus at this stage we are COE still to obtain results on the generating function EN,1 ((−θ, θ); ξ) for general ξ. A step in this direction is obtained by considering the implication of the identity (6.154) [146], [396]. P ROPOSITION 8.4.4 We have the inter-relationship n   CUE COE COE EN,1 (n; (−θ, θ)) = (2(n − l); (−θ, θ)) + EN,1 (2(n − l) − 1; (−θ, θ)) EN,2 l=0

  COE COE (2l; (−θ, θ)) + EN,1 (2l + 1; (−θ, θ)) , × EN,1

(8.143)

COE where EN,1 (−1; (−θ, θ)) := 0, and EN,p (n; (−θ, θ)) = 0 for n > N . Equivalently, in terms of the generating functions CUE EN,2 ((−θ, θ); ξ) :=

± EN,1 ((−θ, θ); ξ) :=

∞

CUE (1 − ξ)n EN,2 (n; (−θ, θ)),

n=0 ∞

  COE COE (1 − ξ)n EN,1 (2n; (−θ, θ)) + EN,1 (2n ± 1; (−θ, θ))

(8.144)

n=0

we have − + CUE ((−θ, θ); ξ) = EN,1 ((−θ, θ); ξ)EN,1 ((−θ, θ); ξ). EN,2

(8.145)

Proof. To derive (8.143), we suppose there are n eigenvalues in the interval (0, s) of the CUE, and ask what (6.154) says about the corresponding number of eigenvalues in (−θ, θ) for the individual COEs in the ensemble COE ∪ COE. Let us suppose there are an even number 2(n − l) (0 ≤ l ≤ n) in (−θ, θ) from one of the individual COEs in COE ∪ COE. Because every second eigenvalue is integrated over in the operation alt, this means there must be either 2l or 2l ± 1 (these latter two possibilities occurring with probability 12 ) eigenvalues from the other COE to leave n eigenvalues in (0, s).

369

EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH

Thus we obtain the probability “ ”” 1 “ COE COE COE COE EN,1 (2l + 1; (−θ, θ)) + EN,1 EN,1 (2(n − l); (−θ, θ)) EN,1 (2l; (−θ, θ)) + (2l − 1; (−θ, θ)) , (8.146) 2 COE where EN,1 (−1; (−θ, θ)) := 0. Similarly, if there is an odd number 2(n − l) − 1 (−1 ≤ l ≤ n) in (−θ, θ) from this same COE, there must be either 2l + 1, or 2l, 2l + 2 (these latter two possibilities both occurring with probability 12 ), so we obtain the probability “ ”” 1 “ COE COE COE COE (2(n − l) − 1; (−θ, θ)) EN,1 (2l + 1; (−θ, θ)) + (2l + 2; (−θ, θ)) . (8.147) EN,1 EN,1 (2l; (−θ, θ)) + EN,1 2

Adding together (8.146) and (8.147) and summing over l gives (8.143). With (8.143) established, multiplying both sides by (1 − ξ)n and summing over n, (8.145) follows immediately.

In addition to the factorization (8.145), we have previously encountered the factorization (8.127). At ξ = 1 we can see that these are in fact the very same factorizations. Thus according to (8.138) we have   θ   − J COE EN,2 ((0, sin2 ); ξ) a=−b=−1/2 = E2N,1 (0; (−θ, θ)) = E2N,1 ((−θ, θ); ξ) , (8.148) 2 ξ=1 ξ=1 where the second equality follows from the definition (8.144). Equivalently, the equation O± (2N +1)

± ((−θ, θ); ξ) = E2 E2N,1

((0, θ); ξ)

(8.149)

holds for ξ = 1, and is consistent with both factorizations (8.128) and (8.145) for general ξ. In fact methods based on expansions in terms of the eigenvalues and eigenvectors of underlying Fredholm integral operators + (see Section 9.6.1) can be used to establish that (8.149) holds for general ξ in the case of E2N,1 . The factor− ± ization (8.128) then establishes (8.149) for E2N,1 . With E2N,1 thus evaluated, we substitute in (8.144), and use this equation to deduce, with ξ¯ := 2ξ − ξ 2 and thus 1 − ξ¯ = (1 − ξ)2 , that O+ (2N +1)

COE E2N,1 ((−θ, θ); ξ) =

(1 − ξ)E2

¯ + EO ((−θ, θ); ξ) 2 2−ξ

−

(2N +1)

¯ ((−θ, θ); ξ)

.

(8.150)

In the bulk scaling limit (bulk density unity) (8.149) reads O± (2N +1)

E1bulk± ((−s, s); ξ) = E2

((0, s); ξ),

(8.151)

where E1bulk± ((−s, s); ξ) :=

∞

  (1 − ξ)n E1bulk (2n; (−s, s)) + E1bulk (2n ± 1; (−s, s)) ,

n=0

while (8.150) reads E1bulk ((−s, s); ξ)

+ ¯ + E O− ((0, s); ξ) ¯ (1 − ξ)E2O ((0, s); ξ) 2 . = 2−ξ

(8.152)

This latter formula is equivalent to the equation E1bulk (n; 2s) = (−1)n

 [n/2]



[(n−1)/2]

−

E2O (l; s) −

l=0

l=0

Noting from (8.105) that ±

pO 2 (n; s) = −

d O± E (p; s) ds p=0 2 n

 + E2O (l; s) .

(8.153)

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CHAPTER 8

this implies the simple formula

 +  d bulk O− E1 (n; 2s) = (−1)n pO 2 ([(n − 1)/2]; s) − p2 ([n/2]; s) . ds But according to (8.2) (n; s) = pbulk 1

d2 bulk E (n; s) + 2pbulk (n − 1; s) − pbulk (n − 2; s), 1 1 ds2 1

(8.154) ±

(k; s) = 0 for k < 0, so in fact the following recurrence for pbulk (n; s) involving pO where pbulk holds. 2 1 1 P ROPOSITION 8.4.5 We have pbulk (n; s) = 1

 − (−1)n d  O+ p2 ([(n − 1)/2]; s/2) − pO ([n/2]; s/2) 2 2 ds bulk bulk +2p1 (n − 1; s) − p1 (n − 2; s),

(8.155)

(k; s) := 0 for k < 0. where pbulk 1 ±

Using (8.155) and knowledge of the moments of pO 2 (k; s) from Table 8.12 we then obtain the statistical characterizations listed in Table 8.14 in the cases n = 1, 2, 3. 8.4.3 A relationship between gap probabilities with orthogonal and symplectic symmetry It turns out that knowledge of the COE gap probabilities is sufficient for the determination of the CSE gap probabilities. This is a consequence of (4.32) [397]. P ROPOSITION 8.4.6 We have CSE COE (j; (−θ, θ)) = E2N,1 (2j; (−θ, θ)) + EN,4

 1  COE COE E2N,1 (2j − 1; (−θ, θ)) + E2N,1 (2j + 1; (−θ, θ)) (8.156) 2

or equivalently COE pCSE N,4 (n; (−θ, θ)) = p2N,1 (2n + 1; (−θ, θ)).

(8.157)

Proof. Multiplying both sides of (8.156) by n − j + 1, summing over j from 0 to n and making use of (8.17) shows

that (8.156) is equivalent to (8.157). To derive (8.157) we note that with eigenvalues fixed at −θ, θ, the only way these eigenvalues can remain unaffected in the even operation of (4.32), and furthermore this operation leave n eigenvalues inside (−θ, θ), is that there originally be 2n + 1 eigenvalues inside (−θ, θ) (n + 1 of which are integrated over in the even operation). According to (4.32) the eigenvalue p.d.f. for COE2N reduces to that for CSEN upon the even operation, so (8.157) follows.

Recalling the definition (8.149), it follows from (8.156) and (8.149) that  1  O+ (2N +1) O− (2N +1) CSE E2 EN,4 ((−θ, θ); ξ) = ((0, θ); ξ) + E2 ((0, θ); ξ) . 2 In the bulk limit (ρ = 1) this reads  − 1  O+ E2 ((0, s); ξ) + E2O ((0, s); ξ) , E4bulk ((0, s); ξ) = 2 while the bulk limit of (8.157) gives (n; s) = 2pbulk (2n + 1; 2s). pbulk 4 1 Note that the latter implies 

∞ 0

sp pbulk (n; s) ds = 2−p 4

 0

∞

sp pbulk (2n + 1; s) ds. 1

(8.158)

(8.159)

(8.160)

(8.161)

371

EIGENVALUE PROBABILITIES — PAINLEVE´ SYSTEMS APPROACH

1. [458] Let U be a N × N unitary matrix, and replace each entry x + iy, x, y ∈ R by its 2 × 2 real matrix representation (1.36) to obtain a 2N × 2N real orthogonal matrix to be denoted Re(U).

E XERCISES 8.4

(i) Show that if the eigenvalues of U are eiθj , (j = 1, . . . , N ), then the eigenvalues of Re(U) are e±iθj , (j = 1, . . . , N ). (ii) With a(eiθ ) = a(e−iθ ) note that N DY j=1

a(eiθj )

E U (N)

=

N DY

a(eiθj )

j=1

E Re(U (N))

,

where on the r.h.s. the average is over one of each of the eigenvalues e±iθj (j = 1, . . . , N ). Substitute (5.95) for the l.h.s. to deduce Ev(Re(U)) = Ev(O+ (N + 1)) ⊕ Ev(O− (N + 1)), where Ev denotes the eigenvalue distribution of the eigenvalues eiθ , 0 < θ < π of the corresponding matrix ensembles. 2

2a −x , x ∈ (−∞, ∞). Under 2. The generalized Hermite √ ensemble is specified by the weight function w2 (x) = |x| e the scaling x → x/ 2N the correlations coincide with those for the spectrum singularity (7.48) with ρ = 1/π [418], and in particular ““ ” 2 t t ” −√ lim EN,2 ,√ ; ξ; |x|2a e−x = E2s.s. ((−t, t); ξ). (8.162) N→∞ 2N 2N

(i) Show that for a = 0 the logarithmic derivative of the generalized Hermite weight has the structure (5.57) with degree f = 2, and so in general this does not define a classical weight. 2

(ii) Apply (8.123) with J{t2i ,t2i+1 } = (−t, t), g(x) = |x|2a e−x and take the scaling limit (8.162) to conclude ˛ ˛ ˛ ˛ ˛ ˛ E2s.s. ((−t, t); ξ)˛ = E2hard ((0, t2 ); ξ)˛ E2hard ((0, t2 ); ξ)˛ . (8.163) ρ=1/π

3.

a→a−1/2

a→a+1/2

(i) Consider the circular β-ensemble. Show from the definitions that for x1 , . . . , xn small ρ(n) (x1 , . . . , xn ) ∼

“ 2π ”βn(n−1)/2 N! n (N − n)!L L

Y

|xk − xj |β

1≤j 0 are arbitrary. To analyze the remainder term for the Christoffel-Darboux sum in the case of the circular Jacobi ensemble we make use of the integral representation Z 1/2 Γ(a + b + 1) eπix(a−b) |1 + e2πix |a+b (1 + te2πix )r dx = 2 F1 (−r, −b; a + 1; t), (9.10) Γ(a + 1)Γ(b + 1) −1/2 which can be deduced by expanding (1 + te2πix )r according to the binomial theorem and integrating term by term using

383

FREDHOLM DETERMINANT APPROACH

(4.4) in the case N = 1. This shows 2 F1 (−n, −b; a

“ “1” ” + 1; t/n) = 1 F1 (−b; a + 1; −t) 1 + O O(t2 ) . n

It remains to establish the bound (9.2). We make use of the fact that ρ(n) is non-negative and, according to Proposition 5.1.2, equal to the determinant of a symmetric matrix. For the determinant of such matrices we have the following inequality, equivalent to a result of Hadamard given in (9.12) below. P ROPOSITION 9.1.4 For A = [aij ]i,j=1,...,n non-negative and symmetric, det A ≤

n 

aii .

(9.11)

i=1

Proof. [393] Consider first the case aii = 0 for some i = 1, . . . , n. Because a matrix is non-negative if and only if its principal minors are non-negative, it follows that aii ajj − |aij |2 ≥ 0 for each j = i and thus all elements in the ith row of A must vanish, so det A = 0, in agreement with (9.11). Hence it suffices now to consider the case that A is positive definite. We begin by performing the Laplace expansion – » 0 α † , det A = a11 det A1 − det α  A1 where α  = [ak1 ]k=2,...,n and A1 = [aij ]i,j=2,...,n . Analogous to a manipulation performed in the proof of Proposition 2.5.1 we have » – – » – » 1 0(N−1)×1 0 α † 0 α † = det A1 det det det 01×(N−1) A−1 α  A1 α  A1 1 α† A−1  ) > 0, = (det A1 )( 1 α where the inequality follows because A1 is a principal minor of A and thus is itself positive definite, and this implies A−1 1 is positive definite. Hence det A < a11 det A1 , and the result now follows by induction.

Note that writing A = B† B for B an n × n square matrix shows (9.11) is equivalent to the inequality n  n   (9.12) |bij |2 , | det B|2 ≤ i=1

j=1

which is Hadamard’s result [541]. In light of the determinant structure of ρ(n) noted above Proposition 9.1.4, and the fact that the diagonal entries of the determinant equal the one-body density, the bound (9.11) tells us that ρ(n) (x1 , . . . , xn ) ≤

n 

ρ(1) (xi ).

(9.13)

i=1

Assuming the region R is such that the integral of the one-body density over R is finite (i.e., contains a finite number of eigenvalues) we see the integral in (9.2) in fact only grow exponentially fast in n. All criteria of Proposition 9.1.2 are therefore met and so it follows that for the scaled classical random matrix ensemble

384

CHAPTER 9

with unitary symmetry E2scaled (J; ξ) = 1 +

  ∞ 1 2 (−ξ)n dx1 · · · dxn det K scaled(xj , xk ) j,k=1,...,n , n! J J n=1

(9.14)

where scale = bulk, soft, hard or s.s. and J = (−t, t), (t, ∞), (0, t), (−t, t), respectively. We recognise this sum as that occurring in the expansion (5.32) of the determinant of a Fredholm integral operator. Hence E2scaled (J; ξ) = det(1 − ξKJscaled), where

(9.15)

 (1 − ξKJscale )f := f (x) − ξ

K scaled(x, y)f (y) dy.

(9.16)

J

With (9.14) as the starting point, and with scaled = bulk, J = (0, t) Jimbo, Miwa, Mˆori and Sato [324] obtained the Painlev´e transcendent evaluation of (9.15) implied by the first equation of (8.111). Furthermore, this work also identified the Fredholm determinant in (9.15) as a τ -function associated with the integrable structure of a particular monodromy preserving deformation of a linear differential equation. In this chapter we will present both the function theoretic properties of (9.15) which lead to its Painlev´e transcendent evaluations, as well as aspects of the underlying integrable systems theory. E XERCISES 9.1

1. [90] Let J(p),I (x1 , . . . , xp ) denote the p-point correlation for the event that there are exactly n particles in the interval I, and these are at the positions x1 , . . . , xp . The quantity J(p),I is referred to as a Janossy density. (i) Consider the p.d.f. (5.139) and let ζj , ζ˜j be linear combinations of {ξl }l=1,...,j , and let ψk , ψ˜k be linear combinations of {ηl }l=1,...,k . With Z ∞ “Z ∞ Z ” w2 (x)ζj (x)ψk (x) dx = δj,k , − w2 (x)ζ˜j (x)ψ˜k (x) dx = δj,k −∞

let KN (x, y) =

−∞

I

PN−1

PN−1 ˜ ˜ ˜ l=0 ζl (x)ψl (y) and KN (x, y) = l=0 ζl (x)ψl (y). Show h i ˜ N (xi , xj ) J(p),I (x1 , . . . , xp ) = det(1 − KN,I ) det K

i,j=1,...,p

,

(9.17)

where KN,I is the integral operator on I with kernel KN (x, y). (ii) Verify that the Laguerre polynomials {L0k (x − t)}k=0,1,... are orthogonal on the interval (t, ∞) with respect to the weight e−x , and use this together with the result of (i) to show that with this choice of w2 and ξl = ηl = xl−1 ˛ ˜ N (x, y) = K (L) (x − t, y − t)˛˛ K , N a=0

(L)

where KN is the Christoffel-Darboux kernel in the Laguerre case with a = 0. Substitute this in (9.17), and relate its hard edge scaled form with p = 2, t = t2 , x1 = t1 , x2 = t2 , to the formula (8.100) with a = 0. 2.

(i) By making use of Propositions 5.2.1 and 5.2.2 with a(x) = 1 − ξχx∈J , deduce that EN,2 (J; ξ; w2 (x)) = det(1 − K), where K is the integral operator on J with kernel ξKN (x, y), as is consistent with the formula (9.15). (ii) Note from (6.27) and (6.76) with a(x) as in (i) that for β = 1, 4 ”2 “ EN,β (J; ξ; e−βV (x) ) = det[12 − ξfβ χJ ], where χJ := diag[χJ , χJ ], and χJ is the indicator function for J.

(9.18)

385

FREDHOLM DETERMINANT APPROACH

3. Consider a two-species system, species a and b, with coordinates {aj } and {bj }. Let ρ(n1 ,n2 ) ({aj }j=1,...,n1 ; {bj }j=1,...,n2 ) denote the corresponding (n1 , n2 )-point correlation, and introduce the generating function EN (Ja , Jb ; ξx , ξy ) :=

N DY E (l,a) (l,b) (1 − ξa χJa )(1 − ξb χJb ) l=1

(cf. the case μ = 0 of (8.7)). Show that N X

EN (Ja , Jb ; ξa , ξb ) = 1 + Z

Z

n1 ,n2 =0 (n1 ,n2 )=(0,0)

Ja

Z

Z

da1 · · ·

×

(−ξa )n1 (−ξb )n2 n1 !n2 !

db1 · · ·

dan1 Ja

Jb

dbn2 ρ((n1 ,n2 )) ({aj }j=1,...,n1 ; {bj }j=1,...,n2 ). Jb

In the cases that the correlations have a determinant structure ρ((n1 ,n2 )) ({aj }j=1,...,n1 ; {bj }j=1,...,n2 ) 2 3 [Kaa (aj , ak )]j,k=1,...,n1 [Kab (aj , bk )] j=1,...,n1 k=1,...,n 2 5 = det 4 [Kba (bj , ak )] j=1,...,n2 [Kbb (bj , bk )]j,k=1,...,n2 k=1,...,n1

deduce from this that EN (Ja , Jb ; ξa , ξb ) = det(1 − K),

(9.19)

where K is the 2 × 2 matrix integral operator on R with kernel – » χx∈Ja ξa Kaa (x, y)χy∈Ja χx∈Ja ξa Kab (x, y)χy∈Jb . K(x, y) = χy∈Jb ξb Kba (y, x)χx∈Ja χy∈Jb ξb Kbb (y, y)χy∈Jb

9.2 NUMERICAL COMPUTATIONS USING FREDHOLM DETERMINANTS The sine kernel refers to the kernel K bulk (x, y) :=

sin πρ(x − y) , π(x − y)

(9.20)

and the task is to compute bulk ). E2bulk ((0, s); ξ) = det(1 − ξK(0,s)

(9.21)

Let us first address this problem in a numerical sense. In Section 8.3.5 the Painlev´e transcendent formula (8.111) was used to generate the first 700 terms in its power series, and this in turn was used to compute Table 8.13, giving statistical properties of {pbulk (n; s)} 2 for n up to 4. Due to the computational expense incurred in computing the power series, to get more terms is difficult, and this restriction in turn implies a loss of accuracy in the determination of pbulk (n; s) for higher 2 values of n (recall from (8.165) that the leading term in the power series of pbulk (n; s) is proportional to 2 sn+(n+2)(n+1) ). It turns out that a numerical scheme based on (9.21) can overcome this problem [79]. The first tabulations of spacing distributions were based on Fredholm determinants [259]. Thus the Fred− holm determinant formula (9.81) below for E2O ((0, s); ξ = 1) was used to obtain a graphically accurate determination of the bulk spacing distribution pbulk (0; s) (recall (8.139)). This was done by first developing 1 the theory of Section 9.6.1 below, and thereby determining the eigenvalues of the Fredholm integral oper-

386

CHAPTER 9

ator in terms of prolate spheroidal functions. By making use of tables of these functions, E1bulk (0; s) was computed, and a numerical differentiation scheme then used to compute pbulk (0; s) according to (8.154). 1 The same strategy was employed in [392] to tabulate pbulk (0; s) (see [425] for a discussion of the source of 2 inaccuracies in this table). Subsequently, as reproduced in [395], the prolate spheroidal functions were used to plot {Eβbulk (n; s)}, β = 1, 2 for successive values of n up to 10. Recently Bornemann [78], [79] has shown how the Fredholm determinant formulas can be used to compute spacing distributions such as {pbulk (n; s)} for successive values of n up to 14 (and beyond if required), to 2 machine precision of 15-digit accuracy. The strategy relates to the approximation of the integral in definition (9.16) by a sum, according to an m-point quadrature rule of order m (i.e., a rule replacing the integral by a weighted sum of m terms, such that it is exact for a polynomial of degree m − 1) with positive weights wj ,  m K(x, y)f (y) dy ≈ wj K(x, yk )f (yk ). J

k=1

The integral operator eigenvalue equation ξKJ ψ = λψ is correspondingly replaced by the system of m linear equations ξ

m

wj K(yj , yk )ψ(yk ) = λψ(yj )

(k = 1, . . . , m),

j=1 −1/2

or equivalently (with ψ → wj

ψ)

ξ

m

1/2

1/2

wj K(yj , yk )wk ψ(yk ) = λψ(yj ).

j=1 1/2

1/2

The characteristic polynomial of the symmetric matrix [ξwj K(yj , yk )wk ]j,k=1,...,m at λ = 1 is the Fredholm determinant of 1 − ξKJ in this approximation, and so 1/2

1/2

det(1 − ξKJ ) ≈ det[δj,k − ξwj K(yj , yk )wk ]j,k=1,...,m .

(9.22)

It is shown in [78] that with the kernel K(x, y) analytic in a complex neighbourhood of (a, b), this approximation has error O(ρ−m ) (ρ > 1) and thus converges exponentially fast. In practice this means doubling m will typically double the number of correct digits. The derivative in s, required by (8.115) and analogous formulas, is carried out by converting the numerical tabulation to an interpolation in Chebyshev points. According to the formulas of Proposition 8.1.2, to compute higher order spacing distributions from the generating function, derivatives in ξ of (9.22) are required. This being an entire function of ξ (it is a polynomial), one can use the formula for the k-th derivative  2π k! f (k) (z) = e−ikθ f (z + reiθ ) dθ, r > 0, 2πrk 0 with the trapezoidal rule approximation, which is known to converge exponentially fast for periodic analytic integrands. For the bulk spacings the choice r = 1 was empirically determined to be the most numerically stable in this regard.

9.3 THE SINE KERNEL We now take up the problem of using (9.21) to obtain the characterization in terms of the σPV equation (8.112). Key equations from [324] for this purpose were isolated by Tracy and Widom [517], following on from the works [394], [152]. These equations relate to a number of quantities associated with the integral

387

FREDHOLM DETERMINANT APPROACH

operator in (9.15). One of these quantities is the resolvent kernel R(x, y), specified by . (9.23) ξKJ (1 − ξKJ )−1 = R(x, y), . where the notation = denotes that the r.h.s. is the kernel of the integral operator on the l.h.s. Our interest is in the case that J is a single interval, J = [a1 , a2 ] say. Closely related to (9.23), and used in the definition of the other quantities entering the coupled equations, is ρ(x, y) specified by . (1 − ξKJ )−1 = ρ(x, y) (9.24) (note that ρ(x, y) = δ(x − y) + R(x, y)). Both (9.23) and (9.24) have meaning for general kernels; however, the other quantities are special to the Christoffel-Darboux type structure of K scale (x, y). D EFINITION 9.3.1 Let φ(x), ψ(x) be such that ξK scaled (x, y) :=

φ(x)ψ(y) − φ(y)ψ(x) . x−y

(9.25)

In terms of φ(x), ψ(x) and (9.24) introduce the quantities Q(x), qj , P (x), pj according to  a2 ρ(x, y)φ(y)dy, qj = Q(aj ) := x→a lim Q(x), Q(x) := (1 − ξKJ )−1 φ := j x∈(a1 ,a2 )

a1

P (x) := (1 − ξKJ )−1 ψ :=



a2

ρ(x, y)ψ(y)dy,

pj = P (aj ) :=

a1

lim

x→aj x∈(a1 ,a2 )

P (x).

The equations of [517] relating to E2bulk ((0, t); ξ) can now be stated. P ROPOSITION 9.3.2 For (9.14) with

  K scaled(x, y) = K bulk(x, y)

ρ=1/π

,

and in the notation of Definition 9.3.1 with a1 = −t, a2 = t, we have d 2q 2 p2 qp , (ii) R(t, t) = p2 + q 2 − , (iii) R(t, t) = 2(R(−t, t))2 , t2 t dt 2q p dp 2qp2 dq =p− , (v) = −q + , (iv) dt t dt t where p := p2 and q := q2 . (i) R(−t, t) =

From these coupled equations a single differential equation for σ(t) := −tR(t/2, t/2) can be obtained, and knowledge of σ(t) allows E2 ((0, t); ξ) to be computed. For this we require the general formula ∂ log det(1 − ξKJ ) = (−1)j−1 R(aj , aj ) ∂aj

(j = 1, 2)

(9.26)

(see Exercises 9.3 q.1). It follows from this that −2R(t, t) =

d log E2 ((−t, t); ξ), dt

which in turn implies

 E2 ((0, t); ξ) = E2 ((−t/2, t/2); ξ) = exp 0

πρt

(9.27)

σ(u) du. u

(9.28)

On the r.h.s. of (9.28) t has been replaced by πρt to account for a general value of ρ (this is valid because ρ is the only length scale in the problem).

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CHAPTER 9

P ROPOSITION 9.3.3 The quantity σ(t) := −tR(t/2, t/2) satisfies the first of the differential equations in (8.112) (which is the σPV equation of (8.15) with ν0 = ν1 = ν2 = ν3 = 0 and t → −2is) subject to the first of the boundary conditions in (8.113). Proof. From equations (i), (ii), (iv) and (v) of Proposition 9.3.2 it is straightforward to deduce the equations d (tR(−t, t)) = p2 − q 2 dt

and

d (tR(t, t)) = p2 + q 2 . dt

Squaring these equations and subtracting gives „ «2 „ «2 d d (tR(−t, t)) − (tR(t, t)) = −4p2 q 2 = −4t2 (R(−t, t))2 , dt dt where to obtain the last equality equation (i) has again been used. The stated equation now follows by using the equations 4t2 (R(−t, t))2 = 2t

d (tR(t, t)) − 2tR(t, t) and dt

4tR(−t, t)

d d2 (tR(−t, t)) = t 2 (tR(t, t)), dt dt

which are consequences of equation (iii), to eliminate tR(−t, t) and its derivative. The boundary condition follows from the leading small t computation of the n = 1 and n = 2 terms in (9.4) with K(x, y) given by K bulk (x, y) and ρ therein set equal to π1 .

9.3.1 Derivation of the coupled equations Two general operator identities are used throughout the derivation. P ROPOSITION 9.3.4 For any operators L and K, [L, (1 − K)−1 ] = (1 − K)−1 [L, K](1 − K)−1 ,

(9.29)

and for any operator K which can be differentiated with respect to a parameter a, dK d (1 − K)−1 = (1 − K)−1 (1 − K)−1 . da da

(9.30)

Proof. The first identity is verified by letting both sides act on an arbitrary function f , expanding the commutators, and comparing the resulting expressions. The second identity has been derived in Exercises 2.5 q.1.

The following simple lemma is also required. P ROPOSITION 9.3.5 Let M denote multiplication by the independent variable, that is, M f (x) = xf (x). Then for an integral operator LJ with kernel L(x, y) supported on the interval J = [a1 , a2 ], . [M, LJ ] = (x − y)L(x, y). Proof. Since Z (M LJ − LJ M )f := x

a2

Z L(x, y)f (y)dy −

a1

Z

a2

a2

L(x, y)yf (y)dy = a1

(x − y)L(x, y)f (y)dy,

a1

the result follows.

After these preliminaries, we begin the derivation proper with an expression for the resolvent kernel [307]. P ROPOSITION 9.3.6 Let R(x, y) be defined as in (9.23). We have R(x, y) =

Q(x)P (y) − P (x)Q(y) , x−y

(9.31)

389

FREDHOLM DETERMINANT APPROACH

where Q and P are specified in Definition 9.3.1, and thus R(x, x) = −Q(x)P  (x) + P (x)Q (x).

(9.32)

Proof. We will compute (9.29) with L = M in two different ways. By Proposition 9.3.5 and (9.25), . [M, ξKJ ] = φ(x)ψ(y) − φ(y)ψ(x). . . Thus [M, ξKJ ] = A1 − A2 where A1 = φ(x)ψ(y), A2 = φ(y)ψ(x), so we have [M, ξKJ ](1 − ξKJ )−1 f = A1 (1 − ξKJ )−1 f − A2 (1 − ξKJ )−1 f Z a2 Z a2 Z a2 Z a2 = φ(x) dy ψ(y) ρ(y, y  )f (y  )dy  − ψ(x) dy φ(y) ρ(y, y  )f (y  )dy  a1 a1 a1 a1 Z a2 Z a2 Z a2 Z a2 = φ(x) dy f (y) dy  ψ(y  )ρ(y, y  ) − ψ(x) dy f (y) dy  φ(y  )ρ(y, y  ), a1

a1

a1

a1

where we have used the symmetry ρ(y, y  ) = ρ(y  , y) which follows since K(x, y) = K(y, x). Hence, using Definition 9.3.1, we have Z Z a2

[M, ξKJ ](1 − ξKJ )−1 f = φ(x)

f (y)P (y) dy − ψ(x)

a1

and so

a2

f (y)Q(y) dy, a1

. [M, ξKJ ](1 − ξKJ )−1 = φ(x)P (y) − ψ(x)Q(y),

which gives

. (1 − ξKJ )−1 [M, ξKJ ](1 − ξKJ )−1 = Q(x)P (y) − P (x)Q(y).

On the other hand, for the l.h.s of (9.29), we have from Proposition 9.3.5 . [M, (1 − ξKJ )−1 ] = (x − y)ρ(x, y) = (x − y)R(x, y). Equating the above two equations gives the stated result.

Equation (i) of Proposition 9.3.2 can now be derived by using (9.31). To see this, we note that with a1 = −t, a2 = t, from the facts that φ(x) is odd, ψ(x) is even and ρ(x, y) = ρ(−x, −y) (which follows since K(x, y) = K(−x, −y)), it follows from Definition 9.3.1 that q2 = −q1 = q,

p2 = p1 = p,

(9.33)

so substitution gives the desired equation. Whereas Proposition 9.3.6 holds for general φ and ψ, the final conclusion of the next result uses a special property of the particular choice   ξ ξ sin πρx, ψ(x) = cos πρx, (9.34) φ(x) = π π which gives the kernel (9.20). This result is presented in preparation for the computation of P  (x) and Q (x) which occur in the expression (9.32) for R(x, x). P ROPOSITION 9.3.7 Let D denote the operator for differentiation with respect to the independent variable. We have . [D, KJ ] = (Dx + Dy )K(x, y) − δ − (y − a2 )K(x, y) + δ + (y − a1 )K(x, y), where



a2

a1

δ − (y − a2 )f (y)dy := f (a2 )



a2

and a1

δ + (y − a1 )f (y)dy := f (a1 ),

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CHAPTER 9

and consequently, in the cases that (Dx + Dy )K(x, y) = 0, . [D, (1 − ξKJ )−1 ] = R(x, a1 )ρ(a1 , y) − R(x, a2 )ρ(a2 , y). Proof. We have Z a2 Z a2 ∂ ∂ K(x, y)f (y) dy − K(x, y) f (y) dy ∂x a1 ∂y a1 Z a2 ” Z a2 “ ∂ ∂ f (y) K(x, y) dy, := K(x, y)f (y) dy − f (a2 )K(x, a2 ) − f (a1 )K(x, a1 ) + ∂y a1 ∂x a1

[D, KJ ]f :=

which gives the first result. Now suppose (Dx + Dy )K(x, y) = 0. The first result then gives Z a2 Z a2 [D, KJ ](1 − ξKJ )−1 f = dy K(x, y)(δ + (y − a1 ) − δ − (y − a2 )) dy  ρ(y, y  )f (y  ) a1 a1 Z a2 Z a2 = K(x, a1 ) ρ(a1 , y  )f (y  )dy  − K(x, a2 ) ρ(a2 , y  )f (y  )dy  . a1

a1

The second result now follows upon applying (9.29) with L = D and recalling (1 − ξKJ )−1 ξKJ = R.

The derivatives P  (x) and Q (x), and consequently R(x, x), can now be computed in terms of quantities in Definition 9.3.1. P ROPOSITION 9.3.8 With K(x, y) given by (9.20) we have Q (x) = πρP (x) + R(x, a1 )q1 − R(x, a2 )q2 , P  (x) = −πρQ(x) + R(x, a1 )p1 − R(x, a2 )p2 and consequently R(aj , aj ) = πρ(p2j + qj2 ) −

(q2 p1 − p2 q1 )2 a 2 − a1

(j = 1, 2).

Proof. From Definition 9.3.1 we have Q (x) = D(1 − ξKJ )−1 φ = (1 − ξKJ )−1 Dφ + [D, (1 − ξKJ )−1 ]φ. The stated formula for Q (x) now follows from the fact that Dφ = πρψ, Proposition 9.3.7, (9.23) and Definition 9.3.1. The formula for P  (x) is derived similarly, and the formula for R(aj , aj ) is derived by substituting for Q (x) and P  (x) in (9.32), and using (9.31) and Definition 9.3.1.

Equation (ii) of Proposition 9.3.2 follows from the formula above for R(aj , aj ) with j = 2, a2 = t and the substitutions (9.33). For the derivation of equation (iii), a preliminary result is needed. ∂ P ROPOSITION 9.3.9 We have ∂a R(x, y) = (−1)j R(x, aj )R(aj , y) and so with (Dx + Dy )K(x, y) = 0, j    ∂ ∂ ∂ −R(x, a2 )R(a2 , y), j = 1, + R(x, y) = + R(x, a1 )R(a1 , y), j = 2, ∂aj ∂x ∂y

for x, y ∈ (a1 , a2 ). Proof. We have ∂ ∂ R(x, y) = ρ(x, y) ∂aj ∂aj so we consider (9.30) with a = aj . Now ∂ ∂ KJ f = ∂aj ∂aj

Z

a2 a1

K(x, y)f (y)dy = (−1)j K(x, aj )f (aj )

391

FREDHOLM DETERMINANT APPROACH

and so

∂ . KJ = (−1)j K(x, aj )δ ∓ (y − aj ) (j = 1, 2). ∂aj

This gives ∂ ρ(x, y) = (−1)j R(x, aj )ρ(aj , y) ∂aj

(9.35)

and the first result follows since ρ(aj , y) = R(aj , y) for y ∈ (a1 , a2 ). The second result follows from the first result, the fact that for x, y ∈ (a1 , a2 ) « „ ∂ ∂ . R(x, y) [D, (1 − ξKJ )−1 ] = + ∂x ∂y (this can be verified by using integration by parts) and the final equation in Proposition 9.3.7.

Equation (iii) of Proposition 9.3.2 can now be derived by first noting     ∂ ∂ ∂ ∂ + R(x, y) R(aj , aj ) = + = (−1)j (R(a1 , a2 ))2 , ∂aj ∂aj ∂x ∂y x=y=aj where the final equality follows from Proposition 9.3.9. This gives    d ∂ ∂ R(t, t) = R(a2 , a2 ) − R(a2 , a2 )  = 2(R(−t, t))2 dt ∂a2 ∂a1 a2 =−a1 =t as required. For the derivation of the final two equations in Proposition 9.3.2, the formulas         ∂qj ∂ ∂ ∂ ∂pj ∂  + + Q(x) P (x) = and = ∂aj ∂x ∂aj ∂aj ∂x ∂aj x=aj x=aj

(9.36)

(9.37)

(9.38)

are used. The partial derivatives with respect to x are given by Proposition 9.3.8. The derivatives with respect to aj are given by the following result, which is valid for all kernels of the form (9.25). P ROPOSITION 9.3.10 We have ∂ Q(x) = (−1)j R(x, aj )qj ∂aj

and

∂ P (x) = (−1)j R(x, aj )pj . ∂aj

Proof. Now ∂ ∂ Q(x) := (1 − K)−1 φ. ∂aj ∂aj But from (9.35) ∂ . (1 − ξKJ )−1 = (−1)j R(x, aj )ρ(aj , y), ∂aj and the first result follows from Definition 9.3.1. The second result follows similarly.

The final two equations of Proposition 9.3.2 now follow by noting dqj ∂qj ∂qj = − , dt ∂a2 ∂a1 dp

and similarly for dtj , substituting the results of Propositions 9.3.10 and 9.3.8 into (9.38) to compute one of the derivatives, and using Proposition 9.3.10 to compute the other derivative. E XERCISES 9.3

1. The objective of this exercise is toRderive the formula (9.26). For an integral operator KJ on a J = [a1 , a2 ] with kernel K(x, y) we have Tr KJ := a12 K(x, x) dx. Use the formula log det A = Tr log A,

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CHAPTER 9

and the power series expansion of log(1 − ξKJ ) to show that ∂ det log(1 − ξKJ ) ∂aj Z a2 Z ∞ ∂ X ξ n a2 =− dx1 · · · dxn K(x1 , x2 )K(x2 , x3 ) · · · K(xn−1 , xn )K(xn , x1 ) ∂aj n=1 n a1 a1 Z a2 Z a2 ∞ X = (−1)j−1 ξn dx2 · · · dxn K(aj , x2 )K(x2 , x3 ) · · · K(xn−1 , xn )K(xn , aj ). n=1

a1

a1

Identify the final expression with (−1)j−1 R(aj , aj ) as required by (9.26). 2. [324], [517] Here the multiple gap probability will be exhibited as an integrable system. (i) Let E2 (0; J) denote the probability that there are no particles in some collection of intervals J. Show that (9.15) still holds, with the domain of integration of the integral operator now J. (ii) In the case J = ρ = 1/π)

Sm

j=1 (a2j−1 , a2j )

R(ai , ai ) = p2i + qi2 +

2m X

modify the proofs of Propositions 9.3.6–9.3.10 to show that (with

(−1)k

k=1 k=i

(qi pk − pi qk )2 , ai − ak

X ∂qi = pi − (−1)k R(ai , ak )qk ∂ai k=1

2m X ∂pi = −qi − (−1)k R(ai , ak )pk , ∂ai k=1

2m

and

k=i

k=i

∂qj = (−1)k R(aj , ak )qk ∂ak

and

∂pj = (−1)k R(aj , ak )pk ∂ak

with R(aj , ak ) =

qj pk − pj qk qk . aj − ak

q2j+1 = x2j+1 ,

p2j = −iy2j ,

(iii) Introduce the notation q2j = −x2j ,

ω(a) = da log(1 − K) = −

p2j+1 = y2j+1 ,

2m 2m X X (−1)i R(ai , ai )dai = Gi dai , i=1

i=1

2m X (xi yk − yi xk )2 , Gi = x2i + yi2 − ai − ak k=1 k=i

and define the canonical symplectic structure by {xj , xk } = 0,

{yj , yk } = 0,

{xj , yk } =

1 δj,k . 2

Verify that the two pairs of equations in (ii) are equivalent to the Hamilton equations da xj = {xj , ω(a)} and

da yj = {yj , ω(a)}.

Furthermore, verify that the Gi s are in involution so that {Gi , Gj } = 0.

393

FREDHOLM DETERMINANT APPROACH

9.4 THE AIRY KERNEL 9.4.1 Painlev´e II equation The Airy kernel refers to the kernel K soft (X, Y ) :=

Ai(X)Ai (Y ) − Ai(Y )Ai (X) , X −Y

(9.39)

and the task is to compute soft E2soft ((s, ∞); ξ) = det(1 − ξK(s,∞) ).

(9.40)

In [518] it is emphasized that a key ingredient in the derivation of differential equations associated with kernels of the form (9.25) is that φ and ψ are related by the coupled first order differential equations m(x)φ (x) = A(x)φ(x) + B(x)ψ(x), m(x)ψ  (x) = −C(x)φ(x) − A(x)ψ(x),

(9.41) √  where m, A, B and C are polynomials. For the Airy kernel (9.39), φ(x) = ξAi(x) and ψ(x) = ξAi (x), and since Ai (x) = xAi(x) we see that the equations (9.41) are satisfied with √

m(x) = 1, A(x) = 0, B(x) = 1, C(x) = −x.

(9.42)

Using (9.41), the working of Section 9.3 can be modified to derive coupled nonlinear equations for quantities associated with the Fredholm determinant of the Airy kernel on the interval (a1 , a2 ) = (s, ∞) [519]. The equations involve two quantities in addition to those of Definition 9.3.1. D EFINITION 9.4.1 Let φ, ψ, P and Q be as defined in Definition 9.3.1. In terms of these quantities we write where f |g :=

 a2 a1

u := φ|Q

and

v := φ|P  = ψ|Q,

f (y)g(y)dy.

P ROPOSITION 9.4.2 For the Airy kernel (9.39), in the notation of Definitions 9.3.1 and 9.4.1 with a1 = s, a2 = ∞, p := p1 , q := q1 and R := R(s, s), we have q  = p − qu, p = sq + pu − 2qv, R = pq  − qp , R = −q 2 , u = −q 2 , v  = −pq. The derivation of these equations will be given in the next subsection. Presently we use of them to derive a single differential equation for R =: R(s; ξ). From this the generating function E2soft ((s, ∞); ξ) that there are no eigenvalues in the interval (s, ∞) can be calculated, since from the general formula (9.26) we have R(s; ξ) = and so E2soft ((s, ∞); ξ)

d log E2 ((s, ∞); ξ) ds   = exp −

∞

 R(t; ξ)dt .

(9.43)

s

In (8.84) of Proposition 8.3.2 the same formula was obtained, but with R(t; ξ) replaced by us (t; 0; ξ). Indeed it follows from Proposition 9.4.2 that R(t; ξ) and us (t; 0; ξ) are the same quantity. P ROPOSITION 9.4.3 As with us (t; 0; ξ) specified in Proposition 8.3.2, the quantity R of Proposition 9.4.2 satisfies the differential equation   (R )2 + 4R (R )2 − sR + R = 0, (9.44)

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CHAPTER 9

and this equation is to be solved subject to the boundary condition R(t; ξ) ∼ ξρsoft (1) (t). t→∞

Proof. Let us refer to the equations of Proposition 9.4.2 according to their order of presentation. Differentiating the first of the stated equations above, and substituting for the first derivatives using the first, second and fifth equations gives q  = sq + q 3 + q(u2 − 2v). But from the first, fifth and sixth equations we see qq  = uu − v  and thus q 2 = u2 − 2v (the constant of integration is zero since all terms vanish as s → ∞). Substituting for u2 − 2v gives q  = sq + 2q 3 ,

(9.45)

which is the special case α = 0 of the Painlev´e II differential equation as listed in (8.9). Using the fourth equation it can be verified from this that R = q 2 − sq 2 − q 4 . The stated differential equation for R can now be verified using this equation together with further use of the fourth equation.

The derivation of (9.44) implies a formula for E2soft ((s, ∞); ξ) involving q = q(t; ξ), which according to (9.45) is a Painlev´e II transcendent, in contrast to R(t; ξ) which is a σPII transcendent. Thus, making use of the fourth equation in Proposition 9.4.2 and integrating by parts in (9.43), one obtains [519]   ∞  soft E2 ((s, ∞); ξ) = exp − (t − s)q 2 (t; ξ) dt , (9.46) s

where q is the solution of the Painlev´e II equation (9.45) satisfying the condition  q(s; ξ) ∼ ξAi(s) as s → ∞.

(9.47)

We remark that it has been proved [294] that (9.45) has a unique solution subject to (9.46) with ξ = 1. Furthermore, this solution exhibits the asymptotic behavior  q(s; 1) ∼ −s/2 (9.48) s→−∞

and consequently it follows from (9.46) that [519] E2soft (0; (s, ∞))

∼

s→−∞

3

es

/12

.

(9.49)

9.4.2 Numerical computation using the PII equation The evaluation (9.46) is well suited to the numerical computation of psoft 2 (n; t), provided n is small [451]. The strategy is first to calculate the asymptotic expansion of q(s) := q(s; ξ) in (9.45) about s = ∞, and to use this to obtain an accurate evaluation of q(s0 ) and q  (s0 ) for some particular s0 . These values are then used to obtain a power series expansion of q(s) about s = s0 , which in turn provides accurate evaluations of q(s1 ) and q  (s1 ) with s1 < s0 , and the procedure continues (cf. Section 8.3.4). P ROPOSITION 9.4.4 Write for the s → ∞ asymptotic expansion of the Airy function Ai(s) ∼

3/2 ∞ e−(2/3)s (−1)n √ 1/4 αn 2 πs ( 2 s3/2 )n n=0 3

 (3) so that [435] αn = (6n − 1)(6n − 5)αn−1 /72n, α0 = 1. Let αn = 0≤k≤l≤n αn−l αl−k αk and specify {an }n=0,1,... by the recurrence 1  5 3 1 n− n− an−2 (9.50) an = α(3) n + nan−1 − 4 8 6 6

395

FREDHOLM DETERMINANT APPROACH

subject to the initial conditions a−2 = a−1 = 0. We have q(s; ξ) ∼ Proof. Put q =

√

3/2 ∞  e−2s (−1)k ak ξAi(s) + ξ 3/2 . 3/2 7/4 32π s ( 2 s3/2 )k k=0 3

(9.51)

ξAi(s) + ξ 3/2 Q(s), where |Q(s)|  Ai(s) for s → ∞. Then Q ∼ sQ + 2(Ai(s))3 .

Since

(9.52)

3/2 ∞ X (−1)n (3) e−2s αn , 2 8π 3/2 s3/4 n=0 ( 3 s3/2 )n

(Ai(s))3 ∼ we see by substituting

3/2 ∞ X e−2s (−1)k ak 3/2 7/4 32π s ( 2 s3/2 )k k=0 3

Q(s) =

in (9.52) and equating like terms that (9.50) results. 3/2

It is empirically observed in [451] that the sum over k in (9.51) is optimally truncated at k ≈ 43 s0 for given s0  0. This then gives accurate numerical evaluations of q(s0 ; ξ) and q  (s0 ; ξ). For a general s0 , knowledge of c0 := q(s0 ; ξ) and c1 := q  (s0 ; ξ) allows the power series expansion q(s; ξ) =

∞

cl (s − s0 )l

(9.53)

l=0

to be computed by recurrence. (k)

P ROPOSITION 9.4.5 Let cn k

(q(s; ξ)) about s0 (note

(1) cn

:=

n

(k−1) j=0 cn−j cj

be the coefficients in the power series expansion of

= cn ). We have (3)

cn+2 =

2cn + s0 cn + cn−1 . (n + 2)(n + 1)

(9.54)

Proof. This follows by direct substitution of (9.53) in (9.45), and equating like powers of (s − s0 ). ∂ According to (8.86), (8.3) and (9.46), with u(t) := q(t; 1), v(t) := ∂ξ q(t; ξ)|ξ=1 , we have  ∞   d exp − (t − s)u2 (t) dt , psoft 2 (0; s) = ds s     ∞  d ∞ soft 2 psoft (1; s) = p (0; s) − 2 (t − s)u(t)v(t) dt exp − (t − s)u (t) dt . 2 2 ds s s

As it is not practical to leave ξ as a variable in the iteration of (9.54) for general c0 = q(s0 ; ξ) and c1 = q  (s0 ; ξ), we calculate the power series for v(s) about s0 from knowledge of the power series of u(s) and the initial values v(s0 ), v  (s0 ), by noting from (9.45) that v  = (s + 6u2 )v. The moments calculated from this procedure [449] are presented in Table 9.1. Numerical computation of the Fredholm determinant (9.40) according to the method of Section 9.2 allows for the high precision computation of {psoft 2 (n; t)} beyond n = 0 and 1 [79], and these too are presented in Table 9.1.

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CHAPTER 9

p.d.f.

mean

variance

skewness

kurtosis

psoft 2 (0; t) psoft 2 (1; t) psoft 2 (2; t) psoft 2 (3; t) psoft 2 (4; t) psoft 2 (5; t)

−1.77108 68074 −3.67543 72971 −5.17132 31745 −6.47453 77733 −7.65724 22912 −8.75452 24419

0.81319 47928 0.54054 50473 0.43348 13326 0.37213 08147 0.33101 06544 0.30094 94654

0.22408 42036 0.12502 70941 0.08880 80227 0.06970 92726 0.05777 55438 0.04955 14791

0.09344 80876 0.02173 96385 0.00509 66000 −0.00114 15160 −0.00405 83706 −0.00559 9855

Table 9.1 Statistical properties of psoft 2 (n; t) for various n.

9.4.3 Further theory In this subsection the theory contained in [517] will be used to derive the equations of Proposition 9.4.2. Now, examination of the theory of Section 9.3.1 shows that for any integral operator on the interval (a1 , a2 ), with kernel KJ of the form (9.25), there are certain inter-relationships between the quantities of Definitions 9.3.1 and 9.4.1 which are always valid. These are referred to as the universal equations. P ROPOSITION 9.4.6 For j, k = 1, 2 we have qj pk − pj qk (j = k), a j − ak

∂ log det(1 − ξKJ ) = (−1)j−1 R(aj , aj ), ∂aj ∂qj ∂pj (iii) = (−1)k R(aj , ak )qk (j =  k), (iv) = (−1)k R(aj , ak )pk (j = k), ∂ak ∂ak ∂v ∂u = (−1)k qk2 , (vi) = (−1)k pk qk , (v) ∂ak ∂ak ∂w (vii) = (−1)k p2k , ∂ak

(i) R(aj , ak ) =

(ii)

(9.55)

where j, k = 1,2; R, aj , pj , qj and KJ are as in Definition 9.3.1; u and v are as in Definition 9.4.1, and a w := ψ|P  := a12 P (y)ψ(y) dy with P and ψ as defined in Definition 9.3.1. Proof. Equation (i) follows from Proposition 9.3.6, equation (ii) is just (9.26), while (iii) and (iv) follow from Proposition 9.3.10. To derive (v), we note that ∂ ∂u = ∂ak ∂ak

Z

a2

Z Q(y)φ(y) dy = (−1)k φ(ak )qk +

a1

a2

φ(y) a1

∂ Q(y) dy. ∂ak

Substituting the value of ∂Q(y)/∂ak from Proposition 9.3.10 and then noting that Z a2 “ Z a2 ” R(y, ak )φ(y) dy = x→a lim ρ(y, x) − δ(y − x) φ(y) dy = Q(ak ) − φ(ak ) k x∈(a1 ,a2 )

a1

a1

gives the stated result. Equations (vi) and (vii) are derived similarly.

In addition to the universal equations of Proposition 9.4.6 there are equations which depend on the specific form of ψ and φ. Let us take up the task of deriving these equations in the case m(x) = 1,

A(x) = α0 + α1 x, 



B(x) = β0 + β1 x,

C(x) = γ0 + γ1 x.

(9.56)

First we seek formulas for P (x) and Q (x) analogous to those given in Proposition 9.3.8 for the sine kernel. For this purpose the result analogous to Proposition 9.3.7 is required.

397

FREDHOLM DETERMINANT APPROACH

P ROPOSITION 9.4.7 With D denoting the operator for differentiation with respect to the independent variable, we have [D, (1 − ξKJ )−1 ] 2   . = α1 Q(x)P (y) + P (x)Q(y) + β1 P (x)P (y) + γ1 Q(x)Q(y) − (−1)k R(x, ak )ρ(ak , y). k=1

Proof. From (9.41) and (9.56) it follows that (Dx + Dy )ξK(x, y) ” B(x) − B(y) C(x) − C(y) A(x) − A(y) “ φ(x)ψ(y) + ψ(x)φ(y) + ψ(x)ψ(y) + φ(x)φ(y) = x−y x−y x−y “ ” = α1 φ(x)ψ(y) + ψ(x)φ(y) + β1 ψ(x)ψ(y) + γ1 φ(x)φ(y). The result follows from this according to the proof of Proposition 9.3.7.

The differentiation formulas for P (x) and Q(x) can now be obtained by following the method of the proof of Proposition 9.3.8. P ROPOSITION 9.4.8 We have Q (x) = α0 Q(x) + α1 Q1 (x) + (α1 v + γ1 u)Q(x) + β0 P (x) + β1 P1 (x) +(α1 u + β1 v)P (x) −

2

(−1)k R(x, ak )qk ,

k=1

P  (x) = −γ0 Q(x) − γ1 Q1 (x) + (α1 w + γ1 v)Q(x) − α0 P (x) − α1 P1 (x) +(α1 v + β1 w)P (x) − where Q1 (x) :=

 a2 a1

2

(−1)k R(x, ak )pk ,

k=1

yρ(x, y)φ(y) dy and P1 (x) :=

 a2 a1

yρ(x, y)ψ(y) dy.

The quantities Q1 (x) and P1 (x) can be written in terms of Q(x), P (x), u, v and w. P ROPOSITION 9.4.9 We have   Q1 (x) = xQ(x) − vQ(x) − uP (x)

  and P1 (x) = xP (x) − wQ(x) − vP (x) .

Proof. From the proof of Proposition 9.3.6 we have . [M, (1 − ξKJ )−1 ] = Q(x)P (y) − P (x)Q(y). Applying this to φ and ψ and rearranging gives the two stated formulas.

Substituting the results of Proposition 9.4.9 in Proposition 9.4.8, then substituting the resulting equations in the formula (9.32) for R(x, x) and taking the limit as x → aj gives the following formula to supplement the equations of Proposition 9.4.6. P ROPOSITION 9.4.10 We have R(aj , aj ) = qj2 (γ0 + γ1 aj − 2γ1 v − 2α1 w) + 2pj qj (γ1 u − α0 + α1 aj − β1 w) +p2j (β0 + β1 aj + 2α1 u + 2β1 v) +

2 k=1 k=j

(−1)k R(aj , ak )(qj pk − pj qk ).

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CHAPTER 9

According to (9.38) and Proposition 9.3.10, the formulas for the derivatives of Q(x) and P (x) give us ∂q ∂p formulas for ∂ajj and ∂ajj . P ROPOSITION 9.4.11 We have ∂qj = qj (α0 + α1 aj + γ1 u − β1 w) + pj (β0 + β1 aj + 2α1 u + 2β1 v) − (−1)k R(aj , ak )qk , ∂aj k=1 2

k=j

∂pj = pj (−α0 − α1 aj − γ1 u + β1 w) + qj (−γ0 − γ1 aj + 2γ1 v + 2α1 w) − (−1)k R(aj , ak )pk . ∂aj k=1 2

k=j

As our final equation we give the analogue of the second equality in (9.36), which follows from the first equality in (9.36), Proposition 9.4.7, the final statement in the proof of Proposition 9.3.9 and the first statement of Proposition 9.3.9. P ROPOSITION 9.4.12 We have 2  2 ∂ R(aj , aj ) = − (−1)k R(aj , ak ) + 2α1 qj pj + β1 p2j + γ1 qj2 . ∂aj k=1 k=j

Derivation of the d.e.’s for the Airy kernel For the Airy kernel, from (9.42) and (9.56), α0 = α1 = 0,

β0 = 1, β1 = 0,

γ0 = 0, γ1 = −1.

(9.57)

Also (a1 , a2 ) = (s, ∞), which means that p2 = q2 = R(a1 , a2 ) = 0 and p1 , q1 depend only on s. The first two equations of Proposition 9.4.2 can thus be seen to follow immediately from Proposition 9.4.11, the third equation from Proposition 9.4.10 and the first two equations, the fourth equation from Proposition 9.4.12, and the fifth and sixth equations from equations (v) and (vi) respectively of Proposition 9.4.7. E XERCISES 9.4

1. [231] The aim of this exercise is to use the two different evaluations of the soft edge gap probability to deduce an identity between Painlev´e II transcendents. (i) Use the first equation in (8.29), the definition of hII in Proposition 8.2.1 and the first of the Hamilton equations in (8.23) to show that 1 “ t ”˛˛  σII (t) = − 1/3 q  + q 2 + . ˛ 2 t→−21/3 t 2 Write q = qα (t) where α is the parameter in the PII equation of (8.9) to conclude from Proposition 8.3.3 and (9.43) that ” “ 1 “  t ”˛˛ 2 (t) + q−1/2 (t) + . (9.58) us (t; 0; ξ) = R (t; ξ) = − 1/3 q−1/2 ˛ 2 2 t→−21/3 t (ii) Use the fourth equation in Proposition 9.4.2 and (9.45) to deduce from (i) that 1 “  t ”˛˛ 2 (t) + q−1/2 (t) + . q02 (t) = 1/3 q−1/2 ˛ 2 t→−21/3 t 2 (In fact an identity of Gambier [254] gives −21/3 q02 (−2−1/3 t) =

d  2 (t) − t q/2 (t) − q/2 dt 2

(9.59)

399

FREDHOLM DETERMINANT APPROACH

valid for both  = ±.)

9.5 BESSEL KERNELS As seen in Chapter 7, there are two kernels which qualify for the title of a Bessel kernel, Ja (X 1/2 )Y 1/2 Ja (Y 1/2 ) − X 1/2 Ja (X 1/2 )Ja (Y 1/2 ) (9.60) 2(X − Y ) (Ja+1/2 (πρx)Ja−1/2 (πρy) − Ja+1/2 (πρy)Ja−1/2 (πρx)) . K s.s. (x, y) := (πρx)1/2 (πρy)1/2 2(x − y) (9.61)

K hard (X, Y ) :=

We will treat each separately. 9.5.1 Spacings at the hard edge Referring to the integral operator on J = (0, t) with kernel K hard (x, y) and resolvent kernel R(x, y), it follows from (9.26) that   s  E2hard ((0, s); ξ) =: E2hard ((0, s); ξ; a) = exp − R(t; ξ) dt , (9.62) 0

where R(t; ξ) := R(t, t). A theory analogous to that of the sine and Airy kernels can be developed to rederive the characterization of R(t; ξ) as a particular σPIII transcendent implied by (8.88) and Proposition 8.3.3. Furthermore, this development leads to an alternative formula expressing E2hard ((0, s); ξ) in terms of a particular transformed Painlev´e V transcendent √ √[520]. For the Bessel kernel (9.60), φ(x) = ξJa ( x) and ψ(x) = xφ (x). From the d.e. satisfied by the Bessel function it follows that (9.41) holds with m(x) = x, A(x) = 0, B(x) = 1 and C(x) = 14 (x − a2 ). Thus A(x), B(x) and C(x) are of the form assumed in (9.56) with a2 1 , γ1 = (9.63) 4 4 but now m(x) = x. In this case the kernel specific formulas based on Proposition 9.4.7 require modification as the commutator [D, (1 − ξKJ )−1 ] no longer has a simple kernel. Instead it is the commutator [M D, (1 − ξKJ )−1 ] which is used to give the desired formulas. α0 = α1 = 0,

β0 = 1, β1 = 0,

γ0 = −

P ROPOSITION 9.5.1 We have [M D, (1 − ξKJ )−1 ] 2   . = α1 Q(x)P (y) + P (x)Q(y) + β1 P (x)P (y) + γ1 Q(x)Q(y) − (−1)k ak R(x, ak )ρ(ak , y). k=1

Proof. Analogous to the calculation of [D, K] in the proof of Proposition 9.3.7 we find . [M D, K] = (xDx + yDy + 1)K(x, y) − a2 δ − (y − a2 )K(x, y) + a1 δ + (y − a1 )K(x, y). But from (9.41) and (9.56) we see that (xDx + yDy + 1)K(x, y) is given by the r.h.s. of the equation in the proof of Proposition 9.4.7. The stated result now follows from the workings of the proof of the final identity in Proposition 9.3.7.

Due to the similarity of this result with the result of Proposition 9.4.7, by inspecting the role of the commutator [D, K] in the derivations of the formulas for R(aj , aj ) and similar given in the previous section, we

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see that these formulas require only minor modification. P ROPOSITION 9.5.2 In the case m(x) = x the formulas of Propositions 9.4.10 and 9.4.12 for R(aj , aj ) and ∂R(aj , aj )/∂aj hold provided a factor of aj is inserted on the l.h.s. in front of R(aj , aj ) and a factor of ak is inserted in the summand on the r.h.s.. Similarly, the formulas of Proposition 9.4.11 for ∂pj /∂aj and ∂pj /∂aj hold provided a factor of aj is inserted on the l.h.s. and a factor of ak is inserted in the summand on the r.h.s. The analogue of the equations of Proposition 9.4.6 obtained for the Airy kernel can now be obtained. P ROPOSITION 9.5.3 Consider the Bessel kernel (9.60). In the notation of Definitions 9.3.1 and 9.4.1 with a1 = 0, a2 = s, p := p2 , q := q2 and R := R(s, s) we have sq  = p + 41 qu, sp = 14 (a2 − s)q + 12 qv − 14 pu, (sR) = 14 q 2 , u = q 2 ,

v  = pq.

(9.64)

Proof. The first and second equations follow from Proposition 9.4.11 with j = 2, modified according to Proposition 9.5.2 and with the substitutions (9.63). The third equation follows from Proposition 9.4.12, while the fourth and fifth equations follow immediately from equations (v) and (vi) of Proposition 9.4.6.

Making use of the equations (9.64), a single non-linear equation can be derived for the quantity R. P ROPOSITION 9.5.4 With R(t; ξ), set σ(t; ξ) = −tR. As with uh (t; a, 0; ξ) in (8.88), σ(t) satisfies the differential equation (8.90) with μ = 0, subject to the boundary condition (8.92). d to both Proof. A procedure similar to that used in the derivation of (9.44) can be followed. We begin by applying s ds

sides of the first equation in (9.64), and eliminating the first derivatives by using the second, fourth and fifth equations. This gives 1 1 2 1 s(sq  ) = (a2 − s)q + (9.65) (u + 8v)q + sq 3 . 4 16 4 Next, by multiplying the first equation in (9.64) by 2q and using the fourth and fifth equations we see q 2 + 2sqq  = 2v  +

1  uu + u , 2

which implies

sq 2 = 2v +

1 2 u + u. 4

Substituting for u2 + 8v thus allows (9.65) to be rewritten as s(sq  ) =

1 2 1 1 (a − s)q − uq + sq 3 . 4 4 2

(9.66)

Multiplying this equation by q  , adding − 21 (sR) to the l.h.s. and − 81 q 2 to the r.h.s. (this is permitted by the third equation of (9.64)) and antidifferentiating shows −sR + (sq  )2 =

1 2 1 (a − s − u)q 2 + sq 4 . 4 4

But from the third and fourth equations u = 4sR, so (q 2 − 1)sR =

1 2 1 (a − s)q 2 + sq 4 − (sq  )2 . 4 4

(9.67)

Use of this equation, together with the third equation of (9.64), gives the stated differential equation for −tR.

The above working shows that the quantity q can also be characterized as the solution of a non-linear equation. First note substituting sR = u/4 in (9.67) gives an equation expressing (q 2 − 1)u in terms of q. Therefore after multiplying (9.66) by (1 − q 2 ) we can substitute for the term involving (q 2 − 1)u on the

401

FREDHOLM DETERMINANT APPROACH

r.h.s. to obtain the differential equation 1 1 s(q 2 − 1)(sq  ) = q(sq  )2 + (s − a2 )q + sq 3 (q 2 − 2). (9.68) 4 4 The corresponding boundary condition, which follows from the boundary condition for σ(s; ξ) := −sR in Proposition 9.5.4 and the third equation in (9.64), is √ ξ q(s; ξ) ∼ a sa/2 . (9.69) s→0+ 2 Γ(1 + a) The equation (9.68) is related to a Painlev´e equation by a fractional linear transformation. Thus making the transformation 1 + y(x) , s = x2 , q(s) = (9.70) 1 − y(x) one finds that y(x) satisfies the Painlev´e V equation in (8.9) with α = −β = a2 /8, γ = 0 and δ = −2. Furthermore, making use of the third equation in (9.64) we deduce from (9.62) that  1 s  s hard E2 ((0, s); ξ; a) = exp − (log )q 2 (t; ξ) dt , (9.71) 4 0 t which shows that q(t; ξ) determines E2hard ((0, s); ξ). 9.5.2 Symmetrical gap about the spectrum singularity In Section 8.3.5 the generating function E2s.s. ((0, t); ξ) for n eigenvalues in the interval (0, t) with a spectrum singularity at the origin was computed. In this section we will compute the generating function E2s.s. ((−t, t); ξ) for E2s.s. (n; ξ), the probability that there are exactly n eigenvalues in the symmetric interval (−t, t) about the spectrum singularity at the origin. Let pbulk 2,n.n. (0; t) denote the probability density function for the spacing between nearest neighbour eigenvalues in the bulk. This is distinct from pbulk (0; t) which represents the p.d.f. for the spacing between consec2 utive eigenvalues in the bulk, and thus the spacing distribution of the average of the spacing between the left neighbor and the right neighbor, only one of which being the nearest neighbor. It follows from the definitions that  d s.s.  pbulk E2 (0; (−t, t)) . 2,n.n. (0; t) = − dt a=1 We consider the integral operator in (9.15) with K(x, y) = K s.s. (x, y)|ρ=1/π . With R(x, y) denoting the kernel of the corresponding resolvent operator and R(t; ξ) := R(t, t), it follows from (9.26) that  πρt   E2s.s. (0; (−t, t)) = exp − 2 R(s; ξ) ds . (9.72) 0

Here, in the terminal of the integral t has been replaced by πρt to reinstate the general density, and use has been made of the fact that R(s; ξ) is even in s. According to (9.61), the kernel K s.s. (x, y)|ρ=1/π is of the type (9.25) with   ξx ξx Ja+1/2 (x), Ja−1/2 (x), φ(x) = ψ(x) = (9.73) 2 2 and the equations (9.41) hold with m(x) = x,

α0 = −a, α1 = 0,

β0 = 0, β1 = 1,

γ0 = 0, γ1 = 1.

In relation to this kernel, the following coupled equations can be derived [218].

(9.74)

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CHAPTER 9

P ROPOSITION 9.5.5 In the notation of Definitions 9.3.1 and 9.4.1 with a1 = −t, a2 = t, p := p2 , q := q2 , R := R(t, t), relating to the kernel (9.61) with ρ = 1/π and a ∈ Z≥0 , we have (ii) tq  = (−a + u − w)q + tp, (iv) (tR) = p2 + q 2 , (vi) w = 2p2 .

(i) tR = 2(−a + u − w)pq + t(p2 + q 2 ) + 2(pq)2 (iii) tp = −tq − (−a + u − w)p, (v) u = 2q 2 , Proof. For a ∈ Z≥0 , we have from (9.73) that φ(−x) = (−1)a−1 φ(x),

ψ(−x) = (−1)a ψ(x),

which gives K s.s. (x, y) = K s.s. (−x, −y) and thus ρ(x, y) = ρ(−x, −y). From Definition 9.3.1 we then see that q1 = (−1)a−1 q2 = (−1)a−1 q,

p1 = (−1)a p2 = (−1)a p,

v = 0.

(9.75)

Equation (i) can now be derived from (9.4.10) with j = 2, modified according to Proposition 9.5.2, by substituting (9.75) and (9.74). Equations (ii) and (iii) follow from Proposition 9.4.11, modified according to Proposition 9.5.2, after noting from equation (i) of Proposition 9.4.6 and (9.75) that R(−t, t) = (−1)a

pq . t

Equation (iv) follows from Proposition 9.4.11, modified according to Proposition 9.5.2, now with use being made of the general formula ˛ “ ∂ ∂ ” d ˛ − . (9.76) f (−t, t) = f (a1 , a2 )˛ dt ∂a2 ∂a1 a2 =−a1 =t Equations (v) and (vi) result from the universal equations (v) and (vii) of Proposition 9.4.6, together with (9.76).

From the equations of Proposition 9.5.5, a procedure similar to that used to derive Proposition 9.3.3 from Proposition 9.3.2 allows σ(s; ξ) = −sR(s/2; ξ) to be specified as the solution of a certain nonlinear equation. P ROPOSITION 9.5.6 The quantity σ(s; ξ) := −sR(s/2; ξ), where R(s; ξ) occurs in (9.72), satisfies the differential equation   2   =0 (9.77) (sσ  )2 + 4(−a2 + sσ  − σ) (σ  )2 − a − (a2 − sσ  + σ) subject to the boundary condition σ(s; ξ) ∼ −ξ

2(s/4)2a+1 Γ(1/2 + a)Γ(3/2 + a)

as s → 0.

(9.78)

Proof. Consider the equations of Proposition 9.5.5. Multiply (ii) by p, multiply (iii) by q, add and use (v) to obtain (pq) = p2 − q 2 =

1  (w − u ). 2

Antidifferentiating gives pq =

1 (w − u) 2

which together with (iv) allows (i) to be rewritten as tR = −2a(pq) − 2(pq)2 + t(tR) .

(9.79)

Another equation relating tR to pq is obtained by squaring (iv), squaring the formula (pq) = p2 − q 2 obtained above, and subtracting. This gives ((pq) )2 − ((tR) )2 = −4(pq)2 . (9.80) Solving (9.79) for pq (it follows from a small t expansion that the negative square root is to be taken) and (pq) , substituting in (9.80) and introducing the notation σ(2t) := −2tR gives (9.77). The boundary condition follows from the fact

403

FREDHOLM DETERMINANT APPROACH

˛ ˛ that R(t; ξ) ∼ ξK s.s. (t, t)˛

ρ=1/π

as t → 0, and the corresponding behavior of K s.s. (t, t) deduced from (7.49).

The second order second degree equation (9.77) involves a square root and so is not a σ-form Painlev´e equation (8.15). However (8.15) lists σPIII ; if instead one considers the Hamiltonian theory of PIII rather than PIII , the equation (9.77) can be recognized in this context and σ(s; ξ) identified as an auxiliary Painlev´e III Hamiltonian (see Exercises 9.5 q.1). 1. [548] The Hamiltonian for the PIII system, in contrast to the PIII  system (recall Section 8.2.1), is given by [429] “ ” tHIII = 2q 2 p2 − 2tq 2 + (2v1 + 1)q − 2t p + (v1 + v2 )tq,

E XERCISES 9.5

where the parameters v1 , v2 relate to the parameters in the PIII equation of (8.9) by α = −4v2 ,

β = 4(v1 + 1),

(i) With hIII (t) := tHIII +

γ = 4,

δ = −4.

1 (2v1 + 1)2 8

show that hIII = −2q 2 + 2p + (v1 + v2 )q,

8(hIII − thIII ) = (4pq − 2v1 − 1)2 .

Differentiate the latter and use the Hamilton equations (8.23) to show −thIII = (4pq − 2v1 − 1)(2pq 2 + 2p − (v1 + v2 )q). (ii) Use the results of (i) to verify that for  = ±1 “ (thIII )2 = [2(hIII − thIII )] 4(hIII )2 + 16[2(hIII − thIII )] q “ 1 ”“ 1 ”” −16(v2 − v1 − 1) 2(hIII − thIII ) − 16 v2 − v1 + ) . 2 2 (iii) For  = ±, show that with v1 = −a −

1 , 2

v2 = a +

1 , 2

σ(t) = 2hIII

“t” − a2 4i

the equation in (ii) reduces to (9.77). 2. Make use of (7.53), (8.101), (9.15) and (9.61) to show ˛ “ ± ˛ ch E2O (s; ξ) = det 1 − ξK±1/2 ˛

(0,s)

” ,

(9.81)

ch |(0,s) is the integral operator on (0, s) with kernel (7.54). where K±1/2

9.6 EIGENVALUE EXPANSIONS FOR GAP PROBABILITIES 9.6.1 Commuting differential operators and E2 (n; J) We have seen that the logarithmic derivatives of the Fredholm determinants (9.15) satisfy second order nonlinear differential equations of Painlev´e type. In this section we will show that the eigenvalues of the integral operators KJscale can be obtained from the eigenvalues of certain second order linear differential operators. As a consequence, the asymptotic form of the eigenvalues for large values of the parameter determining J

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CHAPTER 9

can be determined, and this in turn allows the asymptotic form of E2scale (n; J)/E2scale (0; J) — the ratio of the probability that there are n eigenvalues in J to the probability that there are no eigenvalues in J — to be computed [517]. To appreciate this last point, we first remark that for each of the three cases under consideration (no direct analysis of scale = s.s. is considered) KJscale has distinct eigenvalues which can be labeled by a non-negative integer and so ordered E2scale (n; J) =

∞  (1 − ξλl ).

(9.82)

l=0

Use of this formula in (8.1) gives E2scale (n; J) λj1 · · · λjn . = scale (1 − λj1 ) · · · (1 − λjn ) E2 (0; J) 0≤j 0 for each l and for all values of the parameter specifying J. A lower bound on the λl can also be deduced. For this each integral operator KJscale , scale = bulk, soft, hard, is written as the square of another integral operator, thereby implying λl ≥ 0 for each l. Thus the following results hold [259], [519], [520]. ˜ bulk be the integral operator on (−1, 1) defined by P ROPOSITION 9.6.1 Let K (−1,1)  1 sin(t(x − y)) ˜ bulk [f ] = f (y) dy. K (−1,1) π(x − y) −1 We have ˜ bulk = (V bulk )† V bulk , K (−1,1) (−1,1) (−1,1)

 bulk V(−1,1) [f ] =

 t 1/2 eitxy f (y) dy. −1 2π 1

˜ soft be the integral operator on (0, ∞) defined by Let K (0,∞)  ∞  ∞ ˜ soft [f ] = dy du Ai(x + u + t)Ai(y + u + t)f (y). K (0,∞) 0

0

We have

 ˜ soft = (V soft )2 , K (0,∞) (0,∞)

soft V(0,∞) [f ] =

∞

Ai(x + u + s)f (u) du. 0

˜ hard be the integral operator on (0, 1) defined by Let K (0,1)  1  1 √ √ ˜ hard [f ] = t K dy du Ja ( txu)Ja ( tyu)f (y). (0,1) 4 0 0 We have ˜ hard = (V hard )2 , K (0,1) (0,1)

hard V(0,1) [f ]

√  1 √ t = Ja ( txu)f (u) du. 2 0

405

FREDHOLM DETERMINANT APPROACH

. R

.

Proof. In general, if VJ = V (x, y), then VJ† VJ =

J

V (u, x)V (u, y) du, so the results are immediate.

˜ bulk , K ˜ soft , K ˜ hard result from K bulk |ρ=1/π , K soft , K hard respectively, with the The operators K (−1,1) (0,∞) (0,1) (−t,t) (t,∞) (0,t) kernels written in integral form as given in Exercises 7.1 q.1, after a simple change of variable to scale the t-dependence from the terminals of integration, a transformation which leaves unchanged the eigenvalues. The decompositions of Proposition 9.6.1 lead to the characterizations of the eigenvalues in (9.15) in terms of the eigenvalues of differential operators. This comes about because for each of the integral operators VJ , one can construct a second order linear differential operator of the form L=

d d α(x) + β(x) dx dx

(9.84)

which commutes with VJ . . P ROPOSITION 9.6.2 Let V[a1 ,a2 ] = V (x, y) and let L be given by (9.84). If α(a1 ) = α(a2 ) = 0 and  ∂   ∂ ∂ ∂ α(x) + β(x) V (x, y) = α(y) + β(y) V (x, y), (9.85) ∂x ∂x ∂y ∂y then L commutes with V[a1 ,a2 ] . Proof. We have Z

“ ∂ ” ∂ α(x) + β(x) V (x, y)f (y) dy ∂x ∂x a Z 1a2 “ ” ∂ ∂ = α(y) + β(y) V (x, y)f (y) dy ∂y ∂y a1 Z a2 Z a2 “ ” ∂ ∂ β(y)V (x, y)f (y) dy V (x, y) α(y) f (y) dy + =− ∂y ∂y a1 a1 Z a2 “∂ ” ∂ = V (x, y) α(y) + β(y) f (y) dy, ∂y ∂y a1 a2

LV[a1 ,a2 ] [f ] =

where the second equality follows from (9.85), while the third and fourth equalities follow by integration by parts and use of the assumption that α(y) vanishes at y = a1 , a2 .

One can verify that the assumption on α(x) and condition (9.85) are satisfied with αbulk (x) = x2 − 1, αsoft (x) = x, αhard (x) = x(1 − x),

β bulk (x) = t2 x2 , β soft (x) = −(x + t)x, a2 β hard (x) = − 4x − tx 4 .

Consider first the bulk case. Then the corresponding eigenfunctions Sn (x; t) =: Sn (x) say, are known as the prolate spheroidal functions. They form an orthonormal set on (−1, 1) and are entire functions of x, which are real for real x (in the limit t → 0+ they reduce to the Legendre polynomials, up to normalization). For n even Sn (x) is an even function of x, while for n odd, Sn (x) is an odd function of x. bulk Because Lbulk commutes with V(−1,1) it follows that the Sn (x) =: Sn (x; t) defined as the eigenfuncbulk bulk tions of L are also the eigenfunctions of V(−1,1) . Using the parity property of the Sn (x) gives that the corresponding eigenvalue equation can be written  t 1/2  1 cos xyt Sn (y; t) dy = μn (t)Sn (x; t), n even, 2π −1  t 1/2  1 i sin xyt Sn (y; t) dy = μn (t)Sn (x; t), n odd, 2π −1

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where the μn (t) are the eigenvalues. The fact that Sn (x) is real for real x implies that μn (t) is real for n even and is pure imaginary for n odd. It also shows that the Sn (x) are the eigenfunctions of V † , with eigenvalues ˜ bulk = (V bulk )† V bulk with corresponding eigenvalues |μn (t)|2 =: λn (t). μ ¯n (t), and thus of K (−1,1) (−1,1) (−1,1) The computation of the large t asymptotics of the λn (t) makes use of the Hellmann-Feynman formula [296], [174], which gives a relation between λn (t) and the eigenfunction Sn (x; t) evaluated at the special point x = 1. P ROPOSITION 9.6.3 Let K(x, y; t) be a symmetric (in x and y) kernel on (a1 , a2 ) depending smoothly on t. Let gj (x; t) be an eigenfunction, normalized so that  a2 (gj (x; t))2 dx = 1, a1

with corresponding eigenvalue λj (t). We have  a2   dx gj (x; t) λj (t) = a1

In particular, for the operator μj (t) =

bulk V(−1,1)

a2

dy gj (y; t) a1

∂ K(x, y; t). ∂t

of Proposition 9.6.2 and with a1 = −1, a2 = 1, this gives

μj (t) 2 λj (t) 2 Sj (1; t) or equivalently λj (t) = 2 Sj (1; t). t t

Proof. For the first statement, we follow [519]. Differentiating the eigenvalue equation for gj (x; t) gives Z

a2

a1

∂ K(x, y; t)gj (y; t) dy + ∂t

Z

a2

K(x, y; t) a1

∂ ∂ gj (y; t)dy = λj (t)gj (x; t) + λj (t) gj (x; t). ∂t ∂t

Multiplying both sides by gj (x; t), integrating over x, and using the normalization, the eigenvalue equation and the symmetry in x and y of K gives the desired result. The first equation of the second result follows simply from the first result by noting that ∂ itxy x ∂ itxy e e = , ∂t t ∂x and using integration by parts. The second part follows from the first part and the fact that λj = (−1)j μ2j .

Applying the WKB method of asymptotic analysis to be the eigenvalue equation for the differential oper√ ator L, it was shown by Fuchs [245] (after the identification Sj (x; t) = a−1/2 fj (ax; t), a = t, where fj is the eigenfunction studied in [245]) that the eigenfunctions have the asymptotic behavior ±Sj (1; t) = 2(3j+2)/2 π 1/4 (j!)−1/2 t(j+1/2)/2 e−t μj (t)(1 + o(1)) for j fixed and t → ∞. Substituting in the final equation of Proposition 9.6.3 gives 1 π 1/2 j+1 j+1/2 −2t  λ 8 t (t) = e (1 + o(1)), j λ2j (t) j! and this, when integrated from t to ∞, implies  π 1/2 j+1 ∞ j+1/2 −2t 4π 1/2 j j+1/2 −2t 1 − 1 ∼ 8 8 t t e dt ∼ e  λj (t ) j! j! t (here use has been made of the fact that λj (t) ∼ 1 as t → ∞). Thus one has the following result [245]. ˜ bulk have the asymptotic behavior P ROPOSITION 9.6.4 The eigenvalues λj (t) of the integral operator K (−1,1) 1 − λj (t) ∼ 4π 1/2 8j (j!)−1 tj+1/2 e−2t for j fixed and t → ∞.

(9.86)

407

FREDHOLM DETERMINANT APPROACH

In the soft and hard edge cases the differential operator implied by (9.86) does not lead to special functions previously studied from this viewpoint. Nonetheless an analysis parallel to that sketched above leading to (9.86) can be carried out to obtain the following results [519], [520]. P ROPOSITION 9.6.5 Let i be fixed. One has √  8  t 3/2  π 5i+3  t 3i/2+3/4 soft 2 − − ∼ exp − 1 − λi t→−∞ i! 2 3 2 √ 2π hard i+(a+1)/2 −2 t 4i+2a+2 1 − λi t ∼ e 2 . t→∞ Γ(a + i + 1)i!

(9.87) (9.88)

Substituting (9.86), (9.87), (9.88) for the first term of (9.83) gives the sought asymptotic forms for the l.h.s. of the latter. P ROPOSITION 9.6.6 Let G(x) denote the Barnes G-function. For n fixed E2bulk (n; t) E2bulk (0; t)

G(n + 1)π −n/2 2−n

2

∼

t→∞

−n/2

(πρt)−n

2

/2 nπρt

e

,

 8n  t 3/2  E2soft (n; (t, ∞)) G(n + 1) −3n2 /4 − ∼ , (−t/2) exp 2 3 2 E2soft (0; (t, ∞)) t→−∞ π n/2 2(5n +n)/2 E2hard (n; (0, t); a) E2hard (0; (0, t); a)

G(a + n + 1)G(n + 1) −n −n(2n+2a+1) −n2 /2−an/2 2n√t π 2 t e , G(a + 1)

∼

t→∞

(9.89)

where in the bulk case the setting of considering the interval J = (−t, t) and ρ = 1/π has been changed to the interval J = (0, t) with a general ρ by scaling t. 9.6.2 Further asymptotics The results of Proposition 9.6.6 can be supplemented by specifying the asymptotic behavior of E2bulk (0; t), E2soft (0, (t, ∞)) and E2hard (0; (0, t)) as the size of the interval gets large. This in turn is related to the asymptotic form of the Painlev´e transcendents specifying these quantities. A log-gas argument, given in Section 14.6, predicts that for some unspecified case dependent constants c, log E2bulk (0; t) ∼ −ct2 , t→∞

log E2soft (0, (t, ∞))

∼

t→−∞

log E2hard (0, (0, t)) ∼ −ct.

ct3 ,

t→∞

(9.90)

Rigorous determination of the first of these behaviors, which also specifies the proportionality constant as c = π 2 /8 (assuming ρ = 1), can be found in [543], [129]. The second result is just (9.49) and so for this c = 1/12. The behaviors (9.90) imply specific leading behaviors of the corresponding Painlev´e transcendents, and with this established the differential equations generate unique asymptotic expansions. The latter allow (9.90) to be extended [517], [519], [520]. P ROPOSITION 9.6.7 We have  e−(πρt) /8  1 1− + O((πρt)−4 ) , 2 1/4 t→∞ 8(πρt) (πρt)   csoft −|t|3 /12 3 2025 −9 1 + e + + O(t ) , E2soft (0; (t, ∞)) ∼ t→−∞ (|t|)1/8 26 |t|3 213 t6 √   1  e−t/4+a t a 9a2 hard + O 3/2 E2 (0; (0, t); a) ∼ chard 1 + 1/2 + , t→∞ 128t ta2 /4 8t t 2

E2bulk (0; t)

∼

cbulk

(9.91)

where cbulk , csoft , chard have the values cbulk = 21/6 e3ζ



(−1)

,

csoft = 21/24 eζ



(−1)

,

chard =

G(1 + a) . (2π)a/2

(9.92)

408

CHAPTER 9

Proof. It follows from (8.1) and the first equation in (8.111) that Z

πt

log E2bulk (0; t) = 0

σ(s; 1) ds, s

(9.93)

where σ satisfies the first equation in (8.112). To reproduce the leading behavior (9.90) with c as specified it must be that σ(s; 1) ∼ − s→∞

s2 . 4

In fact the differential equation satisfied by σ has a unique solution of the form σ(s; 1) = a2 s2 + a0 + a−2 s−2 + · · · which from the knowledge that a2 =

− 14

(a2 = 0)

is calculated as

σ(s; 1) = −

“1” s2 5 131 1 1 − − 2 − 4 − 6 +O 8 . 4 4 4s 2s 2s s

Substituting in (9.93) and reinstating the general density ρ gives the first expansion in (9.91). The value of the undetermined constant τbulk can be deduced from a result in the theory of Toeplitz determinants (see Exercise 9.6 q.1). A rigorous determination is also known [363], [163], [127]. An analogous strategy applies in the soft and hard edge cases. Thus according to (9.43) and the first equation in (8.93) Z ∞ Z t ˛ ds ˛ log E2soft (0; (t, ∞)) = − R(s; 1) ds, log E2hard (0; (0, t)) = − σIII  (s)˛ μ=0 , (9.94) s t 0 ξ=1 the σPIII equation in (8.15) with v1 = v2 = where R(s; 1) satisfies the differential equation (9.44) and σIII  (s) satisfies P 2−j , which calculation shows is a. The first of these differential equations has a unique solution of the form ∞ j=0 a2−j s given by “1” 1 9 s2 − + R(s) ∼ + O , 4 8s 64s4 s7 while the second has a unique solution of the form σ(s) = c1 s + c1/2 s1/2 + c0 + c−1/2 s−1/2 + · · · provided the sign in √ a2 = ±a is fixed. Choosing the minus sign (this makes sense physically from the resulting asymptotic expression for log E2 (0; (0, s))) allows the expansion σ(s) =

a a −1/2 a2 −1 s a2 − s1/2 + + s s + ··· + 4 2 4 16 16

(9.95)

to be generated. Substituting as appropriate in (9.94) gives the final two expansions in (9.91), up to the constants. In the soft edge case the constant follows from the result [126], [30] that for c < 0 Z ∞ Z c “ 1 ” 1 3 1 1 1 |c| + log |c|. R(y; 1) dy + R(y; 1) − y 2 + dy = − log 2 − ζ  (−1) + 4 8y 24 12 8 c −∞ For a ∈ Z≥0 , the constant in the hard edge case can be deduced from the integral formula (8.96), as is done for a more general integral formula in (13.52) below. It’s validity for −1 < a < 1 has been proved in [162].

The expansions (9.90) and (9.92) in the hard edge case with a = ± 21 allow us to deduce the large t expansions of Eβbulk (n; t) for β = 1 and 4 [49]. First the case n = 0 will be considered. According to (8.139), E1bulk (0; s) = E2hard (0; (0, (πs/2)2 )|a=−1/2 , where on the l.h.s. the density has been set equal to 1. It thus follows from the final formula in (9.91) that 2  1  e−(πρt) /16−πρt/4  bulk 1/4 . (9.96) 1 + O E1 (0; t) ∼ (2π) G(1/2) t→∞ t (πρt/2)1/8

409

FREDHOLM DETERMINANT APPROACH

Furthermore, it follows from (8.159) that E4bulk (0; t) ∼

t→∞

1 hard E (0; (0, (πt)2 )|a=1/2 . 2 2

(9.97)

The final formula in (9.91) then gives  1  G(3/2) e−(πρt) /4+πρt/2  . 1 + O t→∞ 2(2π)1/4 t (πρt)1/8 2

E4bulk (0; t) ∼

(9.98)

The asymptotic formulas for Eβbulk (0; t), β = 1, 2, 4 are all consistent with the form e−β(πρt) /16+(β/2−1)πρt/2 . (πρt)(1−(2/β)(1−β/2)2 )/4 2

Eβbulk (0; t) ∼ cβ t→∞

(9.99)

Valko and Vir´ag [525] have used the stochastic sine equation (13.179) below to prove this form for general β > 0. In relation to the general n case, the formula (8.153) tells us that  hard n even E2 (n/2; (πρt/2)2 )|a=−1/2 , bulk E1 (n; t) ∼ E2hard ((n − 1)/2; (πρt/2)2 )|a=1/2 , n odd. t→∞ Making use of both (9.90) and (9.92) it follows from this that E1bulk (n; t) G(n/2 + 1/2)G(n/2 + 1) −n/2 −n2 /2 π ∼ 2 (πρt/2)−n(n−1)/4 enπρt/2 . G(1/2) E1bulk (0; t) t→∞ Similarly, from (8.159) we have

 1 hard  E2 (n; (πρt)2 ) . t→∞ 2 a=1/2

E4bulk (n; t) ∼

(9.100)

(9.101)

Recalling (9.97) and making use of (9.90) this gives 2 E4bulk (n; t) G(n + 3/2)G(n + 1) −n −2n(n+1) π 2 ∼ (πρt)−n −n/2 e2nπρt . bulk t→∞ G(3/2) E4 (0; t)

(9.102)

We see from Proposition 9.6.6, (9.100) and (9.102) that the asymptotic formulas for Eβbulk (n; t)/Eβbulk (0; t), β = 1, 2, 4 are all consistent with the form Eβbulk (n; t)

eβnπρt/2

∼ cβ,n , (πρt)βn2 /4+(β/2−1)n/2 Eβbulk (0; t) t→∞

(9.103)

which can in fact be obtained from a log-gas analysis (see Section 14.6.2).

9.6.3 Some gap probabilities at β = 1 with an evenness symmetry In Section 8.4.2 the equation (8.149) relating gap probabilities for the COE (β = 1 quantities) to gap probabilities in the JUE (β = 2 quantities) was shown to hold for ξ = 1. Here (8.149), along with some companion identities, will be derived for general ξ via a calculation which uses evenness symmetry as well as the eigenvalues and eigenvectors of an underlying Fredholm integral operator [395], [197]. First we will derive a determinant formula for the generating function  ∞  ∞ 2N   1 (l) E2N,1 ((−t, t); ξ; w1 ) := dx1 · · · dx2N w1 (xl )(1 − ξχ(−t,t) ) |xk − xj | Zˆ2N −∞ −∞ l=1

1≤j · · · > lp . We seek ld/rd the total weight GN (l(0) ; l) of all allowed nonintersecting ld/rd lattice paths of N segments starting at (0) (0) (0) (0) {li }i=1,...,p , l1 > l2 > · · · > lp , and finishing at {li }i=1,...,p . This can be determined as a special case of a theorem due to Linstr¨om, Gessel and Viennot (see, e.g., [475]) on computing the generating function for the total weight of a general class of nonintersecting lattice paths.

441

LATTICE PATHS AND GROWTH MODELS

y

y

x

x 1

1

2

4

1

Figure 10.1 A configuration of 3 nonintersecting left diagonal/right diagonal lattice paths of 4 steps starting two units apart (leftmost diagram). An equivalent configuration of up/right diagonal lattice paths starting one unit apart is given in the rightmost diagram, and the paths are also encoded as a semi-standard tableau, with the kth column corresponding to the kth lattice path from the right. The total weight of the right steps in the   configuration can be read off from the semi-standard tableau as w1+#1 s · · · wp+#p s .

P ROPOSITION 10.1.1 Let D denote a directed acyclic graph (i.e., no loops), and let u = (u1 , . . . , ur ), v = (v1 , . . . , vr ) be sets of vertices in D so that in connecting {ui } to {vi } by nonintersecting paths along edges of the graph, the only possibility is to connect ui to vi for each i = 1, 2, . . . , r. Let each edge of the graph be weighted, let h(u, v) denote the total weight of all single paths from u to v, and let H({u}, {v}) denote the total weight of all nonintersecting paths starting at {u} and finishing at {v}. Then H({u}, {v}) = det[h(ui , vj )]i,j=1,...,r .

(10.1)

Proof. Following [502], we note that by definition det[h(ui , vj )]i,j=1,...,r =

X

sgn(σ)h(u1 , vσ(1) ) · · · h(ur , vσ(r) )

(10.2)

σ∈Sr

(recall (5.22)). With Pi denoting a path from ui to vσ(i) , and w(Pi ) := h(ui , vσ(i) ) its corresponding weight, this expansion can be viewed as a generating function for (r + 1)-tuples (σ, P1 , . . . , Pr ), assigned weight sgn(σ)w(P1 ) · · · w(Pr ). We want to show that in the sum all such weights of intersecting paths cancel in pairs, while for the nonintersecting paths sgn(σ) = 1. Choose a total ordering of the vertices so that u1 < · · · < ur < v1 < · · · < vr . Consider a (σ, P1 , . . . , Pr ) with at least one pair of intersecting paths, and let v be the smallest vertex in the total ordering such that the paths intersect. Let i and j be the smallest indices such that paths Pi and Pj pass through v, and introduce the notation Pk (→ v) and Pk (v →) to denote the subpaths of Pk from uk to v and v to vσ(k) . Next define (σ  , P1 , . . . , Pr ) such that Pl = Pl for l = i, j and Pi = Pi (→ v)Pj (v →), Pj = Pj (→ v)Pi (v →), σ  = σ ◦ (i, j) (see Figure 10.2). We note that the set of edges of (P1 , . . . , Pr ) is identical to the set of edges of (P1 , . . . , Pr ) and thus w(P1 ) · · · w(Pr ) = w(P1 ) · · · w(Pr ), while sgn(σ  ) = −sgn(σ). Consequently sgn(σ)w(P1 ) · · · w(Pr ) + sgn(σ  )w(P1 ) · · · w(Pr ) = 0.

(10.3)

Since the graph D is assumed acyclic, this new set of paths has the same set of intersection points as before (as illustrated in [502], if D has cycles an intersection between two paths may be mapped to a self-intersection, thus violating this property). Hence this construction is an involution and so gives a unique pairing of intersecting paths each with the property (10.3), demonstrating that their contribution cancels in (10.2).

442

CHAPTER 10

P1’

P2’

P1’

v

P1

P2

P2’

v

P1

P2

Figure 10.2 The intersecting lattice paths from P1 to P1 and P2 to P2 in the second diagram are constructed from the intersecting lattice paths from P1 to P2 and P2 to P1 according to the prescription in the proof of Proposition 10.1.1. It remains to show that the nonintersecting paths have sgn(σ) = 1. In fact it follows from the assumption on {u} and {v} that the only time nonintersecting paths occur is when σ = I (the identity), so this property is immediate.

To apply Proposition 10.1.1 to nonintersecting ld/rd lattice paths, note that the ld and rd segments form the edges of a directed graph, and that the generating function (total weight) for single lattice paths from l(0) to l is  π  N (0) 1 ld/rd (wj− e−iθ + wj+ eiθ )e−i(l−l )θ dθ =: gN (l(0) ; l). (10.4) 2π −π j=1 Thus ld/rd

GN

  ld/rd (0) (l(0) ; l) = det gN (lj , lk )

j,k=1,...,p

.

(10.5)

The case that the initial and final sites are all odd, l(0) = (2p − 1, 2p − 3, . . . , 1),

l = (2l1 − 1, . . . , 2lp − 1),

(10.6)

or all even, so that the initial points are the minimum allowed distance apart (2 lattice sites), has some special features. First it is easy to see that then each configuration of nonintersecting lattice paths is equivalent to a configuration of nonintersecting lattice paths starting one site apart with segments either vertical or diagonal and to the right (see Figure 10.1). u/rd Let GN denote the total weight of the up/right diagonal (abbreviated u/rd) nonintersecting paths in the case that each rd path segment at step j is weighted qj , and each vertical path segment is weighted unity. From Proposition 10.1.1, for general initial conditions   u/rd u/rd (0) GN (m  (0) ; m)  = det gN (mj , mk ) , (10.7) j,k=1,...,p

where, with ej denoting the elementary symmetric functions (polynomials) as defined in (4.132) (en := 0 for n ∈ Z− ),  π  N (0) 1 u/rd (0) gN (m ; m) = (1 + qj eiθ )e−i(m−m )θ dθ 2π −π j=1 = em−m(0) (q1 , . . . , qN ).

(10.8)

In the case that the paths start one unit apart, the number of distinct u/rd nonintersecting paths can be written in terms of the components of m  in a simple form [365].

443

LATTICE PATHS AND GROWTH MODELS

P ROPOSITION 10.1.2 Let m  (0) = (p − 1, p − 2, . . . , 0) and set qj = 1 (j = 1, . . . , p). We have u/rd

GN

(m  (0) ; m)  =

p 

(N + p − i)! m !(p + N − 1 − mi )! i i=1



(mj − mk ).

(10.9)

1≤jy(s) >y(s+1) .

s=1

1≤jx(s+1) >···>x(s) >x(s+1) . 1

1

2

s

s+1

Under the stated limiting conditions, this give (10.68).

One recognizes (10.68) as the eigenvalue p.d.f. (5.196) (in the case w(x) = e−x ) of the GUE minor process [333]. 2

462

CHAPTER 10

10.2.3 Tiling of the Aztec diamond Another tiling problem which gives rise to nonintersecting lattice paths is domino tiling of the so-called Aztec diamond lattice. The latter can be defined as the union of all vertices and edges which lie on lattice squares [m, m + 1] × [l, l + 1], (m, l ∈ Z) within the diamond shaped region {(x, y) : |x| + |y| ≤ n + 1}. A domino tiling is a covering of this lattice by 2 × 1 and 1 × 2 rectangles whose corners lie on the lattice points. To associate the tiling with lattice paths, color the squares in the Aztec diamond alternating white-black, choosing the left-most square in the top half as white, and the left-most square in the bottom half as black. This gives a checkerboard pattern in which the squares alternate between white-black and black-white down any column or across any row. Then for a horizontal domino which covers a white-black (black-white) pair of squares, reading left to right, no (horizontal) segment of a path is marked; for a vertical domino which covers a white-black (black-white) pair of squares, reading top to bottom, an ld (−rd) segment of a path is marked. As seen in Figure 10.11, the tiling then maps uniquely to certain returning nonintersecting rd/−ld/rh paths (recall Section 10.1.4).

Figure 10.11 Mapping between a domino tiling of the Aztec diamond and nonintersecting rd/−ld/rh paths.

From the correspondence the total number of distinct tilings can be calculated [164]. P ROPOSITION 10.2.5 The number of domino tilings of the Aztec diamond lattice bounded by {(x, y) : |x| + |y| ≤ n + 1} is equal to 2n(n+1)/2 . Proof. This follows by setting hk = k − 1, p = n + 1, w = 1 in the formula of Proposition 10.1.11. E XERCISES 10.2

1. [330] In this exercise a grand canonical ensemble of ww,ld/rd lattice paths is studied.

(i) In (10.63) write M → 2M + 1 and note that then Z 1 lim G(M + x, M + y) = cos πt(x − y) M →∞

(ii) With 2w > 1 and θ0 := kernel

0 2 1 arcos 2w , π

Z

θ0

(2w cos πt/2)2N dt. 1 + (2w cos πt/2)2N

take the N → ∞ limit of the result in (i) to obtain as the limiting

cos πt(x − y) dt =

0

sin πθ0 (x − y) . π(x − y)

(iii) Verify that for j, k = 1, . . . , [(M − 1)/2] [(M −1)/2]

X l=1

ww,ld/rd

gN

(2j − 1, 2l − 1) sin

“ π(2l − 1)k π(2j − 1)k πk ”N sin = 2 cos . M M M

463

LATTICE PATHS AND GROWTH MODELS

Conclude that the eigenvalues of ww,ld/rd

[gN are given by {(2 cos

(2j − 1, 2l − 1)]j,l=1,...,[(M −1)/2]

πk N ) }k=1,...,[(M −1)/2] M

and use this to obtain the evaluation [(M −1)/2] “

ZN,M (ζ)[1] =

Y

k=1

“ πk ”N ” 1 + ζ 2 cos . M

(10.69)

10.3 DISCRETE POLYNUCLEAR GROWTH MODEL 10.3.1 Robinson-Schensted-Knuth correspondence A u/rh and u/lh pair of nonintersecting lattice paths, initially equally spaced, can be put into one-to-one correspondence with a matrix of integers, a result (with a pair of paths recorded as a pair of semi-standard tableaux) usually referred to as the Robinson-Schensted-Knuth (RSK) correspondence [246]. The non-negative integer matrix can itself be thought of as recording events in a discrete space and time growth process [329]. Thus consider an n × n square non-negative integer matrix X = [xi,j ]i,j=1,...,n with rows numbered from the bottom, rotated 45◦ anticlockwise. Label the horizontal rows of the rotated matrix by t = 1, 2, . . . , 2n − 1 and the vertical columns by x = 0, ±1, . . . , ±(n−1), where x = 0 corresponds to the diagonal of the original matrix (see Figure 10.12). The entries xi,j in the matrix for successive t values (t = i + j − 1) are heights of “nucleation events” — columns of unit width and height xi,j centered about the corresponding x-coordinate which are placed on top of the profile formed by earlier nucleation events and their growth. Thus at t = 1 there is a nucleation event at x = 0 which consists of a column of width 1 and height x1,1 marked on the line at y = 0 in the xy-plane. In general, as t → t + 1 the profile of all nucleation events so far recorded is to “grow” one unit in the −x direction and one unit in the +x direction. Thus in going from t = 1 to t = 2 the nucleation event centered at x = 0 of height x1,1 now has width 3 units. On top of this profile, centered at x = −1 and x = 1 nucleation events of unit width and height x2,1 , x1,2 , respectively are then drawn. In now going from t = 2 to t = 3 this new profile is to grow one unit to the left and one unit to the right. In so doing we see that an overlap of width one unit and height min(x2,1 , x1,2 ) will occur. This overlap is ignored in the first diagram (profile on y = 0), and recorded instead as a profile on the line immediately below (here y = −1). The process is repeated with these rules until the nucleation event of height xn,n at t = 2n − 1 has been recorded above x = 0 on the first diagram. The boundary of each profile forms a pair of lattice paths — one to the left and one to the right of x = 0. The lattice paths for x < 0 start at x = −(2n − 32 ) and go either up (in integer amounts with each unit regarded as a step) or to the right (in steps of two units) until they reach x = − 21 , while the lattice paths for x > 0 start at x = (2n − 3/2) and go either up or right until they reach x = 21 . In the y-direction the paths start one unit apart at y = 0, . . . , −(n − 1) (see Figure 10.12). Conversely, each such family of nonintersecting paths corresponds to a unique n × n non-negative integer matrix, which can be constructed by reversing the above procedure, so we thus have a bijection between paths and matrices. The paths in the pair are separately equivalent to the lattice path type of Figure 10.3 and thus to a semi-standard tableau, so the bijection is also between non-negative integer matrices and pairs of semi-standard tableaux of the same shape (the latter constraint comes about because the pairs of paths have the same finishing points). The above correspondence between integer matrices and pairs of nonintersecting lattice paths can be extended to a correspondence between weighted integer matrices and pairs of weighted nonintersecting lattice paths. For this each entry xi,j in the integer matrix is weighted by (1 − ai bj )(ai bj )xi,j , with the factor (1 − ai bj ) representing a normalization allowing the weighting to be interpreted as a probability from a ge-

464

CHAPTER 10

b3

0 1

a3 a2

5 4 3 2

1

b1

2 0

2 a1

t=3

t

b2

0 1

a3 a2

4 b1

a12 b2

1

1

y=0

a2

y=−1 0 t=4

x −2 −1 0 1 2

2 b1

3 a2

a3

t=1

b1

a1

4 b1

y=0

a12

0 b2

t=2 b13

b2

y=0

a2

y=−1

a22

0 a12

t=5

y=0

2 b3

a3 a2 3

4 b1 a12 b3

a3 a2

b2 0

y=0 y=−1

Figure 10.12 Mapping from a weighted non-negative integer matrix to a pair of weighted nonintersecting lattice paths.

ometric distribution. Since each entry xi,j corresponds to a nucleation event which adds a unit column of height xi,j to the top profile, the weight can be recorded n on the profile by weighting the left side of the new x x column by bj i,j and the right side by ai i,j (we take i,j=1 (1 − ai bj ) as an overall normalization factor). If columns should overlap as the profiles grow, the resulting column recorded on the profile below is to carry a left side and a right side weight which are exactly equal to those erased in the overlap, thus conserving the total vertical weight in the direction of any side. Crucially the rules of the growth process ensure that at each time step the weight in the direction of any one side is proportional to one particular bj (left sides) or one particular ai (right sides). At the end of the process this means that each vertical step at x = −(2n + 12 − 2j) is weighted by bj , and each vertical step at x = 2n + 12 − 2j by aj (see Figure 10.12). The profile with endpoints at y = −l + 1 (l = 1, . . . , n) will be referred to as the level-l path, and its evolution forms a growth process known as the discrete polynuclear growth model [450], [330]. Of interest is the statistical properties of the profile, in particular its maximum displacement μ1 . To study this quantity the maximum displacements μl of the level-l paths for each l = 1, 2, . . . , n are relevant. The nonintersecting condition implies μ1 ≥ μ2 ≥ · · · ≥ μn so μ = (μ1 , μ2 , . . . , μn ) forms a partition. We know from (10.24) that with each of the vertical steps at x = −(2n + 21 − 2j) weighted by bj , the total weight of all nonintersecting u/rh paths (with horizontal steps two units) initially equally spaced at y = 0, . . . , −(n−1) along x = −(2n− 32 ) and finishing at y = μ1 , μ2 −1, . . . , μn −(n−1) along x = −1/2 is given by the Schur polynomial sμ (b1 , b2 , . . . , bn ). Similarly, with vertical steps at x = (2n + 12 − 2j) each weighted by aj , the total weight of all nonintersecting u/lh (with horizontal steps two units) initially equally spaced at y = 0, . . . , −(n − 1) along x = 2n − 32 and finishing at y = μ1 , μ2 − 1, . . . , μn − (n − 1) along x = 12 is given by the Schur polynomial sμ (a1 , a2 , . . . , an ). It follows that if the non-negative integers xi,j in the matrix X are independent geometric random variables with parameter ai bj and thus Pr(xi,j = k) = (1 − ai bj )(ai bj )k ,

(10.70)

then the probability that such an integer matrix X corresponds to a pair of nonintersecting lattice paths with

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LATTICE PATHS AND GROWTH MODELS

maximum heights μ is given by n 

(1 − ai bj )sμ (a1 , a2 , . . . , an )sμ (b1 , b2 , . . . , bn ).

(10.71)

i,j=1

A fundamental statistical quantity relating to the maximum height μ1 =: h of the level-1 path, namely its cumulative probability density, follows from this by summing over μ : μ1 ≤ l. Thus Pr(h ≤ l) =

n 

(1 − ai bj )



sμ (a1 , a2 , . . . , an )sμ (b1 , b2 , . . . , bn ).

(10.72)

μ: μ1 ≤l

i,j=1

We remark that since the Schur polynomials are symmetric functions this probability is symmetric in {ai } and {bj }, a feature which is not obvious from the definitions. We note too that the normalization condition for this probability implies the Cauchy identity (10.58). From the combinatorial definition of the Schur polynomial in (10.16), the sum over μ in (10.72) can be regarded as a sum over all pairs of weighted semi-standard tableaux of shape μ, content n and with row length no greater than l. Now we know from Figure 10.1 that such tableaux uniquely code weighted ld/rd nonintersecting lattice paths initially two sites apart consisting of at most l paths and n steps. This can be extended to exactly l paths by appending a suitable number of paths starting immediately to the left of those already present and which move only ld (such paths are not recorded as part of the tableau). A pair of tableaux of the same shape, content n and row length no greater than l, then corresponds to l returning ld/rd nonintersecting lattice paths of 2n steps in which both the initial and final spacings are two sites apart. This viewpoint allows Pr(h ≤ l) to be written in terms of a random matrix average [34]. P ROPOSITION 10.3.1 We have Pr(h ≤ l) =

l n    (1 − ai bj ) (1 + aj e−iθk )(1 + bj eiθk )

n  i,j=1

j=1 k=1

CUEl

.

(10.73)

Proof. From the returning paths interpretation, Pr(h

≤ l) =

n Y

ld/rd

(1 − ai bj )G2n

(0)

({lj

= 2(l − j) + 1}j=1,...,l ; {lk = 2(l − k) + 1}k=1,...,l ),

(10.74)

i,j=1

where the weights in G2n are specified by ws− = as , ws+ = 1 (s = 1, . . . , n), ws+ = bs , ws− = 1 (s = n + 1, . . . , 2n). The result now follows from (10.48). ld/rd

10.3.2 Joint probabilities Consider the measure on the space of non-negative integer matrices implied by (10.70). Then, as discussed in Section 10.3.1, the probability an n1 × n2 non-negative integer matrix maps to a pair of semi-standard tableaux with shape μ, one of content n1 , the other of content n2 , under the RSK correspondence is equal to n2 n1  

(1 − ai bj )sμ (a1 , . . . , an1 )sμ (b1 , . . . , bn2 )

(10.75)

i=1 j=1

(cf. (10.71)). Note that this vanishes for (μ) > min(n1 , n2 ). Next we seek the joint probability that an n1 × (n2 + 1) non-negative integer matrix with measure implied by (10.70) corresponds to a pair of semi-standard tableaux with shape μ, contents n1 and n2 + 1, and that the n1 × n2 bottom left sub-block corresponds to a pair of semi-standard tableaux with shape κ and contents n1 and n2 . According to the remark below (10.75), for the probabilities to be nonzero one requires (μ) ≤ min(n1 , n2 + 1) and (κ) ≤ min(n1 , n2 ).

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The growth model picture of the RSK correspondence tells us that μl equals the height of the level-l path at x = ±1/2, while κl equals the heights of the level-l path at x = − 32 . The inequalities (10.104) imply that μ1 ≥ κ1 ≥ μ2 ≥ κ2 ≥ · · · ≥ κn ≥ μn ,

(10.76)

where n = min(n1 , n2 + 1). The nonintersecting paths corresponding to these heights have total weights sμ (a1 , . . . , an1 )

and

sκ (b1 , . . . , bn2 )

to the right and left, respectively. The path segments connecting the level-l path at x = − 23 to x = − 21 have n1 n2 +1 l weight bμnl2−κ +1 . Noting too that the normalization for the paths is j=1 k=1 (1 − aj bk ), the following result is obtained [222]. P ROPOSITION 10.3.2 Consider an n1 ×(n2 +1) non-negative integer matrix with entries chosen according to (10.70). The joint probability that this matrix maps under the RSK correspondence to a tableau of shape μ, and that the n1 × n2 matrix obtained by deleting the final column maps to a tableaux of shape κ is equal to n1 n 2 +1 Pn  (μl −κl ) (1 − aj bk )sμ (a1 , . . . , an1 )sκ (b1 , . . . , bn2 )bn2l=1 , +1 j=1 k=1

where n = min(n1 , n2 + 1), supported on partitions interlaced according to (10.76) and with (μ) ≤ n1 , (κ) ≤ n2 . Suppose now that under the RSK correspondence with weights chosen according to (10.70), the pair of semi-standard tableau corresponding to a particular n1 × n2 matrix (n1 > n2 ) has shape κ. It then follows from Proposition 10.3.2 and (10.71) that the probability an n1 × (n2 + 1) matrix, obtained by adding an extra column to the existing matrix, of having shape μ is P (μ, κ) = χ(μ > κ)

n1 P 2  sμ (a1 , . . . , an1 ) nj=1 (μj −κj )+μn2 +1 (1 − ai bn2 +1 ) , bn2 +1 sκ (a1 , . . . , an1 )

(10.77)

i=1

where χ(μ > κ) = χ(μ1 ≥ κ1 ≥ μ2 ≥ · · · ≥ μn2 ≥ κn2 ≥ μn2 +1 ≥ 0). This in turn allows for an extension of Proposition 10.3.2. P ROPOSITION 10.3.3 Consider an n1 × p (n1 ≥ p) non-negative integer matrix with entries chosen according to (10.70). The joint probability that under the RSK correspondence this matrix is such that the principal n1 × s submatrices (s = 0, . . . , p) corresponds to pairs of semi-standard tableaux with shape μ(s) (s) (note that μi = 0 for i > s) is equal to p n1   i=1 j=1

(1 − ai bj )sμ(p) (a1 , . . . , an1 )

p 

Ps−1

bs

(s) (s−1) )+μ(s) p j=1 (μj −μj

χ(μ(s) > μ(s−1) ).

(10.78)

s=1

Proof. Since in the case n = Q 0 (10.71) can be taken as equal to unity, the sought joint probability is obtained from (s+1) , μ(s) ). (10.77) by forming the product p−1 s=0 P (μ 10.3.3 The exponential limit and continuous RSK With ai = e−αi /L , bi = e−βi /L and the height xi,j at each site scaled by xi,j → Lxi,j we see that the discrete geometrical distribution (10.70) becomes the exponential distribution Pr(xi,j ∈ [y, y + dy]) = (αi + βj )e−(αi +βj )y dy,

y ≥ 0.

(10.79)

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LATTICE PATHS AND GROWTH MODELS

This distribution can be used to define a probability measure on n×n matrices with non-negative real numbers as entries. The RSK correspondence gives a bijective mapping with a pair of u/rh nonintersecting lattice paths, each starting two units apart along y = 0 and finishing at y1 > y2 > · · · > yn > 0 along x = 0, in which the up steps are continuous and weighted by an exponential variable proportional to the increment. Explicitly, with the entries of X distributed according to (10.79), the RSK mapping requires that at x = −(2n + 12 − 2j) −βj v (x = 2n + 12 − 2j) the up increment v of each level-l path with (e−αj v ), and nl ≤ j is to be weighted e there is an overall weighting of the paths by the normalization i,j=1 (αi + βj ). 1 The total weight of a single path with vertical increments nof length vj at x = −(2n+ 2 −2j) (j = l, . . . , n), −βj vj weighted by e with prescribed total displacement j=l vj = y is given by  ∞  ∞ n n   e−βj y n =: ul ({βj }j=l,...,n ; y). (10.80) dvl e−βl vl · · · dvn e−βn vn δ y − vj = μ=l (βj − βμ ) 0 0 j=l

j=l

μ=j

A (semi-)continuous analogue of Proposition 10.1.1 then gives that the corresponding total weight of the set of continuous nonintersecting paths for x < 0 is det[ul ({βj }j=l,...,n ; yk )]k,l=1,...,n . Multiplying this by the weight of the paths for x > 0, and the normalization, we see that the continuous analogue of (10.71) is [222], [87] n 

(αi + βj ) det[ul ({αj }j=l,...,n ; yk )]k,l=1,...,n det[ul ({βj }j=l,...,n ; yk )]k,l=1,...,n

i,j=1

n

= n

i,j=1 (αi

i<j (αj

+ βj )

− αi )(βj − βi )

det[eαj yk ]j,k=1,...,n det[eβj yk ]j,k=1,...,n .

(10.81)

In the special case αi = a + (i − 1)c,

βj = a ˜ + (j − 1)˜ c

(10.82)

e−(a+(l−1)c)yk (1 − e−cyk )n−l , cn−l (n − l)!

(10.83)

(10.80) can be simplified to read ul ({αj }j=l,...,n ; yk ) =

and similarly for ul ({βj }j=l,...,n ; yk ). Substituting in (10.81) and making use of the Vandermonde determinant identity (1.173) we obtain the p.d.f. n n      1 (a + a ˜ + (i − 1)c + (j − 1)˜ c) (c˜ c)j−1 Γ2 (j) i,j=1 j=1 Pn  ×e−(a+˜a) j=1 yj (e−cyj − e−cyi )(e−˜cyj − e−˜cyi ). (10.84) 1≤i<j≤n

˜) → In the case c = c˜, after the change of variables and replacement of parameters e−cyj → yn+1−j , (a + a c(α + 1), (10.84) reduces to the JUE supported on (0, 1) with a = α, b = 0, while with a + a ˜ = 1 and c = c˜ → 0 it reduces to the LUE with a = 0. Thus, with E(0; J; ME) denoting the probability of having no eigenvalues in the interval J of the matrix ensemble ME, in the case c = c˜ we see that  1 1 Pr l1 ≤ log = E(0; (0, s); JUEn | a=α ), b=0 c s where the JUE is supported on (0, 1) and 0 < s < 1. Similarly, in the case c = c˜ → 0, Pr(l1 ≤ s) = E(0; (s, ∞); LUEn |a=0 ). In Exercises 10.3 q.1 it is shown how a modification of (10.81) leads to more general examples of the JUE

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and LUE. The p.d.f. (10.84) can also be obtained from (10.71), by first specializing the parameters −1 (a1 , . . . , aN ) = (z1 , z1 t1 , z1 t21 , . . . , z1 tN ), 1

−1 (b1 , . . . , bN ) = (z2 , z2 t2 , z2 t22 , . . . , z2 tN ), 2

then simplifying the Schur polynomials according to (10.22). After setting t1 = e−c/L , t2 = e−˜c/L , z1 = e−˜a/L , (μi + N − i)/L = yi and taking the limit as L → ∞ we see that (10.71) multiplied by L2N tends to (10.84). This is to be expected as then the geometric distribution reduces to (10.79) with αi and βj therein equal to (10.82). Continuous RSK limits of joint probabilities can also be taken. In particular, substituting in (10.78) ai = (s) (s) bj = e−1/2L , writing (μi + s − i)/L = xi and taking L → ∞ (note that then Pr(xi,j ∈ [x, x + dx]) = −x e dx) gives, upon use of (10.23), a functional form known from the study of the recursive construction of (s) Wishart matrices in Exercises 4.3 q.4. Explicitly, the joint p.d.f. for {xi }i=1,...,s , s = 1, . . . , p is proportional to p  (p) (p) (xl )n1 −p e−xl



(p)

(xi

p−1 

χ(x(s+1) > x(s) ),

(10.85)

s=1

1≤i<j≤p

l=1

(p)

− xj )

where χ(x(s+1) > x(s) ) is specified as in (4.100). (s) In the growth model picture the variables xs correspond to the heights λ1 (n1 , s) on the outer profile. It has been shown in Proposition 7.1.4 that with the horizontal distance along the profile O(N 2/3 ), the correlation (s) between the variables xj , appropriately scaled, is given by the dynamical extension of the Airy kernel (7.19). It thus follows that the distribution function for two heights along the profile with this separation have distribution in terms of the dynamical Airy kernel as implied by the result of Exercises 9.1 q.3.

10.3.4 Correlation functions and the Borodin-Okounkov identity The joint p.d.f. (10.71) for the heights in the polynuclear growth model is, in view of the determinant formula (10.16) for the Schur polynomials, of the form (5.139) so the theory of Section 5.8 applies in relation to the calculation of the correlation functions. In fact, by using the determinant formula (10.27) instead of (10.16), a double contour form of the corresponding correlation kernel in the limit n → ∞ is possible [430]. P ROPOSITION 10.3.4 With μ a partition, write μk − (k − 1) =: nk , and consider the joint p.d.f. on n implied by (10.71). For n → ∞ we have ρ(l) (n1 , . . . , nl ) = det[K(nα , nβ )]α,β=1,...,l , where

 K(k, l) =

Cr1

dα 2πiαk

 Cr2

dβ 1 H(1/β; {bi })H(α; {ai }) . 2πiβ −l+1 α − β H(β; {ai })H(1/α; {bi })

(10.86)

(10.87)

Here r1 > 1 > r2 > 0 and Cr denotes a circle about the origin of radius r, while H(u; {qi }) :=

∞  j=1

1 . 1 − qj u

Proof. In the following some formal manipulations on semi-infinite Toeplitz determinants are carried out; for their justification we refer to [456], [331].

469

LATTICE PATHS AND GROWTH MODELS

In terms of {nk }, after use of (10.27) the joint p.d.f. (10.71) reads ∞ Y

h i (1 − ai bj ) det hnk +j−1 ({ai })

i,j=1

j,k=1,...,n

h i det hnk +j−1 ({bi })

j,k=1,...,n

.

As this is a symmetric function in {nk } which vanishes if nj = nk (j = k), we can relax the ordering constraint n1 > n2 > · · · implied by the definition of {nk }. According to the result of Proposition 5.8.1 with ξj (n) = hn+j−1 ({ai }), ηj (n) = hn+j−1 ({bi }), for n → ∞ the l-point correlation is given by (10.86) with ∞ X

K(p, q) =

cjk hp+j−1 ({ai })hq+j−1 ({bi }),

(10.88)

j,k=1

P −1 where [cjk ]j,k=1,2,... = ([ ∞ . l=−∞ ξj (l)ηk (l)]j,k=1,2,... ) In preparation for computing the matrix inverse, introduce the notation for a semi-infinite Toeplitz matrix h i T [f (z)] = [z j−k ]f (z) j,k=1,2,...

(10.89)

for f (z) a Laurent expandable function. We then have ∞ X l=−∞

ξj (l)ηk (l) =

∞ X

hl−j ({ai })hl−k ({bi }) =

„ h i h i« T H(1/z; {ai }) T H(z; {bi }) ,

(10.90)

jk

l=1

where in obtaining the first equality the change of variables l → l − k − j has been performed and the fact that the summation vanishes for l < 1 used, while for the second equality use has been made of (10.25). Because (10.90) is a decomposition into the product of an upper triangular and a lower triangular Toeplitz matrix, and for such matrices in general the inverse is a Toeplitz matrix of the same type but with reciprocal generating function, it follows h [cjk ]j,k=1,2,... = T

i i h i hX 1 1 ej−l ({bi })ek−l ({ai }) , T = H(z; {bi }) H(1/z; {ai }) j,k=1,2,... l=1 ∞

where {ej } denote the elementary symmetric functions (4.132). This result in turn allows us to compute that ∞ X

cjk β k α−j =

j,k=1

assuming 0 < |β| < 1 < |α|. Consider now the generating function

1 β/α , 1 − β/α H(1/α; {bi })H(β; {ai })

∞ X

(10.91)

αk β −l K(k, l).

k,l=−∞

Substituting (10.88) and making use of (10.91) shows ∞ X

αk β −l K(k, l) =

k,l=−∞

1 H(1/β; {bi })H(α; {ai }) . 1 − β/α H(β; {ai })H(1/α; {bi })

(10.92)

This is equivalent to the sought result (10.87).

Suppose the coordinates {nj } are restricted to be less than or equal to l. According to the general theory of Section 9.1 we have that lim Pr(h ≤ l) = det(1 − KJ ),

n→∞

(10.93)

where KJ is the Wiener-Hopf operator on J = {l + 1, l + 2, . . . , } with kernel K(j, k). On the other hand

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(10.73) gives lim Pr(h ≤ l) =

n→∞

∞ 

l ∞    (1 − ai bj ) (1 + aj e−iθk )(1 + bj eiθk )

i,j=1

j=1 k=1

CUEl

.

(10.94)

Equate (10.93) and (10.94) (with θk → −θk ), and substitute ∞

∞

akl = kck ,

l=1

bkl = kc−k

l=1

for the corresponding power sum symmetric functions. Noting that with these substitutions, and uj = aj , uj = bj , respectively, we have ∞ 

∞   (1 − uj z) = exp − c∓k z k ,

j=1

(10.95)

k=1

and recalling the general identity (5.32) gives an identity expressing a general Toeplitz determinant in terms of a Fredholm determinant. ∞ P ROPOSITION 10.3.5 Let Dn [f ] be specified as in Proposition 14.4.1 below. With c(θ) := n=−∞ cn einθ we have Dn [ec(θ) ] = enc0 +

P∞

p=1

pc−p cp

det(1 − KJ ),

(10.96)

where KJ is the Wiener-Hopf operator on [n + 1, n + 2, . . . ) with kernel   ∞   dα dβ 1 exp ck (β k − αk ) + c−k (α−k − β −k ) . K(k, l) = k −l+1 α−β Cr1 2πiα Cr2 2πiβ k=1

A reformulation of (10.96), and an alternative derivation, are given in Exercises 10.3 q.2. This result was given in the context of random matrix theory by Borodin and Okounkov [86]. Later it was realized that the same identity had occurred in the context of Toeplitz determinant theory in an earlier study by Geronimo and Case [260]. In this context, observe that (10.96) effectively gives higher order corrections to the Szeg¨o formula (14.71) below. E XERCISES 10.3

1. [222] In this exercise a generalization of (10.84) will be derived.

(i) In the case that X is an n × m (n ≥ m) matrix of non-negative real numbers distributed according to (10.79), by taking bm+1 = · · · = bn = 0 in (10.71), and noting that we must then have (μ) ≤ m for this to be nonzero, argue that the continuous RSK correspondence maps the distribution on X to m n Y Y

(αi + βj ) det[ul ({αj }j=l,...,n ; yk )]k,l=1,...,m det[ul ({βj }j=l,...,m ; yk )]k,l=1,...,m ,

i=1 j=1

where ul is specified by (10.80). ˜ + (j − 1)˜ c this (ii) Make use of (10.83) (appropriately modified) to show that with αi = a + (i − 1)c, βj = a simplifies to read m n Y “Y

(a + a ˜ + (i − 1)c + (j − 1)˜ c)

m ”“ Y

i=1 j=1

×

m Y

(1 − e−cyk )n−m e−(a+˜a)

k=1

j=1 Pm

j=1

” 1 1 (c˜ c)j−1 Γ(j)Γ(n − m + j) c(n−m)m

Y

yj

1≤i<j≤m

Relate this to the JUE and the LUE, as done for (10.84).

(e−cyj − e−cyi )(e−˜cyj − e−˜cyi ).

(10.97)

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LATTICE PATHS AND GROWTH MODELS

2. [47], [92] The aim of this exercise is to give a reformulation and alternative derivation of (10.96). (i) Let K = [K(l, k)]k,l=1,... be the transpose of the semi-infinite matrix with entries as specified by (10.87). With φ(z) := 1/(H(1/z; {aj })H(z; {bj })) show that (10.96) can alternatively be written Dn [φ(eiθ )] = D∞ [φ(eiθ )] det(1 − Qn KQn ),

(10.98)

where Qn is the semi-infinite diagonal matrix differing from the identity by the deletion of the first n elements. (ii) With φ− (z) := 1/H(1/z; {aj }), φ+ (z) := 1/H(z; {bj }) so that φ(z) = φ− (z)φ+ (z), verify from (10.92) that ∞ X (φ− /φ+ )p+l−1 (φ+ /φ− )−p−k+1 K(k, l) = p=1

where we have made use of the notation fj = [z j ]f (z). After introducing the further notation H(a) = [aj+k−1 ]j,k=1,2,...

H(˜ a) = [a−j−k+1 ]j,k=1,2,... ,

(10.99)

conclude that K as specified in (i) can be written K = H(φ− /φ+ )H(φ + /φ− ).

(10.100)

(iii) In terms of the notation in (ii), together with T (a) = [aj−k ]j,k=1,2,... , by verifying the formula (ab)l =

∞ X

ap bl−p

p=−∞

show that

T (ab) = T (a)T (b) + H(a)H(˜b).

(10.101)

(iv) Let A be an N × N invertible matrix, let Pn,N be the diagonal matrix with the first n elements 1 and all other entries 0, and set Qn,N = 1N − Pn,N . Then, as already noted in (5.178) but in a different notation, a theorem of Jacobi gives [8] det Pn,N A−1 Pn,N =

det Qn,N AQn,N . det A

(10.102)

(note too that (1.150) corresponds to the case n = 1). Under general conditions this remains valid for N → ∞ and thus semi-infinite matrices. By noting from (10.101) that with A = 1 − K, K as in (10.100), Pn,∞ A−1 Pn,∞ = Pn,∞ T (φ− /φ+ )T (φ+ /φ− )Pn,∞ −1 = Pn,∞ T (φ−1 + )T (φ)T (φ− )Pn,∞ −1 = Tn (φ−1 + )Tn (φ)Tn (φ− ),

where Tn := Pn,∞ T Pn,∞ , deduce (10.98).

10.4 FURTHER INTERPRETATIONS AND VARIANTS OF THE RSK CORRESPONDENCE 10.4.1 Stochastic recurrences The nonintersecting paths underlying the discrete polynuclear growth process can be generated by recurrences which couple together the displacements along level-l and level-(l − 1) at varying positions along the paths. Furthermore, in the case of the top path, this recurrence allows for the derivation of some inter-relations between various growth models, and also for some different interpretations of the process relating to this path.

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CHAPTER 10

Let Xn1 ,n2 denote the first n1 rows and n2 columns of the integer matrix X. Extend this to a square . n1 ,n2 of dimension max(n1 , n2 ) × max(n1 , n2 ) by adjoining rows of zeros to the top (or columns matrix X of zeros to the right as appropriate), then apply the growth algorithm and record λ1 (n1 , n2 ), the maximum displacement of the corresponding level-1 path. Consideration of the rules of the growth process leads to an interpretation of λl (n1 , n2 ) for n1 = n or n2 = n as displacements in the paths relating to X itself. Thus one has that λl (n, j) gives the height of the level-l path at x = −(2n + 12 − 2j), while λl (i, n) gives the height of the level-l path at x = (2n + 12 − 2i). One immediate implication is the summations n  n n  n   λl (n, j) − λl (n, j − 1) = λl (i, n) − λl (i − 1, n) = xi,j , xi,j , (10.103) i=1

l=1

l=1

j=1

where λl (0, j) = λl (i, 0). These equate the increments at a given position to the sum of the nucleation events contributing to those increments. ˜ n1 ,n2 , the same considerations give that λl (n∗ , j) is equal to the height For the paths contributing to X of the level-l paths at x = −(2n∗ + 12 − 2j), while λl (i, n∗ ) is equal to the height of the level-l path at x = (2n+ 12 −2i), where n∗ = max(n1 , n2 ). One notes too that λl (n∗ , j) = λl (n1 , j), λl (i, n∗ ) = λl (i, n2 ). Further relevant points are that the λl (n1 − 1, n2 − 1) occur as the displacement of the level-l path at the origin for times t = 2n∗ − 2 (second-last step), and that λl (i, j) ≥ λl (i, j − 1) ≥ λl+1 (i, j),

λl (i, j) ≥ λl (i − 1, j) ≥ λl+1 (i, j).

(10.104)

These considerations, together with the rules of the growth process, imply recurrences for {λl (i, j)}. Thus the rules of the growth process give that λ1 (n1 , n2 ) is equal to the maximum of the heights in level-1 at x = −1 and x = 1, plus the nucleation height xn1 ,n2 , and so   (10.105) λ1 (n1 , n2 ) = max λ1 (n1 , n2 − 1), λ1 (n1 − 1, n2 ) + xn1 ,n2 , where λ1 (0, j) = λ1 (i, 0) = 0. Similarly, the maximum displacement λl (n1 , n2 ) of the level-l path (l = 2, . . . , n) satisfies   λl (n1 , n2 ) = max λl (n1 , n2 − 1), λl (n1 − 1, n2 ) + x(l−1) (10.106) n1 ,n2 , (l−1)

where xn1 ,n2 is the height of an overlap event (if any) which occurs in the growth of the nucleation events xn1 ,n2 −1 or xn1 −1,n2 and/or corresponding plateaux. This latter height is equal to the minimum of the height in level-(l − 1) at x = ±1, with the height at the origin in the second-last step subtracted so that   x(l−1) n1 ,n2 = min λl−1 (n1 , n2 − 1), λl−1 (n1 − 1, n2 ) − λl−1 (n1 − 1, n2 − 1). Substituting this in (10.106) gives that for l > 1   λl (n1 , n2 ) = max λl (n1 , n2 − 1), λl (n1 − 1, n2 )   + min λl−1 (n1 , n2 − 1), λl−1 (n1 − 1, n2 ) − λl−1 (n1 − 1, n2 − 1).

(10.107)

10.4.2 Last passage percolation and increasing subsequences The process interpretation of the mapping from the non-negative integer matrix X to nonintersecting paths can alternatively be viewed as a certain directed last passage percolation [329]. Let us regard each element xi,j of the matrix X as a waiting time associated with the lattice site (i, j) (1 ≤ i, j ≤ n). The directed last passage percolation is defined by forming a u/rh lattice path from the site (1,1) to the site (n1 , n2 ), such that it maximizes the sum of the waiting times (in general such a path will not be unique). Thus the quantity of

473

LATTICE PATHS AND GROWTH MODELS

interest is



(1)

LX (n1 , n2 ) := max

xi,j ,

(10.108)

(1,1) u/rh (n1 ,n2 )

which is referred to as the last passage time. From this definition we see that   (1) (1) (1) LX (n1 , n2 ) = max LX (n1 , n2 − 1), LX (n1 − 1, n2 ) + xn1 ,n2 (1)

(10.109)

(1)

with LX (0, k) = LX (j, 0) := 0. This is identical to the recurrence (10.105) with (1)

LX (n1 , n2 ) = λ1 (n1 , n2 )

(10.110)

for all 1 ≤ n1 , n2 ≤ n, so the last passage time is equal to the maximum height in the growth process applied ˜ n1 ,n2 . to X (l) The definition (10.108) of the last passage time can be generalized to a quantity LX (n1 , n2 ) which relates to the level-p heights λp (n1 , n2 ) for each p = 1, . . . , l. Thus let (rd∗ )l denote the set of l disjoint rd∗ lattice paths, the latter defined as either a single point, or points connected by segments formed out of arbitrary positive integer multiples of steps to the right and steps up in the rectangle 1 ≤ i ≤ n1 , 1 ≤ j ≤ n2 (the steps are said to be disjoint if they connect no common lattice sites). In terms of this set generalize (10.108) to (l) LX (n1 , n2 ) = max xi,j . (10.111) (rd∗ )l

These quantities obey a recurrence generalizing (10.109),  (l) (l−1) (l−1) (l) LX (n1 , n2 ) + LX (n1 − 1, n2 − 1) = max LX (n1 − 1, n2 ) + LX (n1 , n2 − 1),  (l−1) (l) LX (n1 , n2 − 1) + LX (n1 − 1, n2 ) + xn1 ,n2 , (10.112) (0)

(l−1)

where LX (n, m) := 0, which can be proved by consideration of the union of the paths realizing LX (l) LX in each term [457]. Noting that in general

and

max(al−1 + bl , bl−1 + al ) − max(al−2 + bl−1 , bl−2 + al−1 ) = max(bl − bl−1 , al − al−1 ) + min(al−1 − al−2 , bl−1 − bl−2 )

(10.113)

we see by subtracting from the recurrence (10.112) the same recurrence with l → l − 1, that the recurrence (10.107) results with the identification (l)

(l−1)

λl (n1 , n2 ) = LX (n1 , n2 ) − LX

(n1 , n2 ),

or equivalently (l)

LX (n1 , n2 ) =

l

λp (n1 , n2 ).

(10.114)

p=1 (l)

The non-negative matrix X can be presented as a two-line array, and LX (n1 , n2 ) interpreted in terms of weakly increasing subsequences (see Figure 10.13). In fact it is this viewpoint which is most prevalent in the combinatorics literature, being the one taken in the pioneering works of Schensted and Greene [481], [273]. A two-line array is constructed from the n1 × n2 matrix Xn1 ,n2 of non-negative integers by writing down in sequence, reading along the first row of the matrix, then the second row  and so on (recall that our convention is to count rows from the bottom), xi,j copies of the ordered pair ji (see Figure 10.13 for an example). By construction the entries in the top row of the two-line array are all weakly increasing, but not so

474

CHAPTER 10

0

0

1

3

1

2

1

1

1

2

2

2

2

2

2

3

1

0

2

1

3

3

1

1

1

2

3

3

3

t=0

Q1

Q2

Q3

t=1

Q1

Q2

Q3

t=2

Q1

Q2

Q3

t=3

Q1

Q2

Q3

t=4

Q1

Q2

Q3

t=5

Q1

Q2

Q3

t=6

Q1

Q2

Q3

t=7

Q1

Q2

Q3

t=8

Q1

Q2

Q3

Figure 10.13 A 3 × 3 non-negative integer matrix, with the corresponding queueing “ ” process “ ” “ and ” “two-line ” “ ”array. Note (1) 1 2 2 2 that LX (3, 3) = 8, corresponding to the path through the elements 1 , 1 , 2 , 3 , 33 , while the copies of these ordered pairs forms an increasing subsequence of length 8 in the bottom line of the two-line array.

in general the bottom row. The maximum of the lengths of subsequences in the bottom row which are weakly increasing (i.e., are of the form lj1 ≤ lj2 ≤ · · · ≤ ljm ) is readily seen to be given by the formula (10.108) (1) specifying LX (n1 , n2 ). In this context (10.110), with λ1 (n1 , n2 ) interpreted as the length of the first row of the corresponding semi-standard tableau, is due to Schensted. Furthermore, the maximum of the lengths of (l) l disjoint subsequences in the bottom row which are all weakly increasing is the same thing as LX (n1 , n2 ) and in this context (10.114) is due to Greene.

10.4.3 Queues (1)

Focusing attention on LX (n1 , n2 ) only, a number of further interpretations of the non-negative integer elements of X and the corresponding meaning of the last passage time are possible. One such interpretation is to view the xi,j as service times in a queueing process [28]. Specifically, consider an infinite number of queues Qj (j = 1, 2, . . . ) which initially have n jobs labeled i = 1, 2, . . . in queue Q1 , and zero jobs in all other queues, and suppose that immediately after service in Qj , each job moves to Qj+1 (see Figure 10.13 for an example). Then with xi,j denoting the time it takes server j to process job i (once it reaches the server), we see that the time T (i, j) it takes job i to leave queue Qj satisfies the recurrence (10.105) and is thus equal to λ1 (i, j). Theory related to the queueing problem when all the service times are chosen from the same distribution allows for a characterization of {λ1 (i, N )}i=1,...,n . Thus one has the general result that with the service times all chosen from a distribution of mean μ and variance σ 2 , the limiting scaled exit times Di := lim

N →∞

T (i, N ) − μN √ σ N

(10.115)

475

LATTICE PATHS AND GROWTH MODELS

of job i from queue N have joint p.d.f. specified by [267] Di =

i−1 

sup 0=t0 j so that ρj − ρk + k − j > 0. The integrand is readily computed, showing that (10.139) is equal to h i β −l Pf sgn(k − j)β |ρj −ρk +k−j| . (10.140) j,k=1,...,2l

The above Pfaffian can be evaluated by setting xj = ρj − j, f (xj ) = β Pf

xj

in the general formula

l i h“ f (x ) ”sgn(xj −xk ) Y f (xQ(2j−1) ) j sgn(xj − xk ) = ε(Q), f (xk ) f (xQ(2j) ) j,k=1,...,2l j=1

(10.141)

where the permutation Q is such that xQ(2j−1) > xQ(2j) ,

Q(2j) > Q(2j − 1) (j = 1, . . . , l)

to give the r.h.s. of (10.137). To see the validity of (10.141), note that of the (2l−1)!! permutations in the sum contributing to the Pfaffian, only one term contributes to the r.h.s. with the rest cancelling in pairs. It remains to verify (10.135). Use of an appropriate modification of (10.136) shows that this is equivalent to the integration formula E D sρ (eiθ1 , e−iθ1 , . . . , eiθl , e−iθl , β)

Sp(2l)

=β

P2l+1 j=1

(−1)j−1 ρj

.

(10.142)

Proceeding as in the derivation of (10.138) and (10.139) shows that the l.h.s. is equal to 2 iθj (ρ2l−k+2 +k−1) 3 Z π Z π l e Y (eiθk − e−iθk ) 1 4 e−iθj (ρ2l−k+2 +k−1) 5 dθ · · · dθ det 1 l (2π)l 2l l! −π |1 − βe−iθk |2 −π k=1 β ρ2l−k+2 +k−1 j=1,...,l k=1,...,2l+1 » – 1 [β ρj +2l+1−j ]j=1,...,2l+1 A(2l+1)×(2l+1) = l Pf , [β ρk +2l+1−k ]k=1,...,2l+1 0 2 where

» A(2l+1)×(2l+1) :=

1 2π

Z

π

−π

eiθ − e−iθ (eiθ(ρj −j−ρk +k) − e−iθ(ρj −j−ρk +k) )dθ |1 − βe−iθ |2

The integral can be computed to give for the l.h.s. » [sgn(k − j)β |ρj −ρk +k−j| ]j,k=1,...,2l+1 β −(l+1) Pf [−β ρk +2l+1−k ]k=1,...,2l

– . j,k=1,...,2l+1

[β ρj +2l+1−j ]j=1,...,2l 0

– .

This is precisely the same as (10.140) with l → l + 1, ρ2l+2 = 0, and so reduces to the r.h.s. of (10.142).

We see from the exact results of Proposition 10.5.3 that Pr(h ≤ 2l + 1) = Pr(h ≤ 2l)|β=0 .

(10.143)

To understand this result, note that for matrices symmetric about the antidiagonal, the path (0, 0)u/rh(n, n) which maximizes (0,0)u/rh(n,n) xi,j can likewise be chosen to be symmetric about the antidiagonal. This means that for h to be odd, the value of xi,n+1−i must be odd, as all values of xi,j off the antidiagonal contributing to h occur in pairs. Moreover (10.131) gives   , Pr(xi,n+1−i = 2l + 1) + Pr(xi,n+1−i = 2l ) = Pr(xi,n+1−i = 2l ) β=0



whereas Pr(xi,n+1−i = 2l − 1)|β=0 = 0, so pairing the odd values on the diagonal with the even values in this way we reclaim the setting of Pr(h ≤ 2l)|β=0 as required by (10.143).

483

LATTICE PATHS AND GROWTH MODELS

Matrices zero above the antidiagonal Consider a general n × n non-negative integer matrix X = [xi,j ]i,j=1,...,n and suppose xi,j = 0 for i > n + 1 − j. For i ≤ n + 1 − j let xi,j occur with the geometric probability (10.70). Applying the RSK correspondence in the nonintersecting paths formulation but stopping at t = n rather than t = 2n − 1 gives a bijection between X and weighted w,u/rh(o)/lh(e) nonintersecting lattice paths as specified in Section 10.1.2, (1) with the spacing between levels one unit (see Figure 10.17). As usual LX (n, n) is equal to the maximum ˜ say. On the other hand h ˜ is equal to the number of nonintersecting of the heights in the level-1 path, h, w,u/rd(o)/ld(e) lattice paths dual to the w,u/rh(o)/lh(e) lattice paths, where in both cases the paths start and finish on the same level, with the interspacing between levels one unit. Hence  w,u/rd(o)/ld(e) (0) ˜ ≤ l) = Pr(h (1 − ai bn+1−j )G2n ({rj = l − j + 1}j=1,...,l ; {rj = l − j + 1}j=1,...,l ). 1≤i≤j≤n

t=1 b1 b1

b2

1

t=2

0

2

2

a2

1

x

2

a2 3

b1

3

b2

y=0

1

−2 −1

0

2

a1

2

1

a1

y=0

0

b3 t

a3

a1

1

0 t=3

2

2

b3

a3

a2

4

b1

2

a1 b2

a2

y=0 y=−1

0

Figure 10.17 Mapping from a weighted non-negative integer matrix with entries zero above the antidiagonal (this portion of the matrix is not shown) and weighted nonintersecting lattice paths. The latter, upon rotation by 90◦ clockwise, can be regarded as w,u/rh(o)/lh(e) type paths.

Let us now set bi = an+1−i , ai = qi (i = 1, . . . , n) so that for sites (i, j) with i < n + 1 − j the probabilities are the same as (10.128) determining h . Making use of (10.51) shows  ˜ ≤ l) Pr(h b

i =an+1−i ai =qi

=



l  n   (1 − qi qj ) |1 + qj eiθk |2 k=1 j=1

1≤i≤j≤n

Sp(2l)

.

Comparing with (10.134) we thus have

 ˜ ≤ l) = Pr(h ≤ 2l) Pr(h

β=0

To anticipate this, we note from (10.131) that Pr(xi,n+1−i

  = k)

 β=0

=

.

(1 − qi2 )qik , k even, 0, k odd.

(10.144)

(10.145)

484

CHAPTER 10

Also, as remarked in the discussion of (10.143), in the case [xi,j ] is symmetric about the antidiagonal xi,j = max χi,j xi,j , (10.146) max (1,1)u/rh(n,n)

(1,1)u/rh(l,n+1−l)

where (l, n+ 1 − l) is any point on the antidiagonal while χi,j = 1 for points on the antidiagonal and χi,j = 2 otherwise. Because according to (10.145), in the case β = 0 the points on the antidiagonal are restricted  to  even values, we can redefine χi,j in (10.146) to equal 2 for all points without changing Pr(h ≤ 2l) , β=0

provided we redefine the probability (10.145) so that   = (1 − qi2 )qi2k , Pr(xi,n+1−i = k) β=0

k = 0, 1, . . .

˜ ≤ l) in (10.144) are precisely the same as With this redefinition we see that the probabilities making up Pr(h ˜ which together imply (10.144). those specifying Pr(h ≤ 2l)|β=0 , and furthermore h = 2h,

10.5.2 Matrices symmetric about the diagonal The rules of the growth process tell us that if the non-negative integer matrix X = [xi,j ]i,j=1,...,n maps to the pair of tableaux (P, Q), then the transposed matrix XT := [xj,i ]i,j=1,...,n maps to (Q, P ). Hence the Robinson-Schensted-Knuth correspondence when applied to non-negative integer matrices symmetric about the diagonal gives a bijection with a single semi-standard tableau, or equivalently a single set of u/rh lattice paths, since then P = Q. To obtain a bijection between weighted symmetric matrices and a weighted set of u/rh lattice paths the simplest situation is to require that the sets of paths have the same weighting, and then to weight a single set by the square of the original weighting. We see from Figure 10.12 that this will happen if we restrict the weights so that ai = bi (i = 1, . . . , n), and thus weight each entry xi,j by (1 − ai aj )(ai aj )xi,j . But as in (10.70) and (10.128) we would like to relate these weights to probabilities, for which purpose we should weight only the sites i ≤ j, with the value of xi,j for i > j fixed by symmetry. This can be done by squaring the weights at sites i < j and readjusting the normalizations. The following analogue of Proposition 10.5.2 is then obtained [329]. P ROPOSITION 10.5.4 Consider a non-negative integer matrix X, symmetric about the diagonal with independent entries chosen according to Pr(xi,j = k) = (1 − qi qj )(qi qj )k , i < j,

Pr(xi,i = k) = (1 − qi )qik .

(10.147)

The probability that under the RSK correspondence X maps to a pair of paths with up steps weighted pro√ portional to qi for x = ±(2n + 12 − 2i), and with final displacements μ, is equal to 

n  (1 − qi ) i=1

(1 − qi qj )sμ (q1 , . . . , qn ).

(10.148)

1≤i<j≤n

The probability on the diagonal entries in (10.147) can be generalized [34]. This is possible because in the RSK mapping for a symmetric matrix n j=1

xj,j =

n

(−1)j−1 μj

(10.149)

j=1

(see Exercises 10.5 q.1). Thus if the probability in question is generalized to read Pr(xi,i = k) = (1 − αqi )(αqi )k ,

(10.150)

485

LATTICE PATHS AND GROWTH MODELS

(10.148) should correspondingly be generalized to n 



(1 − αqi )

i=1

Pn

(1 − qi qj )α

j=1 (−1)

j−1

μj

sμ (q1 , . . . , qn ).

(10.151)

1≤i<j≤n

With X symmetric about the diagonal, let us denote the maximum height of the level-1 path by h . Noting that n n (−1)j−1 μj = #(columns of odd length in μ) = μk mod 2, (10.152) j=1

k=1

where l = μ1 , we see from (10.151) that Pr(h ≤ l) =

n  i=1



(1 − αqi )

(1 − qi qj )

1≤i<j≤n



Pn

α

k=1

μk mod 2

sμ (q1 , . . . , qn ).

(10.153)

μ:μ1 ≤l

As with (10.133) this can be written in terms of an average over a classical group. P ROPOSITION 10.5.5 We have n   (1 − αqi ) Pr(h ≤ l) = i=1

n    (1 − qi qj ) det(1l + αU) det(1l + qj U) j=1

1≤i<j≤n

U∈O(l)

.

(10.154)

Proof. As with the proof of Proposition 10.5.3, we follow [224]. Use of (10.56) in (10.154) and comparison with (10.153) shows that (10.154) is equivalent to the matrix integral evaluation D E Pl det(1l + αU)sρ (U) = α j=1 ρj mod 2 ,

(10.155)

U∈O(l)

where sρ (U) denotes the Schur polynomial as a function of all the eigenvalues of U. For definiteness, consider the l even case, l → 2l, and consider separately the components O± (2l) of O(2l). Recalling the eigenvalue p.d.f. for O+ (2l) (2.62), and proceeding as in the derivation of (10.140) shows det(1l + αU)sρ (U)U∈O + (2l) = 21−l Pf[ajk ]j,k=1,...,2l where

(10.156)

“ ” ajk = (1 + α2 )δ(ρj −j)−(ρk −k),odd + 2αδ(ρj −j)−(ρk −k),even sgn(k − j) “1 ” 1 = (1 + α)2 − (1 − α)2 (−1)(ρj −j)−(ρk −k) sgn(k − j). 2 2

The task is therefore to compute the Pfaffian of the matrix with these entries. For this one uses the identity [502] P X Pf(A + B) = (−1) j∈S j−|S|/2 Pf S (A)Pf S¯ (B), (10.157) S⊆{1,2,...,2l} |S| even

where Pf S (A) denotes the Pfaffian of A restricted to rows and columns specified by the index set S, and similarly Pf S¯ (B) (S¯ denotes the complement of S). With h1 i h 1 i (1 + α)2 sgn(k − j) A= , B = − (1 − α)2 (−1)(ρj −j)−(ρk −k) sgn(k − j) , 2 2 j,k=1,...,2l j,k=1,...,2l from the simple identities Pf[sgn(k − j)] = 1,

Pf[aj,k (−1)(ρj −j)−(ρk −k) ] = (−1)

P

(ρj −j)

Pf[aj,k ],

486

CHAPTER 10

we see that

¯

Pf S A = 2−|S|/2 (1 + α)|S| ,

Pf S¯ B = (−2)−|S|/2 (1 − α)2l−|S| (−1)

P

¯ j∈S

ρj −j

.

Application of (10.157) gives “ 1 + α ”|S| “ 1 − α ”2l−|S|

X

21−l Pf(A + B) = 2

2

S⊆{1,2,...,2l} |S| even

2

P

(−1)

¯ j∈S

ρj

.

(10.158)

But in general X

x|S| y 2l−|S| (−1)

S⊆{1,2,...,2l} |S| even

=

=

1“ 2

¯ j∈S

ρj

x|S| y 2l−|S| (−1)

P

¯ j∈S

ρj

(x + (−1)ρj y) +

j=1

X

+

S⊆{1,2,...,2l}

2l 1“ Y

2

X

P

x|S| (−y)2l−|S| (−1)

P

¯ j∈S

ρj

”

S⊆{1,2,...,2l} 2l Y

” (x − (−1)ρj y) .

j=1

Using this result to evaluate (10.158) and substituting in (10.156) gives the matrix integral evaluation P2l

det(1l + αU)sρ (U)U∈O + (2l) = α

j=1

ρj mod 2

P2l

j=1 (ρj +1)mod 2

+α

.

(10.159)

It remains to compute the corresponding formula for the average over O− (2l). The analogue of (10.156) in this case is

(1 − α2 ) [ζ]Pf[ajk + ζbjk ]j,k=1,...,2l 2l−1 = (−1)ρk −k − (−1)ρj −j . Observing

det(1l + αU)sρ (U)U ∈O −(2l) = (cf. (6.159)), where ajk is as in (10.156) while bjk [bjk ] =  uw  T − w  uT ,

 u = [1]j=1,...,2l , w  = [(−1)ρj −j ]j=1,...,2l

(10.160)

(10.161)

shows that [bjk ] has rank 2. It follows that the Pfaffian in (10.160) is linear in ζ, and so the r.h.s. of (10.160) can be rewritten i ” (1 − α2 ) 1 “ h (10.162) Pf [ajk ] + ζ[bjk ] − Pf[ajk ] . l−1 2 ζ With γ, ζ1 , ζ2 arbitrary nonzero constants, the structure (10.161), and applying elementary row and column operations, verifies that this in turn can be rewritten 3 2 « „ [ajk ] ζ1 w  ζ2  u (1 − α2 ) 1 T 5 − γPf[ajk ] . 4 −ζ1 w  0 γ (10.163) Pf 2l−1 ζ1 ζ2 uT −γ 0 −ζ2  Setting ζ1 = 12 (1 − α)2 , ζ2 = (1 + α)2 , adding one half of the final row/column to the second-last row/column, subtracting the second-last row/column from the final row column, and setting γ = (1 + α2 ) allows (10.163) to be recognised as ˛ ” 21−l “ ˛ 2 ] − (1 + α )Pf[a ] Pf[a . (10.164) ˛ jk jk 2l×2l 2(l+1)×2(l+1) 1 − α2 ρ2l+1 =ρ2l+2 =0 Comparing (10.156) and (10.159) tells us that “ P2l ” P2l Pf[ajk ]2l×2l = 2l−1 α j=1 ρj mod 2 + α j=1 (ρj +1)mod 2 . Substituting in (10.164) and simplifying implies the matrix integral evaluation P2l

det(1l + αU)sρ (U)U∈O −(2l) = α

j=1

ρj mod 2

P2l

−α

j=1 (ρj +1)mod 2

.

(10.165)

487

LATTICE PATHS AND GROWTH MODELS

As the matrix integrals over both components O+ (2l) and O− (2l) have now been evaluated, it only remains to take the arithmetic mean of these to obtain (10.155).

We remark that the Pieri formula (12.190) below enables the general α case of (10.155) to be deduced from the result for α = 0. E XERCISES 10.5

1. [360], [221] The objective of this exercise is to derive (10.149).

(i) Note that for the nonintersecting paths interpretation of the RSK correspondence, [xi,j ]i,j=1,...,n symmetric about i = j implies λl (j, i) = λl (i, j), and use this in (10.105) and (10.107) to conclude λ1 (i, i) = λ1 (i, i − 1) + xi,i , λl (i, i) = λl (i, i − 1) + λl−1 (i, i − 1) − λl−1 (i − 1, i − 1),

l > 1.

(ii) Form appropriate linear combinations of the equations in (i), making use of the fact that λl (l, l − 1) = 0, to deduce the equation xi,i =

i X l=1

(−1)l−1 λl (i, i) −

i−1 X

(−1)l−1 λl (i − 1, i − 1).

l=1

Sum this over i from 1 to n and recall that λl (n, n) = μl to deduce (10.149). 2. Sch¨utzenberger’s evacuation operation on a semi-standard tableau P can be described as follows (see, e.g., [246]). Remove the number n0 , say, of the first square in the first row of P , thus creating a blank square. Interchange the place of the blank square and its neighbor immediately to the right, or immediately below, whichever is smaller (if both neighbors have the same number, choose the one below). Repeat this procedure until the blank square is at the end of a row or the end of a column. Then the square in this position of P R is numbered n + 1 − n0 . The blank square is removed from P and the procedure is repeated to complete the filling of P R . (i) Consider the pair of tableaux (P, Q) equivalent to the pair of lattice paths in Figure 10.12. Note that since the matrix X therein is such that X = XR , we must have P R = Q. (ii) Show that the result of (i) is consistent with the construction of P R from the evacuation operation.

10.6 THE HAMMERSLEY PROCESS Closely related to the last passage percolation model associated with the non-negative matrix X is a last passage percolation model associated with random points in the unit square, known as the Hammersley process. The latter is defined by first marking in the unit square points uniformly at random according to 2 a Poisson rate with intensity z 2 , so that the probability the square contains N points is equal to z 2N e−z /N !. From the points one forms a continuous path by joining points with straight line segments of positive slope, and this path is extended to begin at (0, 0) and finish at (1, 1) by adding an extra segment at both ends. With the length of the extended path defined as the number of points it contains, the stochastic variable l is defined as the maximum of the lengths of all possible extended paths (see Figure 10.18). To relate the Hammersley process so defined to the last passage percolation model of Section 10.4.2, consider an n × n matrix X with entries chosen according to the geometric distribution (10.70). Set ai = bj = z/n. Then to leading order in 1/n2 the probability that xij = 1 is equal to z 2 /n2 while to the same order the probability that xij > 1 is zero. If we now think of the lattice sites (i, j) scaled to the points (i/n, j/n) then in the limit n → ∞ we see that a Poisson process with intensity z 2 in the unit square is generated for the distribution of matrix elements with value unity. This latter point follows from the general fact that the Poisson process in question can be realized as the M → ∞ limit of the discrete process of dividing the unit square up into a regular M × M grid and marking a point randomly within each subsquare with probability z 2 /M 2 .

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1

1 Figure 10.18 Eight points in the unit square, and the extended directed paths of maximum length. Since the number of points in these paths is 3, here ln = 3. (1)

Furthermore, one has that in this limit the last passage time LX (n, n) as defined by (10.108) coincides with the definition of l given above. As a consequence the cumulative probability distribution for l can be deduced from knowledge of the formula for the last passage time (denoted h ) in Proposition 10.3.1. P ROPOSITION 10.6.1 For the Hammersley process  Pl  2 Pr(l ≤ l) = e−z e2z j=1 cos θj

U(l)

.

(10.166)

Proof. The relationship between the Hammersley process and the last passage percolation model of Section 10.4.2 tells us that Pr(l

≤ l) = lim Pr(h n→∞

˛ ˛ ≤ l)˛

ai =bj =z/n

.

Computing the limit in (10.73) gives (10.166).

10.6.1 Relationship to random permutations Any particular realization of the Hammersley process containing N points gives a geometrical construction of a random permutation of {1, 2, . . . , N }. To see this label the x-coordinates of the points by 0 < x1 < · · · < xN < 1 and similarly the y-coordinates by 0 < y1 < · · · < yN < 1. Each point will then have a coordinate of the form (xj , yP (j) ), where {P (1), . . . , P (N )} is a permutation of {1, 2, . . . , N }. Because the N points are distributed at random with uniform distribution, the permutation is also random with uniform distribution. In addition, the quantity l has an interpretation in terms of the permutation. Thus the analogue of a continuous path consisting of segments of positive slope is a subsequence 1 ≤ j1 < j2 < · · · < jr ≤ N such that P (j1 ) < P (j2 ) < · · · < P (jr ), which is referred to as an increasing subsequence. The length of an increasing subsequence is defined as the value r. We then see from the definitions that the maximum length of all increasing subsequences of P coincides with l . 10.6.2 Droplet PNG model Associated with the non-negative matrix X, both a last passage percolation model and a discrete polynuclear growth model have been identified. Likewise, associated with random points in the unit square are a last passage percolation model (the Hammersley process) and a particular polynuclear growth model known as the droplet PNG model [450]. To define the latter consider the xt-half-plane t > 0. Let this half plane be filled with points uniformly at random and such that the mean density is unity. Analogous to the entries xij of the matrix X these points are

LATTICE PATHS AND GROWTH MODELS

489

to be thought of as seeds for nucleation events of layered growth of unit height, although now the nucleation event grows continuously to the left and right with unit velocity and is created with zero width. In the droplet model, at (x, t) = (0, 0) a single layer, taken to be at height zero, starts spreading with unit velocity to the left and to the right. All nucleation events and thus subsequent layers are constrained to occur above this initial layer. As the √ initial layer grows √ with unit speed, only nucleation events bounded by the “lightcone” axis v = (t + x)/ 2, u = (t − x)/ 2 are created at a time that their position coordinate makes contact with the ground layer or its growth. The nucleation events (xi , ti ) inside the lightcone create the beginning of a portion of a layer of unit height on top of the ground layer, or existing layers, at position xi . The layers are formed by the growth of the nucleation events with unit velocity to the left and to the right; if two growing portions of a layer collide, then growth at that point ceases and the two portions become one, growing only at the end points of this one portion (see Figure 10.19 for an example). Of interest are the statistical properties of the height at the origin after this growth process has been underway for time t = T .

Figure 10.19 Example of the plateau profile at the time of four successive nucleation events, including the initial event (which is labeled the 0th event and its plateau the 0th level). Note that between the second and third nucleation events, two plateaux on the first level have coalesced.

To √ analyze this quantity, the √ first observation is that only those nucleation events in the region [u = 0, u = T / 2] × [v = 0, v = T / 2] of the lightcone can contribute to the height at x = 0 up to time t = T . Suppose in a realization of the nucleation events there are N points in this region. For a Poisson process of 2 unit density this occurs with probability z 2N e−z /N !, where z 2 = T 2 /2 is the area of the region. One then uses the construction of the previous subsection to associate with the configuration of points a permutation P , and furthermore marks in the world lines of the growth of the nucleation events (see Figure 10.20). The world lines show clearly the layered structure of the growth, and in particular the height at the origin after time T . To relate this height to a property of P , we first note a construction which determines the layer in which each particular nucleation event occurs. This can be done by partitioning the permutation into decreasing subsequences using the leftmost digits at all times. The jth such decreasing subsequence tells us the coordinates of the nucleation events, projected onto the line x = −t, which lie in the jth layer in the growth process. For example, in Figure 10.20 the permutation is 5374162, and the decreasing subsequences formed from the leftmost digits are (531)(742)(6). As is shown in Exercises 10.6 q.1, it is generally true that the number of decreasing subsequences of this type is equal to the length of the longest increasing subsequence of the same permutation. Thus studying the height at the origin in the PNG model after time T is equivalent to studying the maximum path length in the Hammersley process with intensity z 2 = T 2 /2.

10.6.3 Permutation matrices and increasing subsequences We have seen that each configuration of N points in a realization of the Hammersley process is equivalent to a random permutation of {1, 2, . . . , N }, read off from the labels of the points (xj , yP (j) ), j = 1, . . . , N . Also associated with the labels is a permutation matrix defined so that the entry (j, P (j)) of row j is equal to unity, while all other entries in the row are equal to zero. The RSK correspondence applied to the permutation matrix maps to a pair of u/rh and u/lh lattice paths with the constraint that for each allowed position of the up

490

CHAPTER 10

t

7

6

T

5

7 6 5

4 3

2

1

12

3

4 x

Figure 10.20 World lines for the endpoints of the plateaux. Only nucleation events inside the square-shaped region including the lines x = ±t, and the lines from t = T to these lines affect the height at the origin. The nucleation points occur at v-shaped configurations, while the inverted v part of the world lines correspond to the joining of the plateaux originating from two different nucleation events. The labeling on the lines x = t and x = −t allow the world lines to be identified uniquely with a permutation [532].

steps (recall Figure 10.12) there is a total of 1 such step only. Thus there is a correspondence with a pair of single move u/rh and u/lh lattice paths initially equally spaced, with a total of n moves each, or equivalently with a pair of standard tableaux having the same shape and each of content n (recall Section 10.1.3). From the discussion of the paragraph below (10.114) we know that if the pair of tableaux have shape κ, l then κ1 is equal to the longest increasing subsequence length of the permutation, while i=1 κi is equal to the maximum of the lengths of l disjoint increasing subsequences. Denoting by ln the longest increasing subsequence length, it follows that for a random permutation with uniform measure  1 Pr(ln ≤ l) = # (of pairs of standard tableaux with content n, κ1 ≤ l) . (10.167) n! This can be written as a random matrix average [262], [455]. P ROPOSITION 10.6.2 We have Pr(ln ≤ l) = Proof. We note that

l l 2n  1  iθj 2n  n!  e  = 2 cos θj .  n! j=1 (2n)! U(l) U(l) j=1

fnλ := #(standard tableaux of shape λ and content n)

(10.168)

(10.169)

occurs as a particular coefficient in the monomial expansion of the Schur polynomial sλ (w1 , . . . , wn ). Thus we recall from (10.16) that Q sλ is#jdefined as a weighted sum over semi-standard tableau of shape λ and content n with each tableau weighted by n j=1 wj . Since a standard tableau is a special case of a semi-standard tableau in which each number j = 1, . . . , n occurs exactly once, we see immediately that [w1 w2 · · · wn ]sλ (w1 , . . . , wn ) = fnλ ,

(10.170)

491

LATTICE PATHS AND GROWTH MODELS

where [w1 w2 · · · wn ] denotes the coefficient of w1 w2 · · · wn . Hence Pr(ln ≤ l) =

X 1 sκ (a1 , a2 , . . . , an )sκ (b1 , b2 , . . . , bn ). [a1 · · · an b1 · · · bn ] n! κ:κ ≤l

(10.171)

1

But according to (10.72) and (10.73) X

sκ (a1 , a2 , . . . , an )sκ (b1 , b2 , . . . , bn ) =

κ:κ1 ≤l

l n Y DY

(1 + aj e−iθk )(1 + bj eiθk )

E CUEl

j=1 k=1

.

Reading off the coefficients of a1 · · · an b1 · · · bn in the random matrix average gives the first equality in (10.168). The second equality follows by noting l “X

2 cos θj

”2n

=

j=1

2n “ l X 2n ”“ X j=0

j

eiθk

l ”j “ X

k=1

e−iθk

”2n−j

,

(10.172)

k=1

then observing that only the term j = n contributes to the U (l) average.

We note that the result (10.168) could have been derived from the result (10.166) for the Hammersley process. Thus from the definitions, Pr(lN ≤ l) and Pr(l ≤ l) are related by Pr(l ≤ l) = e−z

2

∞ z 2N Pr(lN ≤ l). N!

(10.173)

N =0

Expanding the random matrix average in (10.166) in a power series in z and using this relation leads to the second equality in (10.168). 10.6.4 Random words A permutation can be regarded as a bijective mapping from {1, . . . , n} to {1, . . . , n}. If we consider instead functions which map from {1, . . . , n} to {1, . . . , k}, then presenting the function as a two-line array specifies a so-called word of length n from an alphabet of k letters. Presenting the two-line array for a word as a nonnegative integer matrix gives a 0, 1 matrix of dimension n× k, constrained so that each row contains exactly a single 1. Applying the mapping of Figure 10.15 with this constraint gives a bijection with u/rh nonintersecting lattice paths making k steps paired with ld/lh nonintersecting lattice paths making n steps (as for general 0,1 matrices), but with the latter lattice paths constrained so that at any one step the number of ld steps is exactly one. In terms of tableaux we thus have a bijection with a semi-standard tableau of content k, and a standard tableau of the same shape also of content n. word Denoting by ln,k the longest increasing subsequence length, it follows that for a random word word Pr(ln,k ≤ l) = k −n #( semi-standard tableaux, shape μ, content k)fnμ . μ:μ1 ≤l

Use of (10.13) and (10.169) then shows   word Pr(ln,k ≤ l) = k −n [b1 · · · bn ] sμ (a1 , . . . , ak )sμ (b1 , . . . , bn ) μ:μ1 ≤l

a1 =···=ak =1

l  k n     = k −n [b1 · · · bn ] (1 + ar eiθp ) (1 + bs e−iθp ) p=1

= k −n

l  j=1

(1 + eiθj )k

r=1 l  j=1

s=1

e−iθj

  

U(l) a1 =···=ak =1

n  U(l)

,

(10.174)

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CHAPTER 10

where the second equality follows from the equality between (10.72) and (10.73). 1. Note from the discussion of Section 10.1.3 that, with l(0) = (l − 1, l − 2, . . . , 0) and wl− = 0, wl+ = 1 (l = 1, . . . , n), wl+ = 0, wl− = 1 (l = n + 1, . . . , 2n),

E XERCISES 10.6

s,ld/rd

# (of pairs of standard tableaux with content n, κ1 ≤ l)) = G2n

(l(0) ; l(0) ).

Substitute this in (10.167), and make use of (10.39) to reclaim the first equality in (10.168). 2. [133] Consider a permutation π of {1, 2, . . . , N }, and let (π) denote the maximum length of the increasing subsequences. (i) Show that the minimum number of single integer moves required to return π back to the identity is N −(π). (ii) Suppose the integers in the permutation, reading from left to right, are sorted into piles according to the rules that low numbers are put on top of high numbers on the left-most possible pile, and if this is not possible, a new pile is started to the right of existing piles. Show that the number of piles is equal to (π).

10.7 SYMMETRIZED PERMUTATION MATRICES 10.7.1 Longest increasing and decreasing subsequence lengths of a random involution A permutation matrix X is a real orthogonal matrix and thus has the property X−1 = XT (remember that we are counting the rows from the bottom, so the operation T corresponds to reflection about the bottom left to top right diagonal). Thus permutation matrices with the symmetry X = XT have the property X2 = 1 and therefore correspond to involutions. Further, the number of 1s on the diagonal corresponds to the number of fixed points of the involution. Our interest is in the cumulative distribution for ln,k , the longest increasing subsequence length of an involution of {1, . . . , n} consisting of k 2-cycles. One approach to computing the cumulative distribution for ln,k is to consider the symmetrized Hammersley model underlying random involutions. First one notes that a random permutation matrix with the symmetry X = XT and n − 2k points on the diagonal corresponds to n random points in the unit square symmetric about the diagonal y = x and with n − 2k of the n points on this line. A Poissonized version of this setting is to break the unit square into M × M equal subsquares, and to break the diagonal up into M equal segments. In each subsquare below the diagonal (together with its image above the diagonal) a point is marked with probability z 2 /M 2 , while a point is marked on each segment of the diagonal with probability αz/M . The probability that there are a total of n points in the square with n − 2k of the points on the diagonal is then given by the coefficient of wn γ n−2k in 2  αzwγ M  αz z2 z 2 w2 M /2 + 1− 1− 2 + . M M M M2 Taking the limit M → ∞ it follows from this that the probability is equal to [wn γ n−2k ]eαzγw+z

2

w 2 /2 −z 2 /2−αz

e

where sn,k =

= e−z

2

/2−αz z

n

n!

αn−2k sn,k ,

 n  (2k)! 2k 2k k!

is the number of involutions consisting of k 2-cycles and n − 2k fixed points. With l denoting the maximum of the lengths of the paths consisting of positive sloping segments, it then follows that Pr(l ≤ l) = e

−αz−z 2 /2

[n/2] ∞ z n n−2k α sn,k Pr(ln,k ≤ l). n! n=0 k=0

(10.175)

493

LATTICE PATHS AND GROWTH MODELS

The significance of the relation (10.175) is that the symmetrized Hammersley model can be realized as a limiting case of the growth process associated with matrices symmetric about the diagonal discussed in Section 10.5.2. We see from the first formula in (10.147), and (10.150), that the required limit is to set qi = z/n for each i = 1, . . . , n and take n → ∞. From the evaluation of the cumulative distribution for h given in Proposition 10.5.5, we can then compute the cumulative distribution for l according to   (10.176) Pr(l ≤ l) = lim Pr(h ≤ l) n→∞

qi =z/n

to obtain the following result [34]. P ROPOSITION 10.7.1 We have Pr(l ≤ l) = e−αz e−z

2

/2



det(1l + αU)ezTr U

 U∈O(l)

.

(10.177)

Rotating all the elements of a permutation matrix with the symmetry X = XT by 90◦ anticlockwise gives a permutation matrix with the symmetry X = XR . In the two line array presentation of an involution, this is equivalent to reversing the ordering of the bottom line, and so interchanging increasing subsequences with decreasing subsequences. Thus the maximum length of the increasing subsequences in the permutation corresponding to the rotated matrix ln,k say, is equal to the maximum length of the decreasing subsequences in the underlying involution. As with (10.175), this is related to the maximum of the length of paths consisting of positive sloping segments in the corresponding Hammersley process by Pr(l ≤ l) = e−βz−z

2

/2

[n/2] ∞ z n n−2k β sn,k Pr(ln,k ≤ l), n! n=0

(10.178)

k=0

where now the antidiagonal y = 1 − x (0 < x < 1) contains points with a Poisson intensity βz. The analogue of the formula (10.176), together with the results of Proposition 10.5.3, then give the following random matrix form [34]. P ROPOSITION 10.7.2 We have  Pl 1 ez j=1 2 cos θj , −iθ 2 |1 − βe k | Sp(2l) k=1  Pl  2 . Pr(l ≤ 2l + 1) = e−z /2 ez j=1 2 cos θj Pr(l ≤ 2l) = e−βz−z

2

/2

l 

Sp(2l)

(10.179) (10.180)

10.7.2 Relationship to the flat PNG model The Hammersley process symmetric about the antidiagonal, but with no points thereon, is relevant to the study of the PNG model in which growth is from a flat substrate [450]. Here, in distinction to the case of the droplet PNG model, growth from the nucleation events (which again occur at random with unit mean in space and time) is no longer restricted to happen above an initial plateau, but rather may occur over the length of the whole x-axis. The profile will then on average be flat and the statistical properties of the height fluctuation independent of the position, which can therefore without loss of generality be studied at x = 0. For nucleation √ events√to affect the height at the origin after time 2T they must occur at positions x√ and times t such √ √ that |x| < 2T − t. This is a triangular shaped region, with vertices (x, t) = (− 2T, 0), ( 2T, 0), (0, 2T ) in the xt-plane, in which nucleation events occur uniformly at random with unit density. From the rules of the PNG model one sees that the height at the origin is equal to the maximum number of points in an upward directed √ path formed from the nucleation events, which starts along the line (x, 0) and finishes at the point (0, 2T ). The triangular shaped region of points can be extended to a square √ shaped region √ by reflecting it about the x-axis. The length of the longest upwards-directed path from (0, − 2T ) to (0, 2T ) within the square

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is the same thing as the length of the longest up/right path in a symmetrized Hammersley process. The symmetrization is with respect to the antidiagonal, which itself contains no points. The height h at the origin of the PNG model is equal to half the length of the longest up/right path in the Hammersley process, and thus the cumulative distribution Pr(h ≤ l) is given by (10.179) with z = T and β = 0. E XERCISES 10.7

1. [246] The objective of this exercise is to derive the explicit formula Q 1≤i<j≤p (li − lj ) Qp fnλ = n! , li := λi + p − i, i=1 li !

(10.181)

where p := μ1 and |λ| = n. (i) Argue that

n X

fnλ =

λ

(i) fn−1 ,

(10.182)

i=1 λ

(i) := 0 for λ(i) not a partition, and note that this formula where λ(i) := (λ1 , . . . , λi − 1, . . . , λn ) and fn−1 0n together with the initial condition f0 = 1 specifies fnλ by induction. Q (ii) With Δ := i<j (li − lj ) show that for the r.h.s. of (10.181) to satisfy the formula of (i) we require

nΔ(l1 , . . . , lp ) =

p X

li Δ(l1 , . . . , li − 1, . . . , lp ).

i=1

Verify this formula by establishing the identity “ “p” ” xi Δ(x1 , . . . , xi + t, . . . , xp ) = x1 + · · · + xp + t Δ(x1 , . . . , xp ). 2 i=1

p X

(First observe that the l.h.s. is anti-symmetric in the xi , and hence has as a factor Δ(x1 , . . . , xp ). From this conclude the expression must be a linear function of t.) 2. [7] In this exercise the combinatorial setting will be used to determine the asymptotic behavior of some matrix integrals. (i) Consider (10.178) in the case β = 0. Noting that Pr(l2n,n ≤ 2l) = 1 (n ≤ l), show Pr(l

˛ ˛ ≤ 2l)˛

β=0

1 , s2(l+1),l+1

Pr(l2(l+1),l+1 ≤ 2l) = 1 −

=1−

z 2l+2 + O(z 2l+4 ). (2l + 2)!

Substitute this in (10.179) with β = 0 to conclude E D Pl 2 2l+2 /(2l+2)!+O(z 2l+4 ) = ez /2−z . ez j=1 2 cos θj Sp(2l)

(ii) Repeat the considerations of (i), applied to (10.175) and (10.177) with α = 0, to show ezTr(U) U∈O(2l) = ez

2

/2−z 2l+2 /(2l+2)!+O(z 2l+4 )

.

(iii) Use the fact that the eigenvalues 0 < θ < π for O− (2l + 2) have the same p.d.f. as the eigenvalues in 0 < θ < π for Sp(2l) to deduce from the results of (i) and (ii) that ezTr(U) U∈O± (2l) = ez

2

/2±z 2l /(2l)!+O(z 2l+2 )

.

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LATTICE PATHS AND GROWTH MODELS

10.8 GAP PROBABILITIES AND SCALED LIMITS 10.8.1 Growth in the square Consider the probability (10.73) relating to the maximum height in the polynuclear growth model with a1 = √ √ · · · = an = q, b1 = · · · = bm = q, bm+1 = · · · = bn = 0. In this case Pr(h ≤ l) = (1 − q)

nm

l 

(1 + qeiθk )m (1 + e−iθk )n

k=1

 U(l)

.

Let us now write q = e−1/L , l → Lt and take the limit L → ∞ (this is the Laguerre case of the exponential limit discussed in Section 10.3.3). The result of Exercises 10.3 q.1 implies [329] lim Pr(h ≤ Lt)

L→∞

1 Γ(j)Γ(n − m + j) j=1



t

= m

n−m −x

= Em,2 (0; (t, ∞); x

e

 dx1 · · ·

0

t

dxm 0

m 

xn−m e−xl l

l=1



(xk − xj )2

1≤j · · · > hn−1 ≥ 0. It follows from this that in the particular exponential limit obtained by writing q = e−1/L , l → Lt and taking L → ∞ [27], lim Pr(h ≤ Lt) =

L→∞

(n/2)!

1 n−1 l=1

 l!

t

 dx1 · · ·

0

= En/2,4 (0, (t, ∞); e

dxn/2 0

−x



n/2

t

e−xj

j=1

).



(xk − xj )4

1≤j 0 can be established by the same technique as used in the proof of Proposition 9.1.3. It follows from this that soft l |Ql (x1 ) · · · Ql (xn )|ρhard (n) (Ql (x1 ), . . . , Ql (xn ))|a=l = ρ(n) (x1 , . . . , xn ) + Rn (x1 , . . . , xn ),

(10.201)

where Rnl → 0 as l → ∞ uniformly on xj ∈ [y, ∞) (j = 1, . . . , n). Thus the analogue of the property (9.3) relating to Proposition 9.1.1 is valid. Hence the conclusion of the latter applies, telling us that lim E2hard (Ql (y); l) = 1 +

l→∞

Z ∞ Z ∞ X (−1)n ∞ dx1 · · · dxn ρsoft (n) (x1 , . . . , xn ), n! y y n=1

499

LATTICE PATHS AND GROWTH MODELS

which recalling (9.4) is (10.200).

We can deduce from Proposition 10.8.3 the following limit theorem, which includes (10.198). P ROPOSITION 10.8.4 We have lim E2hard (0; (0, l2 − 2l(l/2)1/3 y + O(l)); l) = E2soft (0; (y, ∞)).

l→∞

(10.202)

Proof. Since Ql (y) = l2 − 2l(l/2)1/3 y + O(l2/3 ), we can rewrite the l.h.s. of (10.202) as lim E2hard (0; (0, Ql (y) + O(l)); l)

l→∞

which in turn can be written as lim E2hard (0; (0, Ql (y)); l) + lim E2hard (0; (Ql (y), Ql (y) + O(l)); l).

l→∞

l→∞

From Proposition 10.8.3 we know that the first term is equal to E2soft (0; (y, ∞)). For the second term, the working of the proof of Proposition 10.8.3 gives that it equals lim E2soft (0; (y − O(l−1/3 ), y))

l→∞

and thus vanishes.

Closely related to the scaled form (10.199) for the cumulative distribution of l is an analogous limit formula for lN ,  l − 2√N  ≤ y = E2soft (0; (y, ∞)). (10.203) lim Pr N 1/6 N →∞ N This result, quantifying the limiting distribution of the largest increasing subsequence length of a large random permutation, was derived in [31] as a corollary of (10.199). Some of the working is given in Exercises 10.8. The analogue of (10.201) can readily be established for β = 1 and 4. This allows us to deduce from (10.193) and (10.197) the limit formulas [35]  l − 2z   l − 2z  soft ˜4soft (0; (y, ∞)), lim Pr ≤ y = E (0; (y, ∞)), lim Pr ≤ y =E 1 z→∞ z→∞ z 1/3 z 1/3 and these in turn tell us that in relation to increasing subsequences √ √ l  l  2k,k − 2 2k 2k,k − 2 2k soft ˜4soft (0; (y, ∞)). ≤ y = E (0; (y, ∞)), lim Pr ≤ y =E lim Pr 1 k→∞ k→∞ (2k)1/6 (2k)1/6 1. [328] Let the sequence {qn }n=0,1,... satisfy the bounds 0 ≤ qn ≤ 1 and be monotonically decreasing so that qn ≥ qn+1 . Let ∞ X ξn qn φ(ξ) := e−ξ n! n=0

E XERCISES 10.8

and for given d > 0 write p √ μ(d) n log n, n = n + (2 d + 1 + 1)

p √ νn(d) = n − (2 d + 1 + 1) n log n.

The objective of this exercise is to show that −d φ(μ(d) ≤ qn ≤ φ(νn(d) ) + Cn−d n ) − Cn

for all n ≥ n0 , where C is some positive constant.

(10.204)

500

CHAPTER 10

(i) Suppose ξ > 0. With wn (ξ) := e−ξ ξ n /n!, and f (x) = x log x + 1 − x, use Stirling’s formula to show wn (ξ) ≤ C exp(−ξf (n/ξ)). (ii) Note that for 0 ≤ x ≤ 2, f (x) ≥ (x − 1)2 /4 and use this in the result of (i) to deduce √ √ [ξ−2 d+1 ξ log ξ]

X

C , ξd

wn (ξ) ≤

n=0

[2ξ] X √ √ n=[ξ+2 d+1 ξ log ξ]

wn (ξ) ≤

C . ξd

Also, use the fact that f (x) ≥ x/10 for x ≥ 2 to show that for ξ > ξ0 > 0 ∞ X

wn (ξ) ≤ C(ξ0 )e−ξ/5 .

n=[2ξ]

Combine these inequalities to conclude that for ξ large enough ∞ X

wn (ξ) −

n=0

X √ √ n:|n−ξ|≤2 d+1 ξ log ξ

wn (ξ) ≤

C . ξd

(iii) Use the assumption that 0 ≤ qn ≤ 1 to deduce from the final result of (ii) that φ(ξ) −

X √ √ n:|n−ξ|≤2 d+1 ξ log ξ

qn wn (ξ) ≤

C . ξd

P Use the assumption that qn ≥ qn+1 and the fact that ∞ n=0 wn (ξ) = 1 to show X qn wn (ξ) ≤ qξ−2√d+1√ξ log ξ , √ √ n:|n−ξ|≤2 d+1 ξ log ξ

and similarly show X √ √ n:|n−ξ|≤2 d+1 ξ log ξ

qn wn (ξ) ≥ q[ξ+2√d+1√ξ log ξ] −

C . ξd

By combining the above three inequalities, deduce (10.204). 2. [31] In (10.173) replace l by 2z + z 1/3 y so that it reads ∞ ” ” “ l − 2z X z 2N “ lN − 2z −z 2 ≤ y = e ≤ y . Pr Pr z 1/3 N! z 1/3 N=0

Now apply (10.204) with qn → qn (ξ), ξ = z 2 to deduce from (10.199) the limit formula (10.203).

10.9 HAMMERSLEY PROCESS WITH SOURCES ON THE BOUNDARY In the droplet PNG model nucleation events occur above the plateau formed by an initial nucleation event at (x, t) = (0, 0). An extension of this model is to allow for growth of the droplet due to nucleation events forming on the left and right boundaries of the droplet. In particular, suppose nucleation events are created at the rate α− dt on the left boundary, and at the rate α+ dt on the right boundary. An equivalent viewpoint is that surrounding the initial nucleation event at (x, t) = (0, 0) are two staircase structures of growing plateaux (downward sloping for x < 0, upward sloping for x > 0) with vertical increments of one unit having intensity α− for x < 0 and α+ for x > 0. The arrival of a layer of the staircase structure at the boundary

501

LATTICE PATHS AND GROWTH MODELS

of the droplet realizes the creation of nucleation events thereon. It turns out that this particular extension of the droplet PNG model is relevant to the study of the KPZ (Kardar-Parisi-Zhang) universality class of critical phenomena [451]. √ To analyze the height at the origin in this model after time t = 2T we note that it is equivalent to the length L(T, α+ , α− ) of the longest up/right path in a Hammersley process with z 2 = T 2 , and furthermore in which there are points on the x-axis (y-axis) of the unit square with Poisson distribution of intensity α+ (α− ). The cumulative distribution of the longest path length L(T, α+ , α− ) can be expressed as a random matrix average by a limiting procedure applied to the discrete PNG model of Section 10.3.1. For this purpose we extend the n × n non-negative integer matrix [xij ]i,j=1,...,n to an (n + 1) × (n + 1) matrix [xij ]i,j=0,...,n , but again with all elements chosen according to the geometric distribution (10.70). As in the discussion at the beginning of Section 10.6, to recover the setting of the Hammersley model in the region of the square away from the boundary we set ai = bj = T /n (i, j = 1, . . . , n) and take n → ∞. The boundary distributions in this limit are obtained by setting a0 = α+ , b0 = α− . However this procedure leaves at the origin a non-negative integer geometric random variable with parameter α+ α− (this distribution will be denoted g(α+ α− )). Denoting by L+ (T, α+ , α− ) the longest up/right path in this process, we see it is related to L(T, α+ , α− ) by the simple relation χ ∈ g(α+ α− ).

L+ (T, α+ , α− ) = L(T, α+ , α− ) + χ,

(10.205)

Furthermore, the limiting procedure applied to (10.73) (appropriately extended to include the variables a0 and b0 ) gives 2 ˜ l, Pr(L+ (T, α+ , α− ) ≤ l) = (1 − α+ α− )e−(α+ +α− )T −T D

(10.206)

where ˜ l := D

l 

(1 + α+ eiθj )(1 + α− e−iθj )e2T

Pl j=1

cos θj

 U(l)

j=1

.

(10.207)

The formulas (10.205) and (10.206) provide us with a formula for the cumulative distribution of L(T, α+ , α− ). P ROPOSITION 10.9.1 We have 2 ˜ l − α+ α− D ˜ l−1 ). Pr(L(T, α+ , α− ) ≤ l) = e−(α+ +α− )T −T (D

(10.208)

Proof. Introducing the generating functions Q(x) =

∞ X

Pr(L(T, α+ , α− ) ≤ l)xl ,

l=0

Q+ (x) =

∞ X

Pr(L+ (T, α+ , α− ) ≤ l)xl ,

l=0

we see from (10.205) that Q+ (x) =

∞ X l=0

xl

l X

Pr(L(T, α+ , α− ) ≤ l − k)Pr(χ = k)

k=0

= (1 − α+ α− )

∞ X l=0

xl

l X

Pr(L(T, α+ , α− ) ≤ l − k)(α+ α− )k =

k=0

1 − α+ α− Q(x), 1 − xα+ α−

where the final equality follows by writing xl = xl−k xk and summing independently over l − k and k. Multiplying both sides of this identity by 1 − xα+ α− and equating like powers of x gives (10.208).

˜ l in (10.208) with α+ = 1/α− = α can be expressed in terms of The quantity D    Pl ˜ l  = e2T j=1 cos θj Dl = D α+ =α− =0

U(l)

502

CHAPTER 10

familiar from (10.166), and monic orthogonal polynomials {πj (eiθ )}j=0,1,... with respect to the weight e2T cos θ ,  π 1 1 πj (eiθ )πk (eiθ )e2T cos θ dθ = 2 δj,k . 2π −π κj P ROPOSITION 10.9.2 We have    ˜ l  = (1 − l)πl (−α)πl (−α−1 ) − απl (−α)πl (−α−1 ) − α−1 πl (−α)πl (−α−1 ) Dl . (10.209) D α+ =1/α− =α

Proof. Let πn∗ (z) := z n πn (1/z) (cf. Definition 8.6.1). It follows from the working of Exercises 5.1 q.6(i) (for the first equality) and the first, third and fourth equations in Proposition 8.6.2 (for the second equality) that E DQ n iθj − x)(e−iθj − y) n j=1 w(θj )(e 1 X 2 π ∗ (x)πn∗ (y) − xyπn (x)πn (y) U (n) DQ E = 2 κk πk (x)πk (y) = n . (10.210) n κn k=0 1 − xy j=1 w(θj ) U (n)

Taking the limit y → 1/x = −1/α in the second equality gives the stated result.

Our interest is in the scaled limit of (10.208) with α+ = 1/α− = α for l = [2T + T 1/3 s],

α = 1 − y/T 1/3,

According to (10.166), (10.199) and (9.46) lim e

T →∞

−T 2

D[2T +T 1/3 s] (t) =

E2soft (0, (s, ∞))



T → ∞. 

= exp −

∞

 (t − s)q 2 (t) dt ,

(10.211)

s

where q(t) is the Painlev´e II transcendent satisfying (9.45) subject to the boundary condition (9.47) with ξ = 1. The remaining task then is to analyze the polynomials πl (−α) etc. in (10.209). A rigorous treatment of this problem has been given by Baik and Rains [35], [33] using Riemann-Hilbert methods. To avoid technicalities we will proceed instead via a heuristic analysis [451], [196]. First we note from the third and fourth equation in Proposition 8.6.2 that πn+1 (z) = zπn (z) + rn+1 πn∗ (z),

∗ πn+1 (z) = rn+1 zπn (z) + πn∗ (z).

(10.212)

Introduce Rn (T ) := (−1)n−1 rn (T ),

Pn (α) = e−T α πn∗ (−α),

Qn (α) = −e−T α (−1)n πn (−α).

(10.213)

Then we see that the equations (10.212) are consistent with the asymptotic forms R[2T +T 1/3 s] (T ) ∼ T −1/3 u(s),

P[2T +T 1/3 s] (1−y/T 1/3) ∼ a(s, y),

Q[2t+T 1/3 s] (1−y/T 1/3 ) ∼ b(s, y), (10.214)

in which u, a, b are coupled by the partial differential equations ∂a ∂b = ub, = ua − yb. (10.215) ∂s ∂s Moreover, in terms of the limiting functions a and b we see from the first equality in (10.210) that  ˜ [2T +T 1/3 s] (T ) /D[2T +T 1/3 s] (T ) g(s, y) := lim e−(α+ +α− )T D T →∞ α+ =1/α− =1−y/T 1/3  s  s = a(u, y)a(u, −y) du = b(u, y)b(u, −y) du, (10.216) −∞

−∞

503

LATTICE PATHS AND GROWTH MODELS

while (10.209) gives g(s, y) = (s − y 2 )a(s, y)a(s, −y) + a(s, −y)

∂ ∂ a(s, y) − a(s, y) a(s, −y). ∂y ∂y

(10.217)

Knowledge of g(s, y) is sufficient for the scaled form of the cumulative distribution of L(t, α+ , α− ), since from (10.208) and (10.211) it follows that  L(T, 1 − y/T 1/3 , 1 + y/T 1/3) − 2√T  F˜y (s) := lim Pr ≤s 1/6 T →∞ T  ∂  g(s, y)E2soft (0, (s, ∞)) . (10.218) = ∂s From the definition (10.210) of φ∗n and the definitions (10.213) and (10.214), we see that a and b are related by 1

a(s, y) = −b(s, −y)e 3 y

3

−sy

.

(10.219)

The task then is to determine a(s, y) or b(s, y). In fact it is more convenient to obtain a system of equations determining both these quantities. The equations (10.215) give their dependence on s, but involve the unspecified limiting function u(s). Thus it remains to determine u(s), partial differential equations for the dependence of a and b on y, and boundary conditions for the partial differential equations. First we will determine equations for the dependence of a and b on y. P ROPOSITION 10.9.3 We have n r t rn t  rn+1 rn t  n+1 t + 2− πn (z) + − 2 πn∗ (z), πn (z) = z z z z z  r   rn+1 rn t  ∗ n+1 t ∗ + rn t πn (z) + − t + πn (z). πn (z) = − z z

(10.220) (10.221)

Proof. The first equation follows immediately from (8.196) with t → (2T )2 , after making use of the second and third

equations in Proposition 8.6.2. The second equation is derived similarly, after deriving an equation for πn∗  (z) analogous to (8.196) [232].

Formal substitution of the asymptotic forms (10.214) in the equations of Proposition 10.9.3 gives the partial differential equations ∂a = u2 a − (u + yu)b, ∂y

∂b = (u − yu)a + (y 2 − s − u2 )b. (10.222) ∂y √ To determine u we read off from (8.192) with t → 2T that rn satisfies the second order difference equation −

n rn = rn+1 + rn−1 T 1 − rn2

with

r0 = 1,

r1 = −

I1 (2T ) , I0 (2T )

(10.223)

which we know is a transformed version of a particular discrete PII equation. With Rn (t) as specified in (10.213), and the scaled form of this quantity as specified by (10.214), this difference equation becomes the differential equation d2 u = su + 2u3 . (10.224) ds2 We recognize (10.224) as the differential equation (9.45) satisfied by the Painlev´e II transcendent q in (10.211). Furthermore, to be compatible with the first of the initial conditions in (10.223), we require for

504

CHAPTER 10

s = −2T 2/3 and T → ∞ that T −1/3 u(s) → −1, or equivalently  u(s) ∼ − −s/2. s→−∞

Up to the minus sign, this is precisely the boundary condition (9.48), and we know from [294] that (10.224) has the unique solution with this property, u(s) = −q(s).

(10.225)

In relation to the boundary condition, we note from (10.219) that a(s, 0) = −b(s, 0). Using this in (10.215) with y = 0 and u given by (10.225) shows   ∞  a(s, 0) = A exp − q(t) dt . s

To determine the constant A we note from (10.227) below that a(s, 0) → 1 as s → ∞ and thus    ∞ q(t) dt . a(s, 0) = −b(s, 0) = exp −

(10.226)

s

With all quantities now completely determined, the results can be summarized as follows [35], [33]. P ROPOSITION 10.9.4 Denote by −q(t) the Painlev´e II transcendent satisfying (10.211), subject to the boundary condition (9.47) with ξ = 1, and define F˜y (s) as the scaled limit in (10.218). We have   ∞  ∂  g(s, y) exp − F˜y (s) = (t − s)q 2 (t) dt , ∂s s where g(s, y) is given in terms of a(s, y) by (10.217). The function a(s, y) in turn is specified as the solution of the coupled partial differential equations ∂a ∂b = qb, = qa − yb, ∂s ∂s ∂b ∂a = q 2 a − (q  + yq)b, = (q  − yq)a + (y 2 − s − q 2 )b, ∂y ∂y subject to the boundary condition (10.226). E XERCISES 10.9

1.

(i) For a general real weight w(θ) show Q iθj )U (n)  n j=1 w(θj )(1 − xe Qn = πn∗ (x).  j=1 w(θj )U (n)

(ii) In the case of the weight w(θ) = e2T cos θ , use the result of (i) and (10.206) to show that πn∗ (−x) =

˜ n | α+ =x D α− =0

˜ n |α =α =0 D + −

=

exT Pr(L+ (T, x, 0) ≤ n) . Pr(L+ (T, 0, 0) ≤ n)

From this conclude πn∗ (−x) → exT as n → ∞ and thus a(s, y) → 1

as s → ∞.

(10.227)

Chapter Eleven The Calogero–Sutherland model Consideration of shifted mean parameter-dependent Gaussian random matrices, or equivalently Hermitian matrices with entries undergoing Brownian motion, leads to the Dyson Brownian motion model of the onecomponent log-gas. In the classical cases, a similarity transformation of the corresponding Fokker-Planck operator gives the Schr¨odinger operator for the Calogero-Sutherland model, which is the name given to the quantum many-body system for particles interacting on a line or a circle via the 1/r2 pair potential. By generalizing these Schr¨odinger operators to include exchange terms, decompositions into more elementary operators can be exhibited, and these operators can be used to establish integrability. For the coupling β = 2 the pair potential term in the Schr¨odinger operator is not present, giving rise to a free Fermi system and allowing the corresponding Green function to be expressed as a determinant. In the Gaussian and cases, this determinant form can be related to certain matrix integrals, including that due to Harish-Chandra, and Itzykson and Zuber. The determinant form also allows for the calculation of dynamical correlation functions, by applying formulas worked out in Chapter 5. These are analyzed in various scaled limits.

11.1 SHIFTED MEAN PARAMETER-DEPENDENT GAUSSIAN RANDOM MATRICES In some energy spectra problems for chaotic quantum systems there is a parameter which varies the spectrum continuously. A well-known example, highlighted in [284], is the spectrum of the hydrogen atom as a function of the magnetic field strength. One approach to modeling such systems is to consider a random matrix H with distribution interpolating between a fixed matrix H0 (when τ = 0) and one of the Gaussian ensembles of Chapter 1 (when τ = ∞). D EFINITION 11.1.1 With H real symmetric (β = 1), Hermitian (β = 2), or self-dual quaternion real (β = 4), the parameter-dependent Gaussian ensembles are defined to have the joint p.d.f. for the independent elements N   1 (0) Pτ (H(0) ; H) = exp − β |Hjk − e−τ Hjk |2 /2|1 − e−2τ | C j,k=1   1 = exp − βTr(H − e−τ H(0) )2 /2|1 − e−2τ | , C where C is the normalization and H(0) is a fixed Hermitian matrix. Comparing Definition 11.1.1 with the definitions given in Chapter 1 of the parameter-independent Gaussian ensembles we see that instead of the elements being chosen with mean zero and fixed standard deviation, they now have a mean determined by H(0) (recall Section 1.8) and a parameter-dependent variance. For H(0) fixed, we see by changing variables that H = |1 − e−2τ |1/2 X + e−τ H(0) ,

(11.1)

where X a member of the Gaussian β-ensemble (β = 1, 2, 4), and thus H is equal to the sum of a random and deterministic matrix. This setting is sometimes referred to as a random matrix model with a source [95].

506

CHAPTER 11

In applications H(0) may itself be a random matrix. For example, let β = 2 and suppose the joint p.d.f. for the elements at τ = 0 is that of the GOE, P0 (H(0) ) =

N    1 (0) 2 (0) 2 1 1/2 −(Hjj ) /2 e e−(Hjk ) 1/2 2π π j=1 j 0, knowledge of the Mellin transform (complex moments)  ∞ m(λ) := sλ−1 p(s) ds 0

as a function in the complex plane gives 1 p(s) = 2πi



c+i∞ c−i∞

s−x m(x) dx

686

CHAPTER 14

via the inverse Mellin transform. Now, for random matrices from U (N ), and with p(s) denoting the distribution of |Λ(−1)|2 , m(a + 1) =

N 

|2 + 2 cos θl |a

l=1

 U(N )

,

(14.109)

and this in turn has an explicit product of gamma function evaluation which follows from (14.103) (for a discussion of computing the corresponding Mellin transform, see [471]). The asymptotic formula (14.104) with β = 2, r = s = 1, b = 0 shows that for N → ∞ N −a m(a + 1) ∼ 2

G2 (a + 1) . G(2a + 1)

(14.110)

An application to zeta function theory of this result has been given by Keating and Snaith [354], in keeping with their hypothesis relating to (14.107). Thus, after recalling the identification (14.108), one anticipates that the U (N ) result (14.110) should relate to the large T value of  2a 1 1 T   1  + it  dt. (14.111) ζ 2 a 2 (log T ) T 0 In zeta function theory this was conjectured to have the structure f (a)A(a), where A(a) is the number theoretic quantity 2 ∞   1 λ  Γ(a + m) 2 −m  1− A(a) = p p m!Γ(a) m=0 primes p

and 42 24024 1 , f (3) = , f (4) = 12 9! 16! (the first two such values are known rigorously), with the other values of f (a) unknown. Indeed, these values are precisely those given by the r.h.s. of (14.110). Thus one is led to interpret this term as due to U (N ) like properties of the Riemann zeros, leading to the conjecture that for general a f (1) = 1,

f (2) =

f (a) = E XERCISES 14.5 N DY

1.

G2 (a + 1) . G(2a + 1)

(i) Note from the results of Exercises 5.1 q.3(i) and (ii) that

eibθl |2 + 2 cos θl |a (1 + xe−iθl )

l=1

E CUEN

=

MN (a + b, a − b, 1) 2 F1 (−N, a + b; b − a − N ; x). MN (0, 0, 1)

(ii) Deduce from this the limiting behavior E DY MN (0, 0, 1) eibθl |2 + 2 cos θl |a (1 + xe−iθl ) = (1 + x)−(a+b) MN (a + b, a − b, 1) l=1 CUEN N

lim

N→∞

and show that this is consistent with the prediction of the Fisher-Hartwig formula (14.100). 2.

(i) Note that with a(θ) of Proposition 14.5.2 specified by g(θ) = 0, R = 2, θ1 = φ1 , θ2 = φ2 , a1 = b1 = 12 , a2 = −b2 = 12 we have DN [ea(θ) ] = eiN(φ2 −φ1 )

N E DY (eiθl − eiφ1 )(e−iθl − e−iφ2 ) l=1

CUEN

.

687

FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS

(ii) Use (5.89) and (5.87) to deduce from this that DN [ea(θ) ] = eiN(φ2 −φ1 )/2

sin((N + 1)(φ2 − φ1 )/2)) . sin((φ2 − φ1 )/2)

As this does not exhibit the N → ∞ behavior (14.100), conclude that the region |Re(ar )| < 12 , |Re(br )| < 1 sufficient for the validity of the latter cannot in general be enlarged in all parameters. 2 3. [332] The objective of this exercise is to use the Szeg¨o asymptotic formula (14.71) to anticipate features of the Fisher-Hartwig asymptotic formula (14.100). (i) In the second expression of Proposition 14.5.2 for a(θ), replace the logarithms by their corresponding power series truncated at the N th term, and read off that the Fourier coefficients of a(θ) are then given by 8 PR −iθr p 1 > χp≤N , p ≥ 1, < gp − p r=1 (ar + br )e p = 0, g0 , ap = > : gp − 1 PR (ar − br )eiθr p χ|p|≤N , p ≤ −1. r=1 |p| (ii) Show from (i) that for N large ∞ X

pap a−p ∼

p=1

∞ X

pgp g−p −

p=1

(ar − br )g+ (θr ) −

r=1 R X

−

R X

R X

(ar + br )g− (θr )

r=1

(ar + br )(ar − br ) log(1 − ei(θr −θr ) ) + log N

r,r =1 r=r

R X

(a2r − b2r ).

r=1

Substitute this in (14.71) to reproduce (14.100) up to the product over the Barnes G-function in E. 4. [368] The aim of this exercise is to determine the distribution of the form factor for the log-gas on a circle. For this purpose, with k = p/N , p ∈ Z consider N DY

ei(u/

E √ √ N ) cos Nkθl +i(v/ N ) sin Nkθl CβEN

l=1

.

(i) Use the cumulant expansion (14.69), (14.70) to deduce that the logarithm of this average has the large N form Z “ 1 ” u2 + v 2 π T 1− . ρ(2) (θ, 0) cos N kθ dθ + O √ 4 N −π (ii) Conclude that the joint distribution of (A, B) =

N N “ 1 X ” 1 X √ cos N kθj , √ sin N kθj N j=1 N j=1

is for large N the Gaussian 2

e−(A

+B 2 )/2σ 2

,

σ2 =

(iii) In the notation of (ii) note that A2 + B 2 =

1 2

Z

π −π

ρT(2) (θ, 0) cos N kθ dθ.

˛ ˛ N 1 ˛ X ikNθj ˛2 e ˛ ˛ N j=1

so that for k = 0

Z ˆ k) = (A2 + B 2 )CβEN = S(N

π −π

ρT(2) (θ, 0) cos N kθ dθ.

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CHAPTER 14

Conclude from the result of (ii) that the distribution of s = A2 + B 2 is, for large N , the exponential ˆ k). distribution with mean S(N 5. [117, 235] With U ∈ U (N ), eigenvalues {eiθl }l=1,...,N , introduce the characteristic polynomial as above (14.106), and in terms of this define PN Z(s) = e−πiN/2 e−i n=1 θn /2 z −N/2 Λ(z). (note that Z(s) is real for |s| = 1). Results from [117] give that for k ∈ Z+ N −k

2

−2k

|Z  (1)|2k U ∈U (N)

∼ ˜bk ,

(14.112)

N→∞

where, with [xj ]f (x) denoting the coefficient of xj in the power series expansion of f (x), “ ” ˜bk = (−1)k(k+1)/2 (2k)![x2k ] e−x/2 x−k2 /2 det[Iα+β−1 (2√x)]α,β=1,...,k (cf. (14.110)). (i) Interchange row β with row k − β + 1 in the above determinant, then compare with (8.97) to conclude that “ ” k ˜bk = (−1) (2k)![x2k ] ex/2 E ˜ hard(k) ((0, 4x); ξ = 1; k) , 2 A(k, k)

A(a, μ) := a!

a Y (j + μ − 1)! . j! j=1

(ii) Make use of (8.88) and Proposition 8.3.3 to conclude from (i) that “ Z 4x k ds ” ˜bk = (−1) (2k)![x2k ] exp − (η(s) + k2 ) A(k, k) s 0 where η(s) satisfies the differential equation “ 1” k2 (η − sη  ) − 2 = 0 (sη  )2 + 4 (η  )2 − 64 4 subject to the requirement that it is even in s with η(0) = −k2 .

14.6 ASYMPTOTIC PROPERTIES OF Eβ (n; J) AND Pβ (n; J) 14.6.1 Large s behavior of Eβ (0; s) In (9.90) results of a Coulomb gas argument predicting that for an eigenvalue free region J, log E2 (0; J) is to leading order in |J| proportional to |J|3 , |J| and |J|2 at the soft edge, hard edge and in the bulk, respectively, were required. Here these results will be derived by combining macroscopic electrostatics with a scaling argument [189]. One considers the setting of the one-component log-gas confined to the half-line x > 0 with background charge density given by the power law ρb (x) = −Axμ . The basic hypothesis is that for large s Eβ (0; s) ∼ e−βE(s) ,

(14.113)

where E(s) is the electrostatic energy due to excluding the mobile positive charge from the interval (0, s). Note that e−βE(s) is the Boltzmann factor for the configuration with energy E(s). Now, let ρ(1) (x) denote the charge density of the positive charge (ρ(x) = 0 for x ∈ [0, s]), and let ρˆ(x) denote the total charge, ρˆ(x) = ρ(x) + ρb (x). The latter is constrained by the condition of global charge neutrality  ∞ ρˆ(x) dx = 0. (14.114) 0

FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS

With φ(x) = −

∞

−∞

ρˆ(y) log |x − y| dy one then has  1 ∞ ρˆ(x)φ(x) dx. E(s) = 2 0

689

(14.115)

For the region (s, ∞), where there are mobile charges, the potential is a constant which can be taken to be zero. Noting that for x ∈ [0, s], ρˆ(x) = ρb (x) = −Axμ , and making use of (14.114) we see that this latter condition is consistent with the homogeniety property ρˆ(sx) = sμ ρˆ(x).

(14.116)

Substituting in (14.115) gives E(s) = Cs2u+2 ,

C = E(1)

(14.117)

and hence (14.113) gives the prediction Eβ (0; s) ∼ e−βCs

2μ+2

1 2,

.

(14.118)

− 12

and 0 so as to give the known asymptotic form of the density at the soft edge, hard edge With μ = and in the bulk, respectively, found in Chapter 7, this gives the corresponding leading behavior of E2 (0; s) as quoted in (9.90). We remark that the stochastic differential equation characterizations (13.186) and (13.180) of the β-ensembles at the soft edge and the bulk can be used to prove (14.118) in those cases, and furthermore to give the value of C [463], [524], [462]. As already noted, this method can also be used to deduce higher order terms, giving for example the large s expansion in the bulk (9.99). 14.6.2 Large s behavior of Eβbulk (n; s) In the infinite log-gas with a constant background charge density −ρ, the leading large s asymptotics of Eβbulk (n; s)—the probability that an interval of length s contains exactly n eigenvalues—can be determined using an extension of the macroscopic electrostatics argument used above [152], [178]. As in (14.113) the basic hypothesis is that for large s Eβbulk (n; s) ∼ e−βδF ,

(14.119)

where here δF is the change in energy caused by changing the particle density so that the interval of length s contains n particles. Analogous to the hypothesis used in Section 14.2, the change in energy δF is taken to consist of two parts — an electrostatic energy V1 and a free energy V2 . These are calculated from the one-body density ρ(1) (x) according to  1 ∞ V1 = (ρ(1) (x) − ρ)φ(x) dx, (14.120) 2 −∞ ∞ where φ(x) = − −∞ (ρ(1) (y) − ρ) log |x − y| dy, and  ∞ !  ρ(1) (x) fβ [ρ(1) (x)] − fβ [ρ] dx. (14.121) V2 = −∞

Since, according to Proposition 4.8.1, the free energy per particle fβ is such that fβ [ρ(1) (x)] − fβ [ρ] =   1 1 β − 2 log(ρ(1) (x)/ρ), we have  1 1 ∞ − V2 = ρ(1) (x) log(ρ(1) (x)/ρ) dx. (14.122) β 2 −∞ The potential φ(x) and density ρ(1) (x) are calculated via two-dimensional macroscopic electrostatics. For

690

CHAPTER 14

n < ρs we suppose the n particles are confined to an interval (−b, b) and within that interval the system behaves like a conductor so φ(x) = −c (constant). The quantities b and c are to be calculated. The remaining particles are confined to an open-ended region (−∞, −t) and (t, ∞), where 2t = s. These regions are conductors with φ(x) = 0. The remaining intervals, (−t, −b) and (b, t), contain no mobile particles and so behave as an insulator on which φ will vary continuously between 0 and −c. The density ρ(1) (x) is further constrained by the conditions  b  ∞ ρ(1) (x) dx = n, (ρ(1) (x) − ρ)dx = 0. (14.123) −b

−∞

Substituting the constraints on φ(x) in (14.120), together with the condition ρ(1) (x) = 0 for b < |x| < t and the first condition in (14.123), then integrating by parts gives  t cn dφ(x) V1 = − + ρ dx. (14.124) x 2 dx b To determine dφ/dx we introduce the complex electric field E(z) = −∂φ(z)/∂x + i∂φ(z)/∂y, so that dφ/dx = −ReE(x) (recall Section 1.4.3). Now E(z) is required to be an analytic function of z in the upper half-plane (its value in the lower half-plane is given by symmetry), with real part which vanishes on the conducting regions of the real line. Furthermore, E(z) must have branch points at z = ±t, ±b, it must vanish as |z| → ∞ (since there is no net charge) and as t → ∞ with b = 0 it must equal πiρ (since then the real line is an insulator with uniform charge density ρ and so φ(z) = πy; recall Section 2.7.1). The unique function with these properties is   2 1/2 z − b2 E(z) = πρ i − 2 , t − z2 where the square root is chosen to be positive real on the real axis between (−t, −b) and (b, t). Taking the real part and changing sign gives that for x ∈ (b, t)  2 1/2 dφ x − b2 = πρ 2 , (14.125) dx t − x2 while the density in the interval (−b, b) is given by 1/2  2  1 ! + b − x2 − E (x) − E (x) = ρ 2 ρ(1) (x) = − . 2πi t − x2

(14.126)

Note that b is specified by the first equation in (14.123) with the substitution of (14.126). An integral identity in [270] allows the resulting equation to be rewritten   (14.127) n = 2ρt E(k  ) − k 2 K(k  ) , where K and E are the complete elliptic integrals of the first and second kinds and k 2 = 1 − k  with k  = b/t. The explicit formula (14.125) allows the evaluation of (14.124), 2

cn π 2 ρ2 2 + (t − b2 ). 2 4 Also, integrating (14.125) from b to t and recalling that φ(t) = 0, φ(b) = −c, gives 1/2  t 2   x − b2   2   c = πρ dx = πρt E (k ) − k K (k ) . t2 − x2 b V1 = −

(14.128)

(14.129)

FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS

691

The integral in (14.121) defining V2 can also be calculated exactly [152]. P ROPOSITION 14.6.1 One has V2 =

1

−

β

1 c, 2

where c = −φ(b) = −φ(x), x ∈ [−b, b]. Proof. Define a function F (z), analytic in the upper half-plane, by F (z) = −

i i 1 (E(z) − E(¯ z )) + (E(z) + E(z)) = ρ + E(z) 2πi 2π π

). Since log(F (z)/ρ) is also analytic in the upper half-plane, and decays of order (on the real axis F (x) = ρ(1) (x) − πi dφ dx 1/|z|2 as |z| → ∞, the residue theorem gives Z ∞ Z ∞ “ ““ i dφ ” i dφ ” ” F (x) log(F (x)/ρ)dx := log ρ(1) (x) − /ρ dx = 0. (14.130) ρ(1) (x) − π dx π dx −∞ −∞ Now for x ∈ / (−t, −b), (b, t) we know that dφ/dx = 0 while for x in these intervals ρ(1) (x) = 0, dφ/dx is positive and ` ´ ` ´ log ρ(1) (x) − πi dφ = log π1 dφ − iπ/2. Taking the real part of (14.130) we therefore have dx dx Z

∞

1 ρ(1) (x) log(ρ(1) (x)/ρ)dx = 2 −∞

„Z

Z t«

−b

+ −t

b

1 dφ dx = − (φ(−b) + φ(b)), dx 2

and the stated result follows upon recalling (14.122).

We see that V1 and V2 are completely specified by (14.128), (14.129) and Proposition 14.6.1. Hence Eβbulk (n; s) is completely specified — all that remains is to calculate the large s behavior of log Eβ (n; s). For this purpose, note that substituting (14.126) in the first equation of (14.123) and expanding for t  b shows  2nt 1/2 b∼ . (14.131) πρ Since k  = b/t, we see that (k  )2 ∼ (2n/πρt). From the known asymptotic expansions for E  (k  ) and K  (k  ) as k  → 0, (14.129) then gives that  8πρt n log +1 . c ∼ πρt − 2 n Substituting this result in the expressions (14.128) and Proposition 14.6.1 for V1 and V2 , then substituting the sum for δF in (14.119) shows   β n β βn  8πρt (πρt)2  + βn + − 1 πρt + 1− − log + 1 , (14.132) log Eβbulk (n; 2t) t,n→∞ ∼ −β 4 2 2 2 2 n tn and this in turn implies Eβbulk (n; 2t) Eβbulk (0; 2t)

∼ c˜β,N t,n→∞ tn

eβnπρt (πρt)βn2 /4+(β/2−1)n/2

(14.133)

for some c˜β,N . We recognize this latter form as precisely the conjectured asymptotic result (9.103) for the same ratio with n fixed, and we recognize (14.132) with n = 0 as agreeing with the first two terms of (9.99). Also of interest is the case 0  ρs − n  ρs because of its relationship to charge fluctuations, and in particular the formula (14.89). This case is considered in Exercises 14.6 q.1.

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14.6.3 Eβ,N (0; J) and pβ,N (0; J) for large N, |J| Log-gas arguments can also be used to study the asymptotics of gap probabilities and spacing distributions when N is large but finite. Here one is typically concerned with large deviations, when the size of the gap or spacing (measured in appropriate units) depends on N . Moreover, a rigorous justification of the log-gas heuristics can in some circumstances be given [329]. The simplest situation of this type is the gap probability Eβ,N (0; (−α, α)) in the finite N circular βensemble. Here 0 < α < π is an angle on the circle, and so on average the interval (−α, α) would contain αN/π eigenvalues if not constrained to be a gap. According to the hypothesis (14.113)  2π−α  β  2π−α  Eβ,N (0; (−α, α)) ∼ exp dθ1 ρˆ(θ1 ) dθ2 ρˆ(θ2 ) log |eiθ1 − eiθ2 | , (14.134) 2 α α where ρˆ(θ) := ρ(1) (θ) − N/2π. An electrostatics evaluation of ρˆ(θ) and the corresponding electrostatic energy is known [147], but we can in fact bypass such a calculation. The reason is that through the Toeplitz determinant asymptotic formula (9.123) we have knowledge of the electrostatic energy in the case β = 2. Since the electrostatic energy is independent of β this gives β α Eβ,N (0; (−α, α)) ∼ exp N 2 log cos . (14.135) 2 2 The next situation to be considered is the right tail large deviation form of the distribution of the largest eigenvalue in the Gaussian β-ensemble, scaled so that the leading order support is the interval (−1, 1). One sees from (8.73) and (8.86) that in the large s asymptotic regime pβ,N (0; (s, ∞)) ∼ ρ(1),N (s).

(14.136)

With s > 1, and thus outside of the interval of leading support, the density is known from (14.82) in the case β = 1, and so we read off that for general β > 0      (14.137) pβ,N (0; (s, ∞)) ∼ exp − βN s s2 − 1 + log(s − s2 − 1) . In the case of the scaled Laguerre ensemble the corresponding asymptotic form of the distribution of the largest eigenvalue outside the leading support is similarly given by (14.83). Finally we consider the left tail large deviation form of the distribution of the largest eigenvalue in the Gaussian β-ensemble, again scaled so that the leading order support is the interval (−1, 1). This is calculated from the log-gas formula pβ,N (0; (s, ∞)) ∼ e−β(U (s)−U(1)) ,

s < 1,

U (s) is the leading order in N portion of the total potential energy for the scaled Gaussian β-ensemble constrained so that the eigenvalues are restricted to be less than s. The working of Exercises 1.4 q.4 gives that such a constraint corresponds to a background density (1.81), and we see from (1.78) that to leading order in N  √2N s 1 CN U (s) = , x2 ρb (x) dx − 4 √2N (s−l) 2 where C is given by (1.80). Computing the integral making use of (4.2) then gives that for large N and s < 1 [124] √  2  s4 5  s + s2 + 3  1 3 1 2 2s 2 2 pβ,N (0; (s, ∞)) ∼ exp − βN − − s 3 + s − s 3 + s − log . 3 27 18 27 2 3 (14.138)

FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS

693

14.6.4 Covariance and variance of spacing distributions in the bulk For a one-dimensional system, the covariance of particular spacing configurations can be related to the variance of related configurations [100]. P ROPOSITION 14.6.2 In the bulk of a one-dimensional system, choose a particular particle at position xi , and denote the positions of the successive particles to the right by xi+1 , xi+2 , . . . . Then, independent of i, pβ (n; s) = δ(si (n) − s),

si (n) := xi+n+1 − xi

(14.139)

(si (n) will be referred to as the nth order spacing), while Cov(si (n), si+r+1 (n )) = Var(si (|r − n| − 1)) + Var(si (r + n + 1)) −Var(si (|r + n − n + 1| − 1)) − Var(si (r))

(14.140)

with the convention that si (−1) = 0. Proof. The first formula is essentially the definition of pβ (n; s). For the second formula, we note from the identity (z1 − z2 )(z3 − z4 ) =

” 1“ (z1 − z4 )2 + (z2 − z3 )2 − (z1 − z3 )2 − (z2 − z4 )2 2

that in general Cov(z1 − z2 , z3 − z4 ) =

” 1“ Var(z1 − z4 ) + Var(z2 − z3 ) − Var(z1 − z3 ) − Var(z2 − z4 ) . 2

The covariance formula (14.140) now follows from the difference formula (14.139) for si (n).

A noteworthy special case of (14.140) is n = n = 0, which gives the covariance of two nearest neighbor spacings (to the right) with r particles in between as  1 Cov(si (0), si+r+1 (0)) = Var(si (r + 1)) − 2Var(si (r)) + Var(si (r − 1)) (14.141) 2 (an alternative derivation of this result is given in Exercises 14.6 q.2). Another special case of interest is r = n = n . This represents the covariance of successive nth order spacings, and (14.140) then reads  1 Var(si (2n + 1)) − 2Var(si (n)) . Cov(si (n), si+n+1 (n)) = (14.142) 2 To make use of the above formulas the value of Var(si (n)) is required. For the random matrix couplings β = 1, 2, 4, and for small values of n, these can be read off from Tables 8.13, 8.14 and the inter-relation (8.161). The corresponding covariances are listed in Table 14.1. In general, since si (n) is not a linear statistic, Var(si (n)) is not easily accessible. However, physically one would expect Var(si (n)) to be asymptotically equal to Var(χ[0,n+1] ) (here we are assuming unit density), where the latter is the variance in the number of particles in an interval of length n + 1. Using the explicit asymptotic formula (14.85), we thus expect for the one-component log-gas that Var(si (n)) ∼

n→∞

2 log n + O(1). π2 β

(14.143)

Substituting in (14.141) then gives 1 1 , π 2 β r2

(14.144)

1 log n. π2 β

(14.145)

Cov(si (0), si+r+1 (0)) ∼ − r→∞

while substituting in (14.142) shows Cov(si (n), si+n+1 (n)) ∼ − n→∞

694

n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8

CHAPTER 14

An (1)

An (2)

An (4)

Bn (1)

Bn (2)

−0.077333811 −0.024900730 −0.011437207 −0.006455611 −0.004124830 −0.002857751 −0.002095049 −0.001601116 −0.0012631931

−0.05550500 −0.013895003 −0.005959174 −0.003288239 −0.0020817292 −0.001435997 −0.0010503631 −0.0008017114 −0.0006320362

−0.034643119 −0.007118315 −0.002983845 −0.001640118

−0.077333811 −0.138572479 −0.1784829481 −0.20745438515 −0.23007507345

−0.05550500 −0.089255975 −0.10983253094 −0.12449482474 −0.13586646270

Table 14.1 Tabulation of An (β) := Cov(si (0), si+n+1 (0)), Bn (β) := Cov(si (n), si+n+1 (n)) for small values of n and the random matrix couplings β = 1, 2 and 4. Note that in general A0 (β) = B0 (β).

Even with r = 3 the formula (14.144) accurately approximates the exact value given in Table 14.1. More generally we would expect the full distribution of si (n) to be asymptotically equal to that of χ[0,n+1] . It should therefore obey the Gaussian law (14.89). This trend is already seen in the small n data of Tables 8.13, 8.14, in that the skewness and kurtosis is typically decreasing as n increases. Further remarks can be made in relation to both (14.144) and (14.145). Consider first (14.145). Together with (14.143) it implies     Cov si−n−1 (n), si (n) + Var(si (n)) + Cov si (n), si+n+1 (n) ∼ O(1). (14.146) This is essentially a charge neutrality result, which says that the charge excess in the interval containing the first n + 1 particles will be compensated by the excess of opposite sign in the two neighboring intervals of n + 1 particles. Regarding (14.144), we note that the asymptotic behavior is identical to the leading non-oscillatory behavior of the charge-charge correlation as given by (14.9). The physical reasoning which led to (14.143) is consistent with this result. An analogous result is that  1 Cov(χ[0,1] , χ[n,n+1] ) = Var(χ[0,n+1] ) − 2Var(χ[0,n] ) − Var(χ[0,n−1] ) , 2 which according to (14.85) also exhibits the asymptotic behavior (14.144). Analytic information is also available on Var(si (n)) for finite n. In particular, French et al. [243] (see also Exercises 14.6 q.3) have refined the physical relationship between Var(si (n)) and Var(χ[0,n+1] ) by compensating for the fact that fixed particles are present at the endpoints in the definition of si (n) but not in χ[0,n+1] . Consequently they have deduced the formula 1 (14.147) Var(si (n)) ≈ Var(χ[0,n+1] ) − . 6 The accuracy of this formula for the log-gas at β = 1, 2 and 4, and small values of n, can be tested by reading off from Tables 8.13, 8.14 and the inter-relation (8.161), and comparing with the corresponding value of Var(χ[0,n+1] ) deduced from (14.84). The results displayed in Table 14.2 demonstrate an accuracy of up to 1 part in 104 . However for larger n the discrepancy increases; for example we have c9 (2) = 0.1585. Related to the topics of this section is the large k form of psoft β (k; s), the latter being the distribution of the

FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS

n

cn (1)

cn (2)

cn (4)

0

0.1606

0.1640

0.1545

1

0.1670

0.1664

0.1603

2

0.1674

0.1665

0.1622

3

0.1672

0.1664

0.1631

4

0.1644

0.1663

0.1635

695

Table 14.2 Tabulation of cn (β) := Var(χ[0,n+1 ) − Var(si (n)) for the random matrix couplings β = 1, 2 and 4. The approximation (14.147) predicts cn (β) ≈ 1/6.

kth largest eigenvalue at the soft edge. To leading order the mean μk of this distribution must satisfy  μk 2 |μk |3/2 , ρsoft k∼ (1) (X) dX ∼ 3π −∞ where the second asymptotic form follows from (13.68). It is proved in [281], for β = 2, that with σk2 proportional to (log k)/(βk 2/3 ), as k → ∞ (Xk + |μk |)/σk ∼ N[0, 1]. This is consistent with the data of Tables 9.1 and 9.2. 14.6.5 The Δ3 statistic In the bulk of the spectrum, after appropriate scaling, the mean eigenvalue spacing is a constant 1/ρ. Thus the mean number of eigenvalues n(x) from some (arbitrary) origin to a point x in the spectrum increases linearly with x. The actual number of eigenvalues between 0 and x, which is a staircase function that jumps one unit at the position of each eigenvalue, will deviate from a straight line. On the other hand, there will be a unique line of best fit according to the criterion of least square deviation. For such a straight line within an interval of length , Dyson and Mehta [153] have introduced this deviation as a statistic, denoted Δ3 (the subscript 3 occurs because two similar statistics, denoted Δ1 and Δ2 , were also introduced), characterizing the eigenvalue spectrum. Explicitly  1 /2 Δ3 := minA,B (n(y) − Ay − B)2 dy, (14.148)  −/2 where n(y) measures the number of eigenvalues from some arbitrary point a large distance from the interval (−/2, /2). Equating the partial derivatives with respect to A and B in (14.148) to zero to calculate the minimum shows   12 /2 1 /2 A= 3 yn(y) dy, B= n(y) dy. l −/2 l −/2 Minor manipulation then gives   /2   1 l/2 12 1 du dv δ(v − u)n(u)n(v) − 3 uvn(u)n(v) − n(u)n(v) Δ3 =  −l/2 l  −/2  /2  /2 2  1 12uv  = 2 n(u) − n(v) . du dv 1 + 2 2 −/2 l −/2

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Thus if we let N (|u − v|) denote the variance of the number of particles in the interval |u − v|, then by taking the ensemble average we see that  /2  /2  1 12uv  Δ3  = 2 du dv 1 + 2 N (|u − v|). 2 −/2 l −/2 Finally, changing variables u − v = y, u + v = z and performing the integration over z give [440]   2   y 3 y Δ3  = − 2 + 1 N (y) dy. (14.149)  0   Substituting the asymptotic expansion (14.85) (which assumes ρ = 1) in this expression shows that for large  9 Bβ 1 . Δ3  ∼ 2 log  − 2 + π β 4π β 2 E XERCISES 14.6

1. [152], [178] In this exercise the asymptotic form of Eβ (n; s) will be computed in the case 0  s − n  s using the theory which led to (14.132). (i) For 0  ρs − n  ρs (s = 2t), k = b/t ∼ 1. Thus use known expansions of E(k) and K(k) for k near 0 and 1 in the appropriate formulas given in the text to deduce that ρb ∼

ρs n − ρs + , 2 log(ρs/(n − ρs))

c ∼ π2

(ρs − n) . 2 log(ρs/(n − ρs))

(ii) Substitute the results of (i) in (14.128) and Proposition 14.6.1 to deduce from (14.119) that in the region in question “ ” π2 β (ρs − n)2 , Eβ (n; s) ∼ exp − 4 log ρs and relate this to (14.89). 2.

(i) Use the formula pβ (n; s) =

n d2 X (n − j + 1)Eβ (j; s), 2 ds j=0

which follows from (8.16), to show that Z

∞

s2 pβ (n; s) ds = 2

0

n X

Z (n − j + 1)Iβ (j),

∞

Iβ (p) :=

Eβ (p; s) ds. 0

j=0

(ii) Use the identities (si (n))2 =

n “X

si+j (0)

”2

j=0

to deduce that Z ∞

=

n “ X

si+j (0)

j=0

s2 pβ (n; s) ds = (n + 1)si (0)2 + 2

0

”2

+2

X

si+j (0)si+k (0)

0≤j x = e−x . N→∞ 72π The aim of this exercise is to give an heuristic prediction of this result. (i) Let ZˆN (x) denote the random variable for the number of consecutive spacings less than x. Note that Pr(ZˆN (x) = 0) = Pr(ZN > x). Integrate the leading term of the expansion (8.165) with ρ = N/2π to deduce that as x → 0, the probability that a single spacing is less than x is equal to

cβ (N x)1+β ,

cβ =

(β/2)β ((β/2)!)3 . 2π(β + 1)!(3β/2)!

(ii) From (i), by assuming that to some degree of approximation such spacings are independent, conclude Pr(ZˆN (x) = 0) ≈ (1 − cβ (N x)1+β )N ≈ e−cβ N(Nx)

1+β

.

Now scale x and set β = 2 to obtain the result. 6. [169] Consider a sequence of 2N + 1 particles on a circle of radius (2N + 1)/2π, label them by their scaled angles “ 1 1” N + > xn > · · · x1 > x0 > x−1 > · · · > − N + 2 2 and consider that statistic δn := xn − x0 − n. For p = ±1, ±2, . . . , ±N define the corresponding power spectrum P(p) =

N ˛2 1 ˛˛ X ˛ δn e−2πipn/(2N+1) ˛ . ˛ 2N + 1 n=−N

As a long wavelength (|p|  N , N  1), continuum approximation write δn ≈ n(1) (y) is the microscopic density, so that Z ˛2 ˛Z ” 1 ˛ N “ x ˛ (n(1) (y) − y) dy e−2πikx dx˛ ˛ 2N −N 0 Z ˛2 1 “ 1 ”2 ˛˛ N ˛ ≈ n(1) (x)e−2πikx dx˛ , ˛ 2N 2πk −N

P(p) ≈

Rx 0

(n(1) (y) − y) dy, where

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where the second line follows by integration by parts and the assumption that N is large. Deduce from this that “ 1 ”2 ˆ S(2πk), P(p) ∼ 2πk where Sˆ refers to the structure function. In the case of the log-gas, note from (14.8) that this implies P(p) ∼

1 , 2π 2 β|k|

thus exhibiting 1/f noise.

14.7 DYNAMICAL CORRELATIONS 14.7.1 Definition In dynamical many-body systems, the current correlations are fundamental quantities closely related to the density correlations. In particular the macroscopic one-body current J(x; τ ) is related to the one-body dynamical particle density ρ(1) (x; τ ) via the continuity equation ∂ ∂ ρ(1) (x; τ ) = − J(x; τ ). (14.151) ∂τ ∂x The macroscopic one-body current is usually defined as the averaged value of the classical microscopic current jτ (x) =

N dxj (τ ) j=1

dτ

δ(x − xj (τ )).

(14.152)

The latter satisfies ∂ ∂ nτ (x) = − jτ (x), ∂τ ∂x

nτ (x) :=

N

δ(x − xj (τ )),

(14.153)

j=1

and thus the continuity equation is true at a microscopic level. However, in the Fokker-Planck description of Brownian motion the classical microscopic current has no immediate meaning because the velocities do not explicitly occur in (11.15). One way to deduce the analogue of (14.152) is to insist on the applicability of the microscopic continu(1) ity equation. First note that for an observable Aτ1 = A({xj }) measured at parameter value τ1 , and an (2)

observable Bτ2 = B({xj }) measured at parameter value τ2 , the average of the product Aτ1 Bτ2 is given by  ∞  ∞  ∞  ∞ 1 (0) (0) (1) (1) (0) dx · · · dxN p0 (x ) dx1 · · · dxN Aτ1 Aτ1 Bτ2  = N ! −∞ 1 −∞ −∞ −∞  ∞  ∞ (2) (2) ×GFP x(0) ; x(1) ) dx1 · · · dxN Bτ2 GFP x(1) ; x(2) ). τ1 ( τ2 −τ1 ( −∞

−∞

(14.154) The delta function initial condition of the Green function and the structure of (11.15) show Gτ1 (x(0) ; x(1) ) = eLτ

N 

(1)

δ(xl

(0)

− xl ),

(14.155)

l=1 (1)

where it is understood that L acts on {xl }, and in (11.15) we have set γ = 1. Substituting this formula in

699

FLUCTUATION FORMULAS AND UNIVERSAL BEHAVIOR OF CORRELATIONS

(0)

(0)

(0)

(14.154) allows the integration over {xl } to be carried out. Then substituting (14.155) with {xl }, {xl } (1) (2) (1) replaced by {xl }, {xl } in the resulting expression and integrating over {xl } we obtain  ∞  ∞ (2) (2) Aτ1 Bτ2  = dx1 · · · dxN Bτ2 eL(τ2 −τ1 )/γ Aτ1 eLτ1 /γ p0 (x(2) ). (14.156) −∞

−∞

Next we note from Exercises 11.1 q.1 (ii) and (11.15) with γ = 1 that eLτ = e−βW/2 e−τ so we can rewrite (14.156) as  ∞  (2) dx1 · · · Aτ1 Bτ2  = −∞

∞

−∞

where A(τ ) := eτ

PN j=1

Π†j Πj /βγ βW/2

e

dxN e−βW/2 B(τ2 )A(τ1 )p0 (x(2) )eβW/2 ,

PN j=1

(2)

Π†j Πj /βγ

Aτ e−τ

PN j=1

Π†j Πj /βγ

(14.157)

(14.158)

and similarly the definition of B(τ ). N The equation (14.158) with Aτ = nτ (x) = j=1 δ(x − xj (τ )) provides a definition of the dynamical microscopic density n(x; τ ) in operator form. Substituting in (14.151) then allows the sought formula for the microscopic current to be obtained. N P ROPOSITION 14.7.1 Let n(x; τ ) be defined by (14.158) with Aτ = nτ (x) = j=1 δ(x − xj (τ )), and similarly define j(x; τ ) with Aτ = jτ (x), jτ (x) to be determined. Then for the continuity equation (14.151) (which now refers to microscopic quantities) to be satisfied we require jτ (x) = −

N i 1 ∂ 1 ∂  . δ(x − xj (τ )) + δ(x − xj (τ )) γβ j=1 i ∂xj i ∂xj

(14.159)

Proof. With n(x; τ ) and nτ as specified it follows from (14.158) that hX i PN † † ∂n(x; τ ) 1 τ PN Π†j Πj , nτ e−τ j=1 Πj Πj /βγ . = e j=1 Πj Πj /βγ ∂τ γβ j=1 N

Comparison with (14.151) and recalling the definition of jτ then shows i ∂ 1 hX † Π Πj , nτ . jτ (x) = − ∂x γβ j=1 N

The stated result now follows from the fact that N X

Π† Πj =

j=1

N X ∂2 + V (x1 , . . . , xN ) ∂x2j j=1

for some V .

14.7.2 Hydrodynamic limit Physical principles involving the continuity equation (14.153) can be used to predict the small k form of the dynamical structure function as given by (13.228) [54]. For a single particle moving in a viscous medium (γ = 1) Newton’s law of motion gives m

dv(τ ) = −v(τ ) + F , dτ

700

CHAPTER 14

where F is the applied force. Hence in an equilibrium situation v(τ ) = F. The many-body analogue of this latter equation is j(x; τ ) = F (x; τ ), where F (x; τ ) now refers to the macroscopic force density. For the log-gas, in the long wavelength regime, the force density will to leading order be of electrostatic origin, implying [145]  ∞  ∂  V (x) − j(x; τ ) = −nτ (x) ρ(1) (x ; τ ) log |x − x | dx . (14.160) ∂x −∞ To proceed further one takes the partial derivative with respect to x on both sides, and substitutes for ∂j(x; τ )/∂x on the l.h.s. using the continuity equation (14.153). Next the resulting equation is linearized by writing ρ(1) (x ; τ ) = ρ(1) (x) + δnτ (x) (recall that (14.160) refers to the long wavelength regime so the microscopic quantity nτ (x) is essentially smoothed) and terms of order (δnτ (x))2 are ignored. Using the equilibrium condition  ∞  ∂  V (x) − ρ ρ(1) (x) log |x − x | dx = 0, ∂x −∞ the linearized equation then reads

 ∞ ∂δnτ (x) ∂2 = −ρ 2 δnτ (x ) log |x − x | dx δτ ∂x −∞  ∞   ∂ ∂  = −ρ δn (x ) log |x − x | dx . τ ∂x −∞ ∂x

(14.161)

Introducing Fourier transforms and recalling (14.3) this gives ∂δˆ nτ (k) = −ρπ|k|δˆ nτ (k). ∂τ nτ (0)e−πρ|k|τ , or equivalently Thus δˆ nτ (k) = δˆ ˆ τ (0)e−πρ|k|τ . n ˆ τ (k) = n

(14.162)

ˆ τ ) = 1 ˆ nτ (k). Substituting (14.162) predicts But for a system confined to a region of length L, S(k; L n0 (k)ˆ that for the general β log-gas, in the long wavelength k → 0 limit, ˆ τ ) ∼ S(k; ˆ 0)e−πρ|k|τ , S(k; which when combined with (14.8) implies (13.228).

(14.163)

Chapter Fifteen The two-dimensional one-component plasma The two-dimensional one-component plasma (2dOCP) consists of log-potential charges of the same sign in a two-dimensional domain which contains a smeared out neutralizing background, and so is the twodimensional version of the one-component log-gas. Although only one value of the coupling allows an exact solution, there are a number of different two-dimensional geometries and boundary conditions for which this exact solution is possible. Here the exact solutions for disk, sphere and antisphere geometries are considered, as well as the exact solution for metallic and Neumann boundary conditions. The first three of these allow for interpretations as eigenvalue p.d.f.’s, and as the modulus squared of the many-body wave function formed by free fermions confined to these surfaces in the presence of a particular magnetic field. Also associated with these three geometries are the zeros of three families of random polynomials, although the correlations are given not by determinants, but rather by permanents. For the 2dOCP at general coupling, macroscopic arguments of the type used in Chapter 14 imply a number of sum rules and asymptotic formulas, which can be illustrated on the exact results. The fast decay of the correlations in the bulk is responsible for sum rules which have no one-component log-gas analogues. In the last section, a classification scheme for random matrix ensembles with complex eigenvalues is considered.

15.1 COMPLEX GAUSSIAN RANDOM MATRICES AND POLYNOMIALS 15.1.1 Eigenvalues of complex random matrices The Boltzmann factor for the 2dOCP in a disk has been calculated in Exercises 1.4 q.3 as proportional to N  j=1

e−πρΓ|rj |

2

/2



|rk − rj |Γ

(15.1)

1≤j k, we rewrite (15.5) to read (Tkk − Tjj )dVjk +

“X

dVjl Tlk −

l P (2l − 1) and using the notation of the statement of (15.213) this reads Zk,(N−k)/2 [u, v] = 2(k−N)/2 Ak,N

k! (k/2)!

X

ε(P )

P ∈SN P (2l)>P (2l−1)

k/2 Y l=1

Y

N/2

αP (2l−1),P (2l)

βP (2l−1),P (2l)

l=k/2+1

= 2(k−N)/2 Ak,N k!((N − k)/2)![ζ k/2 ]Pf[ζαj,l + βj,l ]j,l=1,...,N (cf. (6.159)), where the final equality, which immediately implies (15.208), can be verified directly from the definition of the Pfaffian.

With N even and ZN [u, v] :=

N/2

ZN [u, v] =

k=0

Z2k,(N −2k)/2 [u, v], we read off from (15.213) that [491]

2N (N +1)/4

1 N l=1

Γ(l/2)

Pf[αj,k + βj,k ]j,k=1,...,N .

(15.214)

N/2 Similarly we see that ZN (ζ) := k=0 ζ k Z2k,(N −2k)/2 [1, 1], which is the generating function for the probability p2k,N = Z2k,(N −2k)/2 [1, 1] of there being exactly 2k real eigenvalues, has the Pfaffian form  1  + β ] . (15.215) ZN (ζ) = Pf[ζα   j,k j,k j,k=1,...,N N u=v=1 2N (N +1)/4 l=1 Γ(l/2) If we choose, say, pj (x) = xj in (15.212) all the integrals defining αj,k , βj,k can be computed explicitly

,

756

CHAPTER 15

[155] (see also the proof of Proposition 15.10.4 below). Furthermore, we can check that by the symmetry of the integrand α2j,2k |u=1 = α2j−1,2k−1 |u=1 = 0,

β2j,2k |v=1 = β2j−1,2k−1 |v=1 = 0.

Thus the entries of the Pfaffian in (15.215) vanish in a checkerboard fashion. As in deducing (8.132) from (8.134) we can therefore write the Pfaffian as a determinant of half the size [10]   1 ZN (ζ) = | + β | , (15.216) det ζα N 2j−1,2k u=1 2j−1,2k v=1 j,k=1,...,N/2 2N (N +1)/4 l=1 Γ(l/2) giving a computable formula for the p2k,N . From this viewpoint the general N exact result pN,N = 2−N (N −1)/4 follows by noting from the coefficient of ζ N/2 in (15.215) that pN,N =

2N (N +1)/4

1 N l=1

Γ(l/2)

det[α2j−1,2k |u=1 ]j,k=1,...,N/2 .

Choosing pj (x) = Rj (x), where {Rj (x)} are the skew orthogonal polynomials for the β = 1 Gaussian weight, as discussed in Section 6.3 the determinant becomes diagonal, implying the sought result.

Correlation functions Since the number of real eigenvalues is a variable, the summed up generalized partition function (15.214) is the appropriate quantity to use in the functional derivative formula for the correlation functions. The latter may involve both real and complex eigenvalues. In the case that it involves only real eigenvalues (to be denoted ρr(n) ) the functional differentiation formula reads  δn 1  ZN [u, 1] . (15.217) ρr(n) (x1 , . . . , xn ) = ZN [1, 1] δu(x1 ) · · · δu(xn ) u=1 To compute (15.217), suppose furthermore that the polynomials {pl−1 (x)}l=1,2,... in Proposition 15.10.3 have been chosen so that the matrix [(αj,k + βj,k )|u=v=1 ]j,k=1,...,2N evaluates to the block diagonal structure of (6.2). This is equivalent to supposing that {pl−1 (x)}l=1,2,... are skew orthogonal with respect to the skew product implied by the matrix elements. Now, the only term in the matrix elements dependent on u is αj,k , 2 and this is proportional to the β = 1 inner product (6.61) with e−V (x) = e−x /2 (Gaussian case). As a consequence the functional differentiation formula (15.217) must give a formula for ρr(n) which is structurally identical to that implied by Propositions 6.3.3 and 6.3.2 for ρGOE (n) . Hence [214], [89], [497]   r S (xj , xk ) I˜r (xj , xk ) r (15.218) ρ(n) (x1 , . . . , xn ) = qdet Dr (xj , xk ) S r (xk , xj ) with  e−y2 /2  Φ2k (x)p2k+1 (y) − Φ2k+1 (x)p2k (y) , uk k=0  y ∂ r 1 S (x, y), I˜r (x, y) = sgn(y − x) − S r (x, z) dz. Dr (x, y) = ∂x 2 x N/2−1

S r (x, y) =

In (15.219) uk−1 := (α2k−1,2k + β2k−1,2k )|u=v=1 ,

 Φk (x) =

∞

−∞

sgn(x − y)pk (y)e−y

2

/2

(15.219) (15.220)

dy.

The crucial difference between ρr(n) and ρGOE (n) is the explicit form of the skew orthogonal polynomials,

THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA

757

which in the former case are given by the following result. P ROPOSITION 15.10.4 Consider the matrix X := [(αj,k + βj,k )|u=v=1 ]j,k=1,...,2N . The polynomials {pl (x)}l=0,... with the property that X evaluates to the block diagonal structure (6.2) are specified by p2j+1 (x) = x2j+1 − 2jx2j−1 ,

(15.221)

√ (α2k−1,2k + β2k−1,2k )|u=v=1 = 2 2πΓ(2k − 1).

(15.222)

p2j (x) = x2j , and furthermore

Proof. For general {pj }j=0,1,... set

pj , pk  := αj,k |u=1 + βj,k |v=1 .

For any even p2j and odd p2j+1 , by changing variables x → −x, y → −y in the definition of αj,k , and changing variables θ → π − θ in the definition of βj,k it is easy to see that p2j , p2k  = p2j+1 , p2k+1  = 0. It remains to verify that with pj as stated in (15.221), p2j−1 , p2k  = 0 for j = k, and that for j = k the normalization (15.222) results. This is an immediate corollary of the explicit formula j −2j+k+3/2 j!Γ(k + 1/2), j ≥ k, (15.223) x2j+1 , x2k  = 0, j < k. To derive (15.223), use can be made of recurrences satisfied by the corresponding integrals [215]. However the details are lengthy, and so will not be given in full. Briefly, using integration by parts one finds the recurrences “ 3” , α2j+4,2k+1 |u=1 = (2j + 2)α2j+2,2k+1 |u=1 − 2Γ j + k + 2 “ ” 3 α2j+2,2k+3 |u=1 = (2k + 1)α2j+2,2k+1 |u=1 + 2Γ j + k + , 2 ” “ √ 3 − 2 2π(j + k + 1)!δj+1,k , β2j+4,2k+1 |v=1 = (2j + 2)β2j+2,2k+1 |v=1 + 2Γ j + k + 2 “ √ 3” β2j+2,2k+3 |v=1 = (2k + 1)β2j+2,2k+1 |v=1 − 2Γ j + k + + 2 2π(j + k + 1)!δj+1,k . 2 With γj,k = xj , xk  := αj,k |u=1 + βj,k |v=1 it follows that √ γ2j+4,2k+1 = (2j + 2)γ2j+2,2k+1 − 2 2π(j + k + 1)!δj+1,k (j ≥ 0, k ≥ 1), √ γ2j+2,2k+3 = (2k + 1)γ2j+2,2k+1 + 2 2π(j + k + 1)!δj+1,k (j ≥ 0, k ≥ 0). √ With initial condition γ2,1 = −2 2π, these recurrences can be verified to have the solution (15.223).

Substituting the result of Proposition 15.10.4 in (15.219) shows 2 2 Γ(N − 2; xy)  e−(x +y )/2  1 √ N −1 x2 /2 γ(N/2 − 1/2; x2 /2) √ ( 2y) + exy , (15.224) e sgn(x) S r (x, y) = 2 Γ(N − 1) Γ(N − 1) 2π where use has been made of (15.50) and (15.55). According to (15.218), setting x = y in this gives the density √ of the real eigenvalues. Plots show that to leading order the support of the density is in the region |x| < N and indicate the boundary layer to be O(1). This suggests that for N → ∞ the correlations are well defined in the neighborhood of this edge. Indeed use of (15.51) shows 2  √ √ X + Y  e−Y 1  1 −(X−Y )2 /2  r lim S ( N + X, N + Y ) = √ e 1 − erf √ + √ (1 + erf X) . (15.225) N →∞ 2π 2 2 2 2

758

CHAPTER 15

For X, Y → −∞ (15.225) reveals the bulk limiting form 2 1 lim S r (X, Y ) = √ e−(X−Y ) /2 , 2π

(15.226)

N →∞

√ which with X = Y implies that the density of real eigenvalues is the constant 1/ 2π. One consequence ∞ of this is that the expected number of real eigenvalues, EN := −∞ ρr (x) dx, has the large N behavior  EN ∼ 2N/π. In fact direct integration of (15.224) in the case x = y leads to the evaluation [156]  2 Γ(N + 1/2) 1 (15.227) EN = + 2 F1 (1, −1/2; N ; 1/2), 2 π Γ(N ) and from this the asymptotic expansion of EN can be systematically generated. Application of Proposition 15.10.4 to the calculation of the correlation between complex eigenvalues is given in Exercises 15.10 q.3. Although not considered here, the mixed correlation involving both real and complex eigenvalues can be computed [89] by working similar to that in the proof of Proposition 6.7.2. Consider the probability Pr(Rs ), say, that either there is no real eigenvalue, or all real eigenvalues are less than s. This is given in terms of the generalized partition function (15.214) according to Pr(Rs ) = ZN [χx∈(−∞,s) , 1].

(15.228)

With f r denoting the 2 × 2 matrix integral operator on (−∞, ∞) with kernel equal to the block matrix exhibited by (15.218), and χ(−∞,s) = diag[χ(−∞,s) , χ(−∞,s) ], the reasoning which led to (9.18) implies  2 Pr(Rs ) = det[12 − f r χ(−∞,s) ]. However, from the perspective √ of numerical computation it is easier to work directly with (15.228). A case of particular interest is s = N , which corresponds to the probability that either no eigenvalue is real, or all real eigenvalues are inside the leading order support. The task of computing such probabilities first arose in the context of stability analysis of biological webs [374], [391] (see also Exercises 15.10 q.4). E XERCISES 15.10

1. [170] A polynomial of the form f (z) = z N + a1 z N−1 + · · · + aN ,

aN = eiφ

aN−j = aN a ¯j ,

is said to be self-reciprocal. Such polynomials have a nonzero probability of having zeros on the unit circle, while zeros off the unit circle occur in pairs ρj eiθj , (1/ρj )eiθj . Suppose N is odd, and let there be L zeros {αj = eiδj }j=1,...,L on the unit circle (L odd) and 2M zeros {βj = ρj eiθj , 1/β¯j = (1/ρj )eiθj }j=1,...,M off the unit circle. The Jacobian J for the change of variables from Re a1 , Im a1 , . . . Re a(N−1)/2 , Im a(N−1)/2 , φ specifying the coefficients, to {δj }j=1,...,L , {ρj , θj }j=1,...,M specifying the zeros has been calculated to be equal to M ˛ ˛Y 1 ˛ ˛ Δ(β1 , 1/β¯1 , . . . , βM , 1/β¯M , α1 , . . . , αL )˛, 2M −(N−1)/2 ˛ ρ m m=1 where Δ is as in (15.208). Relate this to the Boltzmann factor for a certain one-component log-potential system is a unit disk with anti-metallic boundary conditions. 2. [155] Let XN be a real N × N matrix. Let α ± iβ, β > 0 be a pair of complex eigenvalues with corresponding y |2 = 1 and c > 0. eigenvectors  x ± ic y , where | x|2 = | (i) Show that  x· y = 0 and XN [x y ] =

[ x

 y]

»

α −βc

β/c α

– .

(15.229)

Show too that the effect of interchanging  x and  y is to replace c → −1/c in (15.229). Conclude that the decomposition (15.229) for | x|2 = | y |2 = 1, the first component of  x, y positive and  x· y = 0 is unique provided c ≥ 1.

759

THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA

(ii) Show from the result of Exercises 1.9 q.3 that there is a Householder matrix U1 such that U1  x = e1 , and y= y  , where  y  is a unit vector with first component 0. Now use the fact that  x· y = 0 to show that U1  construct a Householder transformation U2 of the form – » T 0N−1 1 , 0N−1 V where V is an N − 1 × N − 1 Householder transformation with the property U2  y  = e2 , to deduce that U2 U1 [ x  y] = [e1

e2 ].

(iii) Conclude from the results of (i) and (ii) that for Q = U2 U1 an orthogonal matrix we have – – » » Z Y2×N−2 α b1 , QT , Z := XN = Q 0N−2×2 XN−2 −b2 α √ where b1 , b2 > 0, b1 b2 = β. 3. [214], [155] In this exercise the explicit form of ρc(n) is given. (i) Let f c denote the 2 × 2 matrix integral operator on (−∞, ∞) with kernel equal to f c (w, z) and let g = diag [g(z), g(z)]. With the polynomials {pl−1 (x)}l=1,2,... chosen as in Proposition 15.10.4, and q2j−2 (z) := −p2j−1 (z), q2j−1 (z) := p2j−2 (z), use the method of Proposition 6.1.9 to show Zn [1, 1 + g] = det(12 + f c g), where, with z = x + iy, w = u + iv, and uk as in (15.219), f c (w, z) := 2iev

2

−u2

√ erfc( 2v)

»

¯ z) S c (w, S c (w, z)

−S c (w, ¯ z¯) −S c (w, z¯)

– ,

S c (w, z) :=

N X pj−1 (w)qj−1 (z) j=1

u[(j−1)/2]

.

(ii) Use the theory of the paragraph including (6.31) to deduce from (i) that ρc(n) ((x1 , y1 ), . . . , (xn , yn )) = qdet[f c ((xj , yj ), (xk , yk ))]j,k=1,...,n and thus in particular r ρc(1) ((x, y))

=

√ 2 Γ(N − 1, x2 + y 2 ) 2y 2 ye erfc( 2y). π Γ(N − 1)

(15.230)

4. Let  y(t) be an N × 1 column vector, and let B be an N × N random real Gaussian matrix in which the entries are chosen from N[0, σ]. Consider the matrix linear differential equation d y(t) y (t). = (−1N + B) dt (i) Show that for  y(t) → 0 as t → ∞, the real parts of all eigenvalues of B must be less than 1.

√ (ii) Deduce from (15.225) and (15.230) √ that for σ = 1 and N large the spectral radius is equal to N + O(1). Use this to show that for σ = / N , and with N → ∞, the condition of (i) holds provided 0 <  < 1. 5. [371], [215] Let S be an element of the GOE. Let A be an antisymmetric real Gaussian matrix with joint distribution of its elements proportional to exp(−Tr A2 /2). With 0 < τ < 1 and c := (1 − τ )/(1 + τ ) define random matrices X according to √ 1 X = √ (S + cA). (15.231) b (i) When τ = 0, b = 1 observe that X is a real Gaussian matrix with distribution (15.207), while when τ = 1, b = 1, X is a member of the GOE.

760

CHAPTER 15

(ii) Write S and A in terms of X and XT , and also show from (15.231) that √ √ 2 (dX) = 2N(N−1)/2 ( c)N(N−1)/2 ( b)−N (dS)(dA), to deduce from the joint distribution of the elements of S and A that the joint distribution of the elements of X is equal to “ “ ”” b Aτ,b exp − Tr XXT − τ Tr X2 (15.232) 2(1 − τ ) (cf. (15.15)) where √ √ 2 2 Aτ,b = ( c)−N(N−1)/2 ( b)N (2π)−N /2 . (iii) Denote (15.208) by Pk,(N−k)/2 ({λj }j=1,...,k ; {xj ± iyj }j=1,...,(N−k)/2 ), which corresponds to the eigenvalue p.d.f. in the case √ τ = 0, b = 1, conditioned so that there are exactly k real eigenvalues. For these matrices scale X → bX/(1 − τ )1/2 to obtain that for matrices with p.d.f. A0,1 bN

2

/2

(1 − τ )−N

2

/2 −bTr XXT /2(1−τ )

e

,

(15.233)

the eigenvalue p.d.f. is equal to bN/2 (1 − τ )−N/2 “ √ ” √ √ ×Pk,(N−k)/2 { bλj /(1 − τ )1/2 }j=1,...,k ; { bxj /(1 − τ )1/2 ± i byj /(1 − τ )1/2 }j=1,...,(N−k)/2 . By comparing (15.233) to (15.232) deduce that for matrices with p.d.f. (15.232) the eigenvalue p.d.f, conditioned so that there are exactly k eigenvalues, is equal to (N−k)/2 k “ τb “ X ”” X Aτ,b (1 − τ )N(N−1)/2 exp λ2j + 2 (x2j − yj2 ) A0,1 2(1 − τ ) j=1 j=1 √ √ √ 1/2 ×Pk,(N−k)/2 ({ bλj /(1 − τ ) }j=1,...,k ; { bxj /(1 − τ )1/2 ± i byj /(1 − τ )1/2 }j=1,...,(N−k)/2 ) √ √ ˛ ˛ 2(N−k)/2 ( b)N(N+1)/2 ( 1 + τ )N(N−1)/2 ˛ ˛ = ˛Δ({λl }l=1,...,k ∪ {xj ± iyj }j=1,...,(N−k)/2 )˛ QN N(N+1)/4 k!((N − k)/2)! 2 Γ(l/2) l=1 (N−k)/2 “r 2b ” P(N −k)/2 2 Pk Y 2 (yj −x2 j) ×e−b j=1 λj /2 eb j=1 erfc yj . 1−τ j=1

15.11 CLASSIFICATION OF NON-HERMITIAN RANDOM MATRICES In our studies of Hermitian random matrices in Chapters 1 and 3, a total of ten distinct ensembles were identified on the basis of symmetry constraints. Furthermore, these were shown to be identical to the Hermitian part of the ten infinite families of symmetric spaces. The task of classifying non-Hermitian random matrices according to symmetries has been undertaken in [59], [379]. D EFINITION 15.11.1 Let A be a square matrix and let p, c, q and k be unitary matrices of the same size as A and with p2 = 1,

cT c† = ±1,

q2 = 1,

¯ = ±1. kk

The matrix A is said to have a symmetry of P -type, C-type, Q-type and K-type respectively if A = −pAp† ,

A = ±cAT c† ,

A = −qA† q† ,

¯ †, A = kAk

respectively. For A to have two or more of these symmetries, the symmetries are required to commute. In addition to the ten classes of Hermitian random matrices already catalogued, in [379] consideration

761

THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA

of these symmetries gave rise to twenty types of non-Hermitian random matrices. Three of these are the real, complex and real quaternion Ginibre ensembles, corresponding to k = 1, no symmetry and k = Z2N , respectively. We will single out just three others, obtained by requiring that the Ginibre matrices anticommute with the P -type symmetry   1p Op×q Oq×p −1q (with the sizes doubled in the quaternion case). This gives the non-Hermitian matrices   Op×p Ap×q Bq×p Oq×q

(15.234)

with the elements of A := Ap×q and B := Bq×p real, complex and real quaternion. With the matrices A and B chosen according to a Gaussian measure, this class of random matrices have been applied to studies of QCD [439], [9]. Following these references, we’ll take up the problem of computing the eigenvalue p.d.f. in the case the elements of (15.234) are complex. In numerical linear algebra, for square matrices A, B the decomposition A = QTA Z† ,

B = QTB Z†

is called the QZ decomposition. Here Q, Z are unitary matrices, while TA , TB are upper triangular matrices with diagonal elements (TA )jj , (TB )jj , such that (TA )jj /(TB )jj are the eigenvalues of B−1 A. The matrices Q and Z are unique provided the entries of the first row are chosen to be positive. A variation of this is the joint decomposition of p × q, q × p (p ≥ q) matrices A, B given by [439] B = V† TB U† ,

A = UTA V,

(15.235)

where TA , TB are p × q, q × p upper triangular matrices and U, V are unitary. With the diagonal entries of TA , TB denoted {xk }, {yk }, respectively, and the nonzero eigenvalues of the product AB denoted {−zk2} so that the eigenvalues of (15.234) are {±izk }, we have −zk2 = xk yk . The matrix V is unique provided the entries of the first row are chosen to be positive, while U requires this condition, but then is only unique up to multiplication on the right by   1q Oq×(p−q) O(p−q)×q S for S a (p − q) × (p − q) unitary matrix. For A and B having all entries independent standard complex Gaussians, we want to compute the distribution of the independent eigenvalues {izk } of (15.234). The first step is to compute the Jacobian for the change of variables implied by (15.235). ˜ A, T ˜ B denote the strictly upper triangular part of TA , TB . We have P ROPOSITION 15.11.2 Let T (dA)(dB) =

q  l=1

|xl |2(p−q)



|zk2 − zj2 |2

1≤j k and j ≤ k, respectively. Furthermore, for j > q, k ≤ q only the term (TA )kk (U† dU)jk of (15.237) contributes. Recalling that (TA )jj = xj , (TB )jj = yj , xj yj = −zj2 , we see by taking the wedge product of the contributing terms that (15.236) results.

In terms of the decomposition (15.235) the joint p.d.f. for A and B is proportional to e−

PN

j=1 (|xj |

2

+|yj |2 )−

P

˜

j