An introduction to random matrices Greg W.Anderson University of Minnesota Alice Guionnet ENS Lyon Ofer Zeitouni University of Minnesota and Weizmann Institute of Science CAMBRIDGE UNIVERSITY PRESS
An Introduction to Random Matrices Greg W. Anderson University of Minnesota Alice Guionnet ENS Lyon Ofer Zeitouni University of Minnesota and Weizmann Institute of Science
Contents Preface page xiii 1 Introduction 1 2 Real and Complex Wigner matrices 6 2.1 Real Wigner matrices:traces,moments and combinatorics 6 2.1.1 The semicircle distribution,Catalan numbers,and Dyck paths 1 2.1.2 Proof#1 of Wigner's Theorem 2.1.1 10 2.1.3 Proof of Lemma 2.1.6:Words and Graphs 11 2.1.4 Proof of Lemma 2.1.7:Sentences and Graphs 17 2.1.5 Some useful approximations 21 2.1.6 Maximal eigenvalues and Furedi-Komlos enumeration 23 2.1.7 Central limit theorems for moments 29 2.2 Complex Wigner matrices 35 2.3 Concentration for functionals of random matrices and logarithmic Sobolev inequalities 38 2.3.1 Smoothness properties of linear functions of the empirical measure 38 2.3.2 Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities 39 2.3.3 Concentration for Wigner-type matrices 42 2.4 Stieltjes transforms and recursions 43 vii
Contents Preface page xiii 1 Introduction 1 2 Real and Complex Wigner matrices 6 2.1 Real Wigner matrices: traces, moments and combinatorics 6 2.1.1 The semicircle distribution, Catalan numbers, and Dyck paths 7 2.1.2 Proof #1 of Wigner’s Theorem 2.1.1 10 2.1.3 Proof of Lemma 2.1.6 : Words and Graphs 11 2.1.4 Proof of Lemma 2.1.7 : Sentences and Graphs 17 2.1.5 Some useful approximations 21 2.1.6 Maximal eigenvalues and F¨uredi-Koml´os enumeration 23 2.1.7 Central limit theorems for moments 29 2.2 Complex Wigner matrices 35 2.3 Concentration for functionals of random matrices and logarithmic Sobolev inequalities 38 2.3.1 Smoothness properties of linear functions of the empirical measure 38 2.3.2 Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities 39 2.3.3 Concentration for Wigner-type matrices 42 2.4 Stieltjes transforms and recursions 43 vii
viii CONTENTS 2.4.1 Gaussian Wigner matrices 46 2.4.2 General Wigner matrices 47 2.5 Joint distribution of eigenvalues in the GOE and the GUE 51 2.5.1 Definition and preliminary discussion of the GOE and the GUE 51 2.5.2 Proof of the joint distribution of eigenvalues 54 2.5.3 Selberg's integral formula and proof of(2.5.4) 59 2.5.4 Joint distribution of eigenvalues-alternative formu- lation 65 2.5.5 Superposition and decimation relations 66 2.6 Large deviations for random matrices 71 2.6.1 Large deviations for the empirical measure 72 2.6.2 Large deviations for the top eigenvalue 82 2.7 Bibliographical notes 86 w Hermite polynomials,spacings,and limit distributions for the Gaus- sian ensembles 91 3.1 Summary of main results:spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles 91 3.1.1 Limit results for the GUE 91 3.1.2 Generalizations:limit formulas for the GOE and GSE 94 3.2 Hermite polynomials and the GUE 95 3.2.1 The GUE and determinantal laws 95 3.2.2 Properties of the Hermite polynomials and oscillator wave-functions 100 3.3 The semicircle law revisited 103 3.3.1 Calculation of moments of Ly 103 3.3.2 The Harer-Zagier recursion and Ledoux's argument 105 3.4 Quick introduction to Fredholm determinants 108 3.4.1 The setting,fundamental estimates,and definition of the Fredholm determinant 108 3.4.2 Definition of the Fredholm adjugant,Fredholm resolvent,and a fundamental identity 111
viii CONTENTS 2.4.1 Gaussian Wigner matrices 46 2.4.2 General Wigner matrices 47 2.5 Joint distribution of eigenvalues in the GOE and the GUE 51 2.5.1 Definition and preliminary discussion of the GOE and the GUE 51 2.5.2 Proof of the joint distribution of eigenvalues 54 2.5.3 Selberg’s integral formula and proof of (2.5.4) 59 2.5.4 Joint distribution of eigenvalues - alternative formulation 65 2.5.5 Superposition and decimation relations 66 2.6 Large deviations for random matrices 71 2.6.1 Large deviations for the empirical measure 72 2.6.2 Large deviations for the top eigenvalue 82 2.7 Bibliographical notes 86 3 Hermite polynomials, spacings, and limit distributions for the Gaussian ensembles 91 3.1 Summary of main results: spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles 91 3.1.1 Limit results for the GUE 91 3.1.2 Generalizations: limit formulas for the GOE and GSE 94 3.2 Hermite polynomials and the GUE 95 3.2.1 The GUE and determinantal laws 95 3.2.2 Properties of the Hermite polynomials and oscillator wave-functions 100 3.3 The semicircle law revisited 103 3.3.1 Calculation of moments of L¯N 103 3.3.2 The Harer–Zagier recursion and Ledoux’s argument 105 3.4 Quick introduction to Fredholm determinants 108 3.4.1 The setting, fundamental estimates, and definition of the Fredholm determinant 108 3.4.2 Definition of the Fredholm adjugant, Fredholm resolvent, and a fundamental identity 111
CONTENTS 车 3.5 Gap probabilities at 0 and proof of Theorem 3.1.1. 116 3.5.1 The method of Laplace 117 3.5.2 Evaluation of the scaling limit-proof of Lemma 3.5.1 119 3.5.3 A complement:determinantal relations 122 3.6 Analysis of the sine-kernel 123 3.6.1 General differentiation formulas 123 3.6.2 Derivation of the differential equations:proof of Theorem 3.6.1 128 3.6.3 Reduction to Painleve V 130 3.7 Edge-scaling:Proof of Theorem 3.1.4 134 3.7.1 Vague convergence of the rescaled largest eigen- value:proofof Theorem 3.1.4 135 3.7.2 Steepest descent:proof of Lemma 3.7.2 136 3.7.3 Properties of the Airy functions and proofof Lemma 3.7.1 141 3.8 Analysis of the Tracy-Widom distribution and proof of Theorem 3.1.5 144 3.8.1 The first standard moves of the game 146 3.8.2 The wrinkle in the carpet 147 3.8.3 Linkage to Painleve II 148 3.9 Limiting behavior of the GOE and the GSE 150 3.9.1 Pfaffians and gap probabilities 150 3.9.2 Fredholm representation of gap probabilities 158 3.9.3 Limit calculations 163 3.9.4 Differential equations 172 3.10 Bibliographical notes 183 4 Some generalities 188 4.1 Joint distribution of eigenvalues in the classical matrix ensembles 189 4.1.1 Integration formulas for classical ensembles 189 4.1.2 Manifolds,volume measures,and the coarea formula 195
CONTENTS ix 3.5 Gap probabilities at 0 and proof of Theorem 3.1.1. 116 3.5.1 The method of Laplace 117 3.5.2 Evaluation of the scaling limit – proof of Lemma 3.5.1 119 3.5.3 A complement: determinantal relations 122 3.6 Analysis of the sine-kernel 123 3.6.1 General differentiation formulas 123 3.6.2 Derivation of the differential equations: proof of Theorem 3.6.1 128 3.6.3 Reduction to Painlev´e V 130 3.7 Edge-scaling: Proof of Theorem 3.1.4 134 3.7.1 Vague convergence of the rescaled largest eigenvalue: proof of Theorem 3.1.4 135 3.7.2 Steepest descent: proof of Lemma 3.7.2 136 3.7.3 Properties of the Airy functions and proof of Lemma 3.7.1 141 3.8 Analysis of the Tracy-Widom distribution and proof of Theorem 3.1.5 144 3.8.1 The first standard moves of the game 146 3.8.2 The wrinkle in the carpet 147 3.8.3 Linkage to Painlev´e II 148 3.9 Limiting behavior of the GOE and the GSE 150 3.9.1 Pfaffians and gap probabilities 150 3.9.2 Fredholm representation of gap probabilities 158 3.9.3 Limit calculations 163 3.9.4 Differential equations 172 3.10 Bibliographical notes 183 4 Some generalities 188 4.1 Joint distribution of eigenvalues in the classical matrix ensembles 189 4.1.1 Integration formulas for classical ensembles 189 4.1.2 Manifolds, volume measures, and the coarea formula 195
CONTENTS 4.1.3 An integration formula of Weyl type 201 4.1.4 Applications of Weyl's formula 208 4.2 Determinantal point processes 217 4.2.1 Point processes-basic definitions 217 4.2.2 Determinantal processes 222 4.2.3 Determinantal projections 225 4.2.4 The CLT for determinantal processes 229 4.2.5 Determinantal processes associated with eigenvalues 230 4.2.6 Translation invariant determinantal processes 234 4.2.7 One dimensional translation invariant determinantal processes 239 4.2.8 Convergence issues 243 4.2.9 Examples 245 4.3 Stochastic analysis for random matrices 250 4.3.1 Dyson's Brownian motion 251 4.3.2 A dynamical version of Wigner's Theorem 264 4.3.3 Dynamical central limit theorems 275 4.3.4 Large deviations bounds 279 4.4 Concentration of measure and random matrices 284 4.4.1 Concentration inequalities for Hermitian matrices with independent entries 284 4.4.2 Concentration inequalities for matrices with non independent entries 289 4.5 Tridiagonal matrix models and the B ensembles 305 4.5.1 Tridiagonal representation of B ensembles 305 4.5.2 Scaling limits at the edge of the spectrum 309 4.6 Bibliographical notes 320 5 Free probability 325 5.1 Introduction and main results 326 5.2 Noncommutative laws and noncommutative probability spaces 328
x CONTENTS 4.1.3 An integration formula of Weyl type 201 4.1.4 Applications of Weyl’s formula 208 4.2 Determinantal point processes 217 4.2.1 Point processes – basic definitions 217 4.2.2 Determinantal processes 222 4.2.3 Determinantal projections 225 4.2.4 The CLT for determinantal processes 229 4.2.5 Determinantal processes associated with eigenvalues 230 4.2.6 Translation invariant determinantal processes 234 4.2.7 One dimensional translation invariant determinantal processes 239 4.2.8 Convergence issues 243 4.2.9 Examples 245 4.3 Stochastic analysis for random matrices 250 4.3.1 Dyson’s Brownian motion 251 4.3.2 A dynamical version of Wigner’s Theorem 264 4.3.3 Dynamical central limit theorems 275 4.3.4 Large deviations bounds 279 4.4 Concentration of measure and random matrices 284 4.4.1 Concentration inequalities for Hermitian matrices with independent entries 284 4.4.2 Concentration inequalities for matrices with non independent entries 289 4.5 Tridiagonal matrix models and the β ensembles 305 4.5.1 Tridiagonal representation of β ensembles 305 4.5.2 Scaling limits at the edge of the spectrum 309 4.6 Bibliographical notes 320 5 Free probability 325 5.1 Introduction and main results 326 5.2 Noncommutative laws and noncommutative probability spaces 328