An Introduction to Random Matrices

1 An Introduction to Random Matrices 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 copyright information here To Meredith, Ben and Naomi Contents Preface page xiii 1 Introduction 1. 2 Real and Complex Wigner Matrices 6. Real Wigner Matrices : traces, moments and combinatorics 6. The semicircle distribution, Catalan numbers, and Dyck paths 7. Proof #1 of Wigner's Theorem 10. Proof of Lemma : Words and Graphs 11. Proof of Lemma : Sentences and Graphs 17. Some useful approximations 21. Maximal eigenvalues and Fu redi-Komlo s enumeration 23. Central limit theorems for moments 29.

2 Complex Wigner Matrices 35. Concentration for functionals of Random Matrices and logarithmic Sobolev inequalities 38. Smoothness properties of linear functions of the empirical measure 38. Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities 39. Concentration for Wigner-type Matrices 42. Stieltjes transforms and recursions 43. vii viii C ONTENTS. Gaussian Wigner Matrices 46. General Wigner Matrices 47. Joint distribution of eigenvalues in the GOE and the GUE 51. Definition and preliminary discussion of the GOE. and the GUE 51. Proof of the joint distribution of eigenvalues 54. Selberg's integral formula and proof of ( ) 59.

3 Joint distribution of eigenvalues - alternative formu- lation 65. Superposition and decimation relations 66. Large deviations for Random Matrices 71. Large deviations for the empirical measure 72. Large deviations for the top eigenvalue 82. Bibliographical notes 86. 3 Hermite polynomials, spacings, and limit distributions for the Gaus- sian ensembles 91. Summary of main results: spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles 91. Limit results for the GUE 91. Generalizations: limit formulas for the GOE and GSE 94. Hermite polynomials and the GUE 95. The GUE and determinantal laws 95. Properties of the Hermite polynomials and oscillator wave-functions 100.

4 The semicircle law revisited 103. Calculation of moments of L N 103. The Harer Zagier recursion and Ledoux's argument 105. Quick Introduction to Fredholm determinants 108. The setting, fundamental estimates, and definition of the Fredholm determinant 108. Definition of the Fredholm adjugant, Fredholm resolvent, and a fundamental identity 111. C ONTENTS ix Gap probabilities at 0 and proof of Theorem 116. The method of Laplace 117. Evaluation of the scaling limit proof of Lemma 119. A complement: determinantal relations 122. Analysis of the sine-kernel 123. General differentiation formulas 123. Derivation of the differential equations: proof of Theorem 128. Reduction to Painleve V 130.

5 Edge-scaling: Proof of Theorem 134. Vague convergence of the rescaled largest eigen- value: proof of Theorem 135. Steepest descent: proof of Lemma 136. Properties of the Airy functions and proof of Lemma 141. Analysis of the Tracy-Widom distribution and proof of Theorem 144. The first standard moves of the game 146. The wrinkle in the carpet 147. Linkage to Painleve II 148. Limiting behavior of the GOE and the GSE 150. Pfaffians and gap probabilities 150. Fredholm representation of gap probabilities 158. Limit calculations 163. Differential equations 172. Bibliographical notes 183. 4 Some generalities 188. Joint distribution of eigenvalues in the classical matrix ensembles 189.

6 Integration formulas for classical ensembles 189. Manifolds, volume measures, and the coarea formula 195. x C ONTENTS. An integration formula of Weyl type 201. Applications of Weyl's formula 208. Determinantal point processes 217. Point processes basic definitions 217. Determinantal processes 222. Determinantal projections 225. The CLT for determinantal processes 229. Determinantal processes associated with eigenvalues 230. Translation invariant determinantal processes 234. One dimensional translation invariant determinantal processes 239. Convergence issues 243. Examples 245. Stochastic analysis for Random Matrices 250. Dyson's Brownian motion 251. A dynamical version of Wigner's Theorem 264.

7 Dynamical central limit theorems 275. Large deviations bounds 279. Concentration of measure and Random Matrices 284. Concentration inequalities for Hermitian Matrices with independent entries 284. Concentration inequalities for Matrices with non independent entries 289. Tridiagonal matrix models and the ensembles 305. Tridiagonal representation of ensembles 305. Scaling limits at the edge of the spectrum 309. Bibliographical notes 320. 5 Free probability 325. Introduction and main results 326. Noncommutative laws and noncommutative probability spaces 328. C ONTENTS xi Algebraic noncommutative probability spaces and laws 328. C - probability spaces and the weak-* topology 332.

8 W - probability spaces 341. Free independence 351. Independence and free independence 351. Free independence and combinatorics 356. Consequence of free independence: free convolution 362. Free central limit theorem 371. Freeness for unbounded variables 372. Link with Random Matrices 377. Convergence of the operator norm of polynomials of inde- pendent GUE Matrices 396. Bibliographical Notes 412. Appendices 417. A Linear algebra preliminaries 417. Identities and bounds 417. Perturbations for normal and Hermitian Matrices 418. Noncommutative Matrix L p -norms 419. Brief review of resultants and discriminants 420. B Topological Preliminaries 421. Generalities 421.

9 Topological Vector Spaces and Weak Topologies 424. Banach and Polish Spaces 425. Some elements of analysis 426. C Probability measures on Polish spaces 427. Generalities 427. Weak Topology 429. D Basic notions of large deviations 431. E The skew field H of quaternions, and matrix theory over F 434. Matrix terminology over F, and factorization theorems 435. xii C ONTENTS. The spectral theorem and key corollaries 437. A specialized result on projectors 438. Algebra for curvature computations 439. F Manifolds 441. Manifolds embedded in Euclidean space 442. Proof of the coarea formula 446. Metrics, connections, curvature, hessians, and the Laplace-Beltrami operator 449.

10 G Appendix on Operator Algebras 454. Basic definitions 454. Spectral properties 456. States and positivity 458. von Neumann algebras 459. Noncommutative functional calculus 461. H Stochastic calculus notions 463. References 468. General Conventions 484. Preface The study of Random Matrices , and in particular the properties of their eigenval- ues, has emerged from the applications, first in data analysis and later as statisti- cal models for heavy nuclei atoms. Thus, the field of Random Matrices owes its existence to applications. Over the years, however, it became clear that models related to Random Matrices play an important role in areas of pure mathematics.