An Introduction to Random Matrices

, by Greg W. Anderson , Alice Guionnet , Ofer Zeitouni

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ISBN: 9780521194525 | 0521194520
Cover: Hardcover
Copyright: 12/21/2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

Preface	p. xiii
Introduction	p. 1
Real and complex Wigner matrices	p. 6
Real Wigner matrices: traces, moments and combinatorics	p. 6
The semicircle distribution, Catalan numbers and Dyck paths	p. 7
Proof #1 of Wigner's Theorem 2.1.1	p. 10
Proof of Lemma 2.1.6: words and graphs	p. 11
Proof of Lemma 2.1.7: sentences and graphs	p. 17
Some useful approximations	p. 21
Maximal eigenvalues and Füredi-Komlós enumeration	p. 23
Central limit theorems for moments	p. 29
Complex Wigner matrices	p. 35
Concentration for functionals of random matrices and logarithmic Sobolev inequalities	p. 38
Smoothness properties of linear functions of the empirical measure	p. 38
Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities	p. 39
Concentration for Wigner-type matrices	p. 42
Stieltjes transforms and recursions	p. 43
Gaussian Wigner matrices	p. 45
General Wigner matrices	p. 47
Joint distribution of eigenvalues in the GOE and the GUE	p. 50
Definition and preliminary discussion of the GOE and the GUE	p. 51
Proof of the joint distribution of eigenvalues	p. 54
Selberg's integral formula and proof of (2.5.4)	p. 58
Joint distribution of eigenvalues: alternative formulation	p. 65
Superposition and decimation relations	p. 66
Large deviations for random matrices	p. 70
Large deviations for the empirical measure	p. 71
Large deviations for the top eigenvalue	p. 81
Bibliographical notes	p. 85
Hermite polynomials, spacings and limit distributions for the Gaussian ensembles	p. 90
Summary of main results: spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles	p. 90
Limit results for the GUE	p. 90
Generalizations: limit formulas for the GOE and GSE	p. 93
Hermite polynomials and the GUE	p. 94
The GUE and determinantal laws	p. 94
Properties of the Hermite polynomials and oscillator wave-functions	p. 99
The semicircle law revisited	p. 101
Calculation of moments of L_N	p. 102
The Harer-Zagier recursion and Ledoux's argument	p. 103
Quick introduction to Fredholm determinants	p. 107
The setting, fundamental estimates and definition of the Fredholm determinant	p. 107
Definition of the Fredholm adjugant, Fredholm resolvent and a fundamental identity	p. 110
Gap probabilities at 0 and proof of Theorem 3.1.1	p. 114
The method of Laplace	p. 115
Evaluation of the scaling limit: proof of Lemma 3.5.1	p. 117
A complement: determinantal relations	p. 120
Analysis of the sine-kernel	p. 121
General differentiation formulas	p. 121
Derivation of the differential equations: proof of Theorem 3.6.1	p. 126
Reduction to Painlevé V	p. 128
Edge-scaling: proof of Theorem 3.1.4	p. 132
Vague convergence of the largest eigenvalue: proof of Theorem 3.1.4	p. 133
Steepest descent: proof of Lemma 3.7.2	p. 134
Properties of the Airy functions and proof of Lemma 3.7.1	p. 139
Analysis of the Tracy-Widom distribution and proof of Theorem 3.1.5	p. 142
The first standard moves of the game	p. 144
The wrinkle in the carpet	p. 144
Linkage to Painlevé II	p. 146
Limiting behavior of the GOE and the GSE	p. 148
Pfaffians and gap probabilities	p. 148
Fredholm representation of gap probabilities	p. 155
Limit calculations	p. 160
Differential equations	p. 170
Bibliographical notes	p. 181
Some generalities	p. 186
Joint distribution of eigenvalues in the classical matrix ensembles	p. 187
Integration formulas for classical ensembles	p. 187
Manifolds, volume measures and the coarea formula	p. 193
An integration formula of Weyl type	p. 199
Applications of Weyl's formula	p. 206
Determinantal point processes	p. 214
Point processes: basic definitions	p. 215
Determinantal processes	p. 220
Determinantal projections	p. 222
The CLT for determinantal processes	p. 227
Determinantal processes associated with eigenvalues	p. 228
Translation invariant determinantal processes	p. 232
One-dimensional translation invariant determinantal processes	p. 237
Convergence issues	p. 241
Examples	p. 243
Stochastic analysis for random matrices	p. 248
Dyson's Brownian motion	p. 249
A dynamical version of Wigner's Theorem	p. 262
Dynamical central limit theorems	p. 273
Large deviation bounds	p. 277
Concentration of measure and random matrices	p. 281
Concentration inequalities for Hermitian matrices with independent entries	p. 282
Concentration inequalities for matrices with dependent entries	p. 287
Tridiagonal matrix models and the ß ensembles	p. 302
Tridiagonal representation of ß ensembles	p. 303
Scaling limits at the edge of the spectrum	p. 306
Bibliographical notes	p. 318
Free probability	p. 322
Introduction and main results	p. 323
Noncommutative laws and noncommutative probability spaces	p. 325
Algebraic noncommutative probability spaces and laws	p. 325
C-probability spaces and the weak-topology	p. 329
W*-probability spaces	p. 339
Free independence	p. 348
Independence and free independence	p. 348
Free independence and combinatorics	p. 354
Consequence of free independence: free convolution	p. 359
Free central limit theorem	p. 368
Freeness for unbounded variables	p. 369
Link with random matrices	p. 374
Convergence of the operator norm of polynomials of independent GUE matrices	p. 394
Bibliographical notes	p. 410
Appendices	p. 414
Linear algebra preliminaries	p. 414
Identities and bounds	p. 414
Perturbations for normal and Hermitian matrices	p. 415
Noncommutative matrix L^p-norms	p. 416
Brief review of resultants and discriminants	p. 417
Topological preliminaries	p. 418
Generalities	p. 418
Topological vector spaces and weak topologies	p. 420
Banach and Polish spaces	p. 422
Some elements of analysis	p. 423
Probability measures on Polish spaces	p. 423
Generalities	p. 423
Weak topology	p. 425
Basic notions of large deviations	p. 427
The skew field H of quaternions and matrix theory over F	p. 430
Matrix terminology over F and factorization theorems	p. 431
The spectral theorem and key corollaries	p. 433
A specialized result on projectors	p. 434
Algebra for curvature computations	p. 435
Manifolds	p. 437
Manifolds embedded in Euclidean space	p. 438
Proof of the coarea formula	p. 442
Metrics, connections, curvature, Hessians, and the Laplace-Beltrami operator	p. 445
Appendix on operator algebras	p. 450
Basic definitions	p. 450
Spectral properties	p. 452
States and positivity	p. 454
von Neumann algebras	p. 455
Noncommutative functional calculus	p. 457
Stochastic calculus notions	p. 459
References	p. 465
General conventions and notation	p. 481
Index	p. 484
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An Introduction to Random Matrices

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