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.
Author(s): Greg W. Anderson, Alice Guionnet, Ofer Zeitouni
Series: Cambridge Studies in Advanced Mathematics 118
Publisher: CUP
Year: 2009
Language: English
Pages: 508
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 15
1 Introduction......Page 17
2.1 Real Wigner matrices: traces, moments and combinatorics......Page 22
2.1.1 The semicircle distribution, Catalan numbers and Dyck paths......Page 23
2.1.2 Proof #1 of Wigner’s Theorem 2.1.1......Page 26
2.1.3 Proof of Lemma 2.1.6: words and graphs......Page 27
2.1.4 Proof of Lemma 2.1.7: sentences and graphs......Page 33
2.1.5 Some useful approximations......Page 37
2.1.6 Maximal eigenvalues and F¨uredi–Koml´os enumeration......Page 39
2.1.7 Central limit theorems for moments......Page 45
2.2 Complex Wigner matrices......Page 51
2.3.1 Smoothness properties of linear functions of the empirical measure......Page 54
2.3.2 Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities......Page 55
2.3.3 Concentration for Wigner-type matrices......Page 58
2.4 Stieltjes transforms and recursions......Page 59
2.4.1 Gaussian Wigner matrices......Page 61
2.4.2 General Wigner matrices......Page 63
2.5 Joint distribution of eigenvalues in the GOE and the GUE......Page 66
2.5.1 Definition and preliminary discussion of the GOE and the GUE......Page 67
2.5.2 Proof of the joint distribution of eigenvalues......Page 70
2.5.3 Selberg’s integral formula and proof of (2.5.4)......Page 74
2.5.4 Joint distribution of eigenvalues: alternative formulation......Page 81
2.5.5 Superposition and decimation relations......Page 82
2.6 Large deviations for random matrices......Page 86
2.6.1 Large deviations for the empirical measure......Page 87
A large deviation upper bound......Page 93
A large deviation lower bound......Page 94
2.6.2 Large deviations for the top eigenvalue......Page 97
2.7 Bibliographical notes......Page 101
3.1.1 Limit results for the GUE......Page 106
3.1.2 Generalizations: limit formulas for the GOE and GSE......Page 109
3.2.1 The GUE and determinantal laws......Page 110
3.2.2 Properties of the Hermite polynomials and oscillator wave-functions......Page 115
3.3 The semicircle law revisited......Page 117
3.3.1 Calculation of moments of LN......Page 118
3.3.2 The Harer–Zagier recursion and Ledoux’s argument......Page 119
3.4.1 The setting, fundamental estimates and definition of the Fredholm determinant......Page 123
3.4.2 Definition of the Fredholm adjugant, Fredholm resolvent and a fundamental identity......Page 126
Multiplicativity of Fredholm determinants......Page 129
3.5 Gap probabilities at 0 and proof of Theorem 3.1.1......Page 130
3.5.1 The method of Laplace......Page 131
3.5.2 Evaluation of the scaling limit: proof of Lemma 3.5.1......Page 133
3.5.3 A complement: determinantal relations......Page 136
3.6.1 General differentiation formulas......Page 137
3.6.2 Derivation of the differential equations: proof of Theorem 3.6.1......Page 142
3.6.3 Reduction to Painlev´e V......Page 144
3.7 Edge-scaling: proof of Theorem 3.1.4......Page 148
3.7.1 Vague convergence of the largest eigenvalue: proof of Theorem 3.1.4......Page 149
3.7.2 Steepest descent: proof of Lemma 3.7.2......Page 150
3.7.3 Properties of the Airy functions and proof of Lemma 3.7.1......Page 155
3.8 Analysis of the Tracy–Widom distribution and proof of Theorem 3.1.5......Page 158
3.8.2 The wrinkle in the carpet......Page 160
3.8.3 Linkage to Painlev´e II......Page 162
Pfaffian integration formulas......Page 164
Determinant formulas for squared gap probabilities......Page 168
Matrix kernels and a revision of the Fredholm setup......Page 171
Main results......Page 174
3.9.3 Limit calculations......Page 176
Statements of main results......Page 177
Proofs of bulk results......Page 179
Block matrix calculations......Page 186
Proof of Theorem 3.1.6......Page 188
Proof of Theorem 3.1.7......Page 192
3.10 Bibliographical notes......Page 197
4 Some generalities......Page 202
4.1.1 Integration formulas for classical ensembles......Page 203
The Gaussian ensembles......Page 204
Laguerre ensembles and Wishart matrices......Page 205
Jacobi ensembles and random projectors......Page 206
The classical compact Lie groups......Page 207
4.1.2 Manifolds, volume measures and the coarea formula......Page 209
4.1.3 An integration formula of Weyl type......Page 215
4.1.4 Applications of Weyl’s formula......Page 222
4.2 Determinantal point processes......Page 230
4.2.1 Point processes: basic definitions......Page 231
4.2.2 Determinantal processes......Page 236
4.2.3 Determinantal projections......Page 238
4.2.4 The CLT for determinantal processes......Page 243
4.2.5 Determinantal processes associated with eigenvalues......Page 244
The Airy process......Page 246
4.2.6 Translation invariant determinantal processes......Page 248
4.2.7 One-dimensional translation invariant determinantal processes......Page 253
4.2.8 Convergence issues......Page 257
4.2.9 Examples......Page 259
The biorthogonal ensembles......Page 260
Birth–death processes conditioned not to intersect......Page 261
4.3 Stochastic analysis for random matrices......Page 264
4.3.1 Dyson’s Brownian motion......Page 265
4.3.2 A dynamical version of Wigner’s Theorem......Page 278
4.3.3 Dynamical central limit theorems......Page 289
4.3.4 Large deviation bounds......Page 293
4.4 Concentration of measure and random matrices......Page 297
4.4.1 Concentration inequalities for Hermitian matrices with independent entries......Page 298
Entries satisfying Poincare’s
inequality......Page 299
Matrices with bounded entries and Talagrand’s method......Page 301
The setup with M = Rm and µ=Lebesgue measure......Page 303
The setup with M a compact Riemannian manifold......Page 311
Applications to random matrices......Page 314
4.5 Tridiagonal matrix models and the β ensembles......Page 318
4.5.1 Tridiagonal representation of β ensembles......Page 319
4.5.2 Scaling limits at the edge of the spectrum......Page 322
4.6 Bibliographical notes......Page 334
5 Free probability......Page 338
5.1 Introduction and main results......Page 339
5.2.1 Algebraic noncommutative probability spaces and laws......Page 341
5.2.2 C-probability spaces and the weak*-topology......Page 345
C-probability spaces......Page 347
Weak*-topology......Page 352
5.2.3 W-probability spaces......Page 355
Laws of self-adjoint operators......Page 359
5.3.1 Independence and free independence......Page 364
Basic properties of non-crossing partitions......Page 370
Free cumulants and freeness......Page 371
5.3.3 Consequence of free independence: free convolution......Page 375
Multiplicative free convolution......Page 381
5.3.4 Free central limit theorem......Page 384
5.3.5 Freeness for unbounded variables......Page 385
5.4 Link with random matrices......Page 390
5.5 Convergence of the operator norm of polynomials of independent GUE matrices......Page 410
5.6 Bibliographical notes......Page 426
A.1 Identities and bounds......Page 430
A.2 Perturbations for normal and Hermitian matrices......Page 431
A.3 Noncommutative matrix Lp-norms......Page 432
A.4 Brief review of resultants and discriminants......Page 433
B.1 Generalities......Page 434
B.2 Topological vector spaces and weak topologies......Page 436
B.3 Banach and Polish spaces......Page 438
C.1 Generalities......Page 439
C.2 Weak topology......Page 441
D Basic notions of large deviations......Page 443
E The skew field H of quaternions and matrix theory over F......Page 446
E.1 Matrix terminology over F and factorization theorems......Page 447
E.2 The spectral theorem and key corollaries......Page 449
E.3 A specialized result on projectors......Page 450
E.4 Algebra for curvature computations......Page 451
F Manifolds......Page 453
F.1 Manifolds embedded in Euclidean space......Page 454
“Critical” vocabulary......Page 456
Lie groups and Haar measure......Page 457
F.2 Proof of the coarea formula......Page 458
F.3 Metrics, connections, curvature, Hessians, and the Laplace–Beltrami operator......Page 461
Curvature of classical compact Lie groups......Page 465
G.1 Basic definitions......Page 466
G.2 Spectral properties......Page 468
G.3 States and positivity......Page 470
G.4 von Neumann algebras......Page 471
G.5 Noncommutative functional calculus......Page 473
H Stochastic calculus notions......Page 475
References......Page 481
General conventions and notation......Page 497
Index......Page 500
