Random matrices: high dimensional phenomena

✍ Scribed by Gordon Blower

• Publisher: Cambridge University Press
• Year: 2009
• Tongue: English
• Leaves: 449
• Series: London Mathematical Society Lecture Note Series 367
• Edition: 1
• Category: Library

✦ Synopsis

This book focuses on the behavior of large random matrices. Standard results are covered, and the presentation emphasizes elementary operator theory and differential equations, so as to be accessible to graduate students and other non-experts. The introductory chapters review material on Lie groups and probability measures in a style suitable for applications in random matrix theory. Later chapters use modern convexity theory to establish subtle results about the convergence of eigenvalue distributions as the size of the matrices increases. Random matrices are viewed as geometrical objects with large dimension. The book analyzes the concentration of measure phenomenon, which describes how measures behave on geometrical objects with large dimension. To prove such results for random matrices, the book develops the modern theory of optimal transportation and proves the associated functional inequalities involving entropy and information. These include the logarithmic Sobolev inequality, which measures how fast some physical systems converge to equilibrium.

✦ Table of Contents

Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Introduction......Page 13
1.1 Weak convergence on compact metric spaces......Page 16
1.2 Invariant measure on a compact metric group......Page 22
1.3 Measures on non-compact Polish spaces......Page 28
1.4 The Brunn-Minkowski inequality......Page 34
1.5 Gaussian measures......Page 37
1.6 Surface area measure on the spheres......Page 39
1.7 Lipschitz functions and the Hausdorff metric......Page 43
1.8 Characteristic functions and Cauchy transforms......Page 45
2.1 The classical groups, their eigenvalues and norms......Page 54
2.2 Determinants and functional calculus......Page 61
2.3 Linear Lie groups......Page 68
2.4 Connections and curvature......Page 75
2.5 Generalized ensembles......Page 78
2.6 The Weyl integration formula......Page 84
2.7 Dyson's circular ensembles......Page 90
2.8 Circular orthogonal ensemble......Page 93
2.9 Circular symplectic ensemble......Page 95
3.1 Relative entropy......Page 96
3.2 Concentration of measure......Page 105
3.3 Transportation......Page 111
3.4 Transportation inequalities......Page 115
3.5 Transportation inequalities for uniformlyconvex potentials......Page 118
3.6 Concentration of measure in matrix ensembles......Page 121
3.7 Concentration for rectangular Gaussian matrices......Page 126
3.8 Concentration on the sphere......Page 135
3.9 Concentration for compact Lie groups......Page 138
4.1 Logarithmic energy and equilibrium measure......Page 144
4.2 Energy spaces on the disc......Page 146
4.3 Free versus classical entropy on the spheres......Page 154
4.4 Equilibrium measures for potentials on the real line......Page 159
4.5 Equilibrium densities for convex potentials......Page 166
4.6 The quartic model with positive leading term......Page 171
4.7 Quartic models with negative leading term......Page 176
4.8 Displacement convexity and relative free entropy......Page 181
4.9 Toeplitz determinants......Page 184
5.1 Convergence to arclength......Page 189
5.2 Convergence of ensembles......Page 191
5.3 Mean field convergence......Page 195
5.4 Almost sure weak convergence for uniformly convex potentials......Page 201
5.5 Convergence for the singular numbers from the Wishart distribution......Page 205
6.1 Variation of functionals and gradient flows......Page 208
6.2 Logarithmic Sobolev inequalities......Page 215
6.3 Logarithmic Sobolev inequalities for uniformlyconvex potentials......Page 218
6.4 Fisher's information and Shannon's entropy......Page 222
6.5 Free information and entropy......Page 225
6.6 Free logarithmic Sobolev inequality......Page 230
6.7 Logarithmic Sobolev and spectral gap inequalities......Page 233
6.8 Inequalities for Gibbs measures onRiemannian manifolds......Page 235
7.1 Group representations......Page 239
7.2 Young diagrams......Page 241
7.3 The Vershik distribution......Page 249
7.4 Distribution of the longest increasing subsequence......Page 255
7.5 Inclusion-exclusion principle......Page 262
8.1 Determinantal random point fields......Page 265
8.2 Determinantal random point fields on the real line......Page 273
8.3 Determinantal random point fields and orthogonal polynomials......Page 282
8.4 De Branges's spaces......Page 286
8.5 Limits of kernels......Page 290
9.1 Integrable operators and Hankel integral operators......Page 293
9.2 Hankel integral operators that commute with second order differential operators......Page 301
9.3 Spectral bulk and the sine kernel......Page 305
9.4 Soft edges and the Airy kernel......Page 311
9.5 Hard edges and the Bessel kernel......Page 316
9.6 The spectra of Hankel operators andrational approximation......Page 322
9.7 The Tracy-Widom distribution......Page 327
10.1 The Costin-Lebowitz central limit theorem......Page 333
10.2 Discrete Tracy-Widom systems......Page 339
10.3 The discrete Bessel kernel......Page 340
10.4 Plancherel measure on the partitions......Page 346
10.5 Fluctuations of the longest increasing subsequence......Page 355
10.6 Fluctuations of linear statistics over unitary ensembles......Page 357
11.1 Some inductive limit groups......Page 364
11.1.2 Infinite torus......Page 365
11.1.4 Infinite orthogonal group......Page 367
11.2 Hua-Pickrell measure on the infinite unitary group......Page 369
11.3 Gaussian Hilbert space......Page 377
11.4 Gaussian measures and fluctuations......Page 381
12.1 Tensor products of Hilbert space......Page 385
12.2 Hermite polynomials and Mehler's formula......Page 387
12.3 The Ornstein-Uhlenbeck semigroup......Page 393
12.4 Hermite polynomials in higher dimensions......Page 396
13.1 The Ornstein-Uhlenbeck process......Page 404
13.2 The logarithmic Sobolev inequality for the Ornstein-Uhlenbeck generator......Page 408
13.3 The matrix Ornstein-Uhlenbeck process......Page 410
13.4 Solutions for matrix stochastic differential equations......Page 413
13.5 The Burgers equation......Page 420
14.1 Noncommutative probability spaces......Page 423
14.2 Tracial probability spaces......Page 426
14.3 The semicircular distribution......Page 430
References......Page 436
Index......Page 445