Random Matrices: High Dimensional Phenomena Book

Introduction 1

1 Metric measure spaces 4

1.1 Weak convergence on compact metric spaces 4

1.2 Invariant measure on a compact metric group 10

1.3 Measures on non-compact Polish spaces 16

1.4 The Brunn-Minkowski inequality 22

1.5 Gaussian measures 25

1.6 Surface area measure on the spheres 27

1.7 Lipschitz functions and the Hausdorff metric 31

1.8 Characteristic functions and Cauchy transforms 33

2 Lie groups and matrix ensembles 42

2.1 The classical groups, their eigenvalues and norms 42

2.2 Determinants and functional calculus 49

2.3 Linear Lie groups 56

2.4 Connections and curvature 63

2.5 Generalized ensembles 66

2.6 The Weyl integration formula 72

2.7 Dyson's circular ensembles 78

2.8 Circular orthogonal ensemble 81

2.9 Circular symplectic ensemble 83

3 Entropy and concentration of measure 84

3.1 Relative entropy 84

3.2 Concentration of measure 93

3.3 Transportation 99

3.4 Transportation inequalities 103

3.5 Transportation inequalities for uniformly convex potentials 106

3.6 Concentration of measure in matrix ensembles 109

3.7 Concentration for rectangular Gaussian matrices 114

3.8 Concentration on the sphere 123

3.9 Concentration for compact Lie groups 126

4 Free entropy and equilibrium 132

4.1 Logarithmic energy and equilibrium measure 132

4.2 Energy spaces on the disc 134

4.3 Free versus classical entropy on the spheres 142

4.4 Equilibrium measures for potentials on the real line 147

4.5 Equilibrium densities for convex potentials 154

4.6 The quartic model with positive leading term 159

4.7 Quartic models with negative leading term 164

4.8 Displacement convexity and relative free entropy 169

4.9 Toeplitzdeterminants 172

5 Convergence to equilibrium 177

5.1 Convergence to arclength 177

5.2 Convergence of ensembles 179

5.3 Mean field convergence 183

5.4 Almost sure weak convergence for uniformly convex potentials 189

5.5 Convergence for the singular numbers from the Wishart distribution 193

6 Gradient flows and functional inequalities 196

6.1 Variation of functionals and gradient flows 196

6.2 Logarithmic Sobolev inequalities 203

6.3 Logarithmic Sobolev inequalities for uniformly convex potentials 206

6.4 Fisher's information and Shannon's entropy 210

6.5 Free information and entropy 213

6.6 Free logarithmic Sobolev inequality 218

6.7 Logarithmic Sobolev and spectral gap inequalities 221

6.8 Inequalities for Gibbs measures on Riemannian manifolds 223

7 Young tableaux 227

7.1 Group representations 227

7.2 Young diagrams 229

7.3 The Vershik Ω distribution 237

7.4 Distribution of the longest increasing subsequence 243

7.5 Inclusion-exclusion principle 250

8 Random point fields and random matrices 253

8.1 Deterrninantal random point fields 253

8.2 Deterrninantal random point fields on the real line 261

8.3 Deterrninantal random point fields and orthogonal polynomials 270

8.4 De Branges's spaces 274

8.5 Limits of kernels 278

9 Integrable operators and differential equations 281

9.1 Integrable operators and Hankel integral operators 281

9.2 Hankel integral operators that commute with second order differential operators 289

9.3 Spectral bulk and the sine kernel 293

9.4 Soft edges and the Airy kernel 299

9.5 Hard edges and the Bessel kernel 304

9.6 The spectra of Hankel operators and rational approximation 310

9.7 The Tracy-Widom distribution 315

10 Fluctuations and the Tracy-Widom distribution 321

10.1 The Costin-Lebowitz central limit theorem 321

10.2 Discrete Tracy-Widom systems 327

10.3 The discrete Bessel kernel 328

10.4 Plancherel measure on the partitions 334

10.5 Fluctuations of the longest increasing subsequence 343

10.6 Fluctuations of linear statistics over unitary ensembles 345

11 Limit groups and Gaussian measures 352

11.1 Some inductive limit groups 352

11.2 Hua-Pickrell measure on the infinite unitary group 357

11.3 Gaussian Hilbert space 365

11.4 Gaussian measures and fluctuations 369

12 Hermite polynomials 373

12.1 Tensor products of Hilbert space 373

12.2 Hermite polynomials and Mehler's formula 375

12.3 The Ornstein-Uhlenbeck semigroup 381

12.4 Hermite polynomials in higher dimensions 384

13 From the Ornstein-Uhlenbeck process to the Burgers equation 392

13.1 The Ornstein-Uhlenbeck process 392

13.2 The logarithmic Sobolev inequality for the Ornstein-Uhlenbeck generator 396

13.3 The matrix Ornstein-Uhlenbeck process 398

13.4 Solutions for matrix stochastic differential equations 401

13.5 The Burgers equation 408

14 Noncommutative probability spaces 411

14.1 Noncommutative probability spaces 411

14.2 Tracial probability spaces 414

14.3 The semicircular distribution 418

References 424

Index 433