Contents
Preface ix
Acknowledgments x
Chapter 1. Preparatory material 1
§1.1. A review of probability theory 2
§1.2. Stirling’s formula 35
§1.3. Eigenvalues and sums of Hermitian matrices 39
Chapter 2. Random matrices 55
§2.1. Concentration of measure 56
§2.2. The central limit theorem 79
§2.3. The operator norm of random matrices 105
§2.4. The semicircular law 134
§2.5. Free probability 155
§2.6. Gaussian ensembles 182
§2.7. The least singular value 209
§2.8. The circular law 223
Chapter 3. Related articles 235
§3.1. Brownian motion and Dyson Brownian motion 236
§3.2. The Golden-Thompson inequality 252
§3.3. The Dyson and Airy kernels of GUE via semiclassical
analysis 259
§3.4. The mesoscopic structure of GUE eigenvalues 265
vii
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