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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