Contents
Preface ix
Chapter 1. Introduction 1
1.1. Objectives and Problems 1
1.2. Example 13
1.3. Comments and Problems 21
Part 1. Classical Ensembles 33
Chapter 2. Gaussian Ensembles: Semicircle Law 35
2.1. Technical Means 35
2.2. Deformed Semicircle Law 43
2.3. The Case of Random
H(0)
54
2.4. Problems 59
Chapter 3. Gaussian Ensembles: Central Limit Theorem for Linear
Eigenvalue Statistics 69
3.1. Covariance for Traces of the Resolvent 69
3.2. Central Limit Theorem for Linear Eigenvalue Statistics of
Differentiable Test Functions 74
3.3. Central Limit Theorem for (ϕ(M))jj 90
3.4. Problems 94
Chapter 4. Gaussian Ensembles: Joint Eigenvalue Distribution and Related
Results 101
4.1. Joint Eigenvalue Probability Density 101
4.2. Orthogonal Polynomial Techniques 107
4.3. Simplest Applications 113
4.4. Comments and Problems 118
Chapter 5. Gaussian Unitary Ensemble 129
5.1. Hermite Polynomials 129
5.2. Bulk of the Spectrum 131
5.3. Edges of the Spectrum 147
5.4. Problems 152
Chapter 6. Gaussian Orthogonal Ensemble 159
6.1. Correlation and Cluster Functions 159
6.2. Bulk of the Spectrum 166
6.3. Edges of the Spectrum 171
6.4. Problems 175
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