Spectral theory of random matrices

Abstract

CONTENTS Introduction Chapter I. Distribution of the eigenvalues and eigenvectors of random matrices § 1. Polar decomposition of random matrices § 2. Symmetric and Hermitian random matrices § 3. Non-symmetric random matrices § 4. Reduction of random matrices to triangular form § 5. Gaussian random matrices § 6. Unitary random matrices § 7. Orthogonal random matrices Chapter II. Limit theorems for the spectral functions of Hermitian random matrices § 1. Limit theorems of general form for the spectral functions of random matrices § 2. The canonical spectral equation § 3. Wigner's semicircle law § 4. A refinement of Wigner's semicircle law Chapter III. Limit theorems for the spectral functions of non-self-adjoint random matrices § 1. The V-transform of spectral functions § 2. Limit theorems like the law of large numbers for normalized spectral functions of non-self-adjoint random matrices with independent entries § 3. The regularized V-transform for spectral functions § 4. An estimate of the rate of convergence of the Stieltjes transforms of spectral functions to the limit function § 5. Estimates of the deviations of spectral functions from the limit functions § 6. The circle law § 7. The elliptic law Chapter IV. Limit theorems for the eigenvalues of random matrices § 1. Integral representations for random determinants § 2. Limit theorems for the eigenvalues of symmetric random matrices § 3. Limit theorems for the eigenvalues of non-symmetric random matrices § 4. Limit theorems for the distributions of the spacings of random matrices References