Spectral theory of random matrices.
(Spektral’naya teoriya sluchajnykh matrits.)

*(Russian)*Zbl 0656.15012
Teoriya Veroyatnostej i Matematicheskaya Statistika, 40. Moskva: Nauka. 376 p. R. 4.6 (1988).

In this monograph the author investigates the distribution of the eigenvalues and eigenvectors of random matrices and of matrix random processes with additive independent increments, under the assumption that the distribution of the matrices is absolutely continuous with respect to Haar measure on several matrix groups. He also proves limit theorems for spectral functions, eigenvalues and eigenvectors of \(n\times n\) random matrices as \(n\to \infty\). Among the deepest results in the book is a proof of an “elliptic law” which is a more precise version of the “semicircle law” of E. P. Wigner [SIAM Rev. 9, No.1, 1-23 (1967; Zbl 0144.482)]. Another result is an estimate for the resolvent of a covariant matrix of large size which has the remarkable property that, under the Kolmogorov condition, it is asymptotically normal. The contents are as follows:

Chapter 1. Distributions of eigenvalues and eigenvectors of random matrices. Separate results are given for real symmetric, Hermitian, unitary, etc. matrices. Chapter 2. Distributions of eigenvalues and eigenvectors of matrix random processes. Among the topics dealt with here are perturbation formulas for the resolvent and the eigenvalues and eigenvectors of random matrices, Kolmogorov equations for the density of eigenvalue distributions and stochastic differential equations.

Chapter 3. Limit theorems for spectral functions of self-adjoint random matrices. He proves limit theorems for smoothed normed spectral functions, for the traces of powers of random matrices, for the spectral functions of random Jacobi matrices and also of Hermitian covariance matrices. These results are derived with the help of limit theorems for the Stieltjes transform. Chapter 4. Limit theorems for spectral functions of non-selfadjoint random matrices. With the help of the so-called V- transform these problems are reduced to the selfadjoint case. Chapter 5. Limit theorems for distributions of eigenvalues and eigenvectors of random matrices. The principal tool used in this chapter is the method of integral transforms of random determinants.

The book is based on the author’s own considerable research in the field as well as work of Wigner, Lipschitz, Dyson, Meta, Marchenko and Pastur, etc.

Chapter 1. Distributions of eigenvalues and eigenvectors of random matrices. Separate results are given for real symmetric, Hermitian, unitary, etc. matrices. Chapter 2. Distributions of eigenvalues and eigenvectors of matrix random processes. Among the topics dealt with here are perturbation formulas for the resolvent and the eigenvalues and eigenvectors of random matrices, Kolmogorov equations for the density of eigenvalue distributions and stochastic differential equations.

Chapter 3. Limit theorems for spectral functions of self-adjoint random matrices. He proves limit theorems for smoothed normed spectral functions, for the traces of powers of random matrices, for the spectral functions of random Jacobi matrices and also of Hermitian covariance matrices. These results are derived with the help of limit theorems for the Stieltjes transform. Chapter 4. Limit theorems for spectral functions of non-selfadjoint random matrices. With the help of the so-called V- transform these problems are reduced to the selfadjoint case. Chapter 5. Limit theorems for distributions of eigenvalues and eigenvectors of random matrices. The principal tool used in this chapter is the method of integral transforms of random determinants.

The book is based on the author’s own considerable research in the field as well as work of Wigner, Lipschitz, Dyson, Meta, Marchenko and Pastur, etc.

Reviewer: F.J.Gaines

##### MSC:

15B52 | Random matrices (algebraic aspects) |

15A18 | Eigenvalues, singular values, and eigenvectors |

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

60J99 | Markov processes |

60F17 | Functional limit theorems; invariance principles |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |