##
**Random matrices.
3rd ed.**
*(English)*
Zbl 1107.15019

Pure and Applied Mathematics (Amsterdam) 142. Amsterdam: Elsevier (ISBN 0-12-088409-7/hbk). xviii, 688 p. (2004).

The author’s famous book Random Matrices was first published in 1967. The second edition appeared in 1990 (see Zbl 0780.60014). And the one under review is the third edition, which adopts recent developments in this field. It is to be noted that the previous editions have contributed to the developments as standard references.

The consistent feature of the book through the three editions is the presentations of various analytic results by means of explicit calculations. The third edition consists of 27 chapters. The last 8 chapters are new in this edition.

In Chapter 1, the motivation of the study of random matrices is explained in connection with the distributions of the energy levels of complex nuclei, where a large number of particles are interacting according to unknown laws. The similarity between the distribution of eigenvalues of random matrices and the nontrivial zeros of the Riemann zeta function is also mentioned. Gaussian ensembles of Hermitian matrices such as Gaussian orthogonal, Gaussian unitary and Gaussian symplectic ensembles [GOE, GUE and GSE] and so on are defined by the invariance properties and the statistical independence of the matrix elements (Chapter 2) and the probability densities of eigenvalues of the ensembles are derived (Chapter 3). After preparing a method of integration (Chapter 5), the correlation functions and the level spacings for GUE, GOE and GSE are obtained (Chapters 6 to 8).

Chapters 4 and 9 deals with statistical mechanical interpretations of the Gaussian ensembles. The circular ensembles of unitary matrices are introduced and their properties are studied in Chapters 10 to 12. Chapters 13 to 15 deal with Gaussian ensembles of Hermitian matrices other than GOE, GUE and GSE.

Chapter 16 deals with various statistical quantities for sequences of the eigenvalues of random matrices. A detailed presentation of Selberg’s integral is given in Chapter 17. Chapter 18 to 20 concern the level spacing functions. The functions related to the Fredholm determinant of the sine kernel satisfy Painlevé equations. This fact is proved and used to yield asymptotic expansion of these functions in Chapter 21. Moments of characteristic polynomials of random matrices are expressed by determinants or Pfaffians in Chapter 22.

Chapter 23 is devoted to the theory of coupled random matrices, which appears in the study of 2-dimensional quantum gravity. In Chapter 24, the distribution of eigenvalues on the edge of the semi-circle are examined in the limit \(N \to \infty\). Random perturbations and restricted trace ensembles are treated in chapters 25 and 27, respectively. Probability densities for determinants are calculated in Chapter 26 and Section 15.4.

The book is supplemented by appendices of 151 pages, notes, detailed references and author and subject indices.

The composition of this edition reflects the history of this fields and the book. However, it might not be the best way to non-expert readers’ understanding. This book can be recommended to all the researchers who are interested in the fields.

The consistent feature of the book through the three editions is the presentations of various analytic results by means of explicit calculations. The third edition consists of 27 chapters. The last 8 chapters are new in this edition.

In Chapter 1, the motivation of the study of random matrices is explained in connection with the distributions of the energy levels of complex nuclei, where a large number of particles are interacting according to unknown laws. The similarity between the distribution of eigenvalues of random matrices and the nontrivial zeros of the Riemann zeta function is also mentioned. Gaussian ensembles of Hermitian matrices such as Gaussian orthogonal, Gaussian unitary and Gaussian symplectic ensembles [GOE, GUE and GSE] and so on are defined by the invariance properties and the statistical independence of the matrix elements (Chapter 2) and the probability densities of eigenvalues of the ensembles are derived (Chapter 3). After preparing a method of integration (Chapter 5), the correlation functions and the level spacings for GUE, GOE and GSE are obtained (Chapters 6 to 8).

Chapters 4 and 9 deals with statistical mechanical interpretations of the Gaussian ensembles. The circular ensembles of unitary matrices are introduced and their properties are studied in Chapters 10 to 12. Chapters 13 to 15 deal with Gaussian ensembles of Hermitian matrices other than GOE, GUE and GSE.

Chapter 16 deals with various statistical quantities for sequences of the eigenvalues of random matrices. A detailed presentation of Selberg’s integral is given in Chapter 17. Chapter 18 to 20 concern the level spacing functions. The functions related to the Fredholm determinant of the sine kernel satisfy Painlevé equations. This fact is proved and used to yield asymptotic expansion of these functions in Chapter 21. Moments of characteristic polynomials of random matrices are expressed by determinants or Pfaffians in Chapter 22.

Chapter 23 is devoted to the theory of coupled random matrices, which appears in the study of 2-dimensional quantum gravity. In Chapter 24, the distribution of eigenvalues on the edge of the semi-circle are examined in the limit \(N \to \infty\). Random perturbations and restricted trace ensembles are treated in chapters 25 and 27, respectively. Probability densities for determinants are calculated in Chapter 26 and Section 15.4.

The book is supplemented by appendices of 151 pages, notes, detailed references and author and subject indices.

The composition of this edition reflects the history of this fields and the book. However, it might not be the best way to non-expert readers’ understanding. This book can be recommended to all the researchers who are interested in the fields.

Reviewer: Hiroshi Tamura (Kanazawa)

### MSC:

15B52 | Random matrices (algebraic aspects) |

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |