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**Random matrices.
Rev. and enlarged 2. ed.**
*(English)*
Zbl 0780.60014

Boston, MA: Academic Press, Inc. xviii, 562 p. (1991).

In Chapters 2-11, 13-15 and 18, the book mainly deals with Gaussian ensembles of various types such as Gaussian orthogonal ensembles, Gaussian symplectic ensembles, Gaussian unitary ensembles, Gaussian ensembles of antisymmetric Hermitian matrices, Gaussian ensembles of other Hermitian matrices as well as with circular ensembles of various types such as orthogonal, unitary or symplectic ensembles. Joint probability density functions for the eigenvalues of the respective matrices are derived as well as the correlation functions and the cluster functions.

Much motivation for considering random matrices is derived from nuclear physics and many results are interpreted within this area via making the hypothesis that the characteristic energies of chaotic systems behave locally as if they were the eigenvalues of a matrix with randomly distributed elements. Chapter 1 introduces into the theory of nuclear spectra from this point of view and Chapter 2 discusses the usual symmetries that a quantum system might possess. The joint density for the various matrix elements of the Hamiltonian is derived. The transition from matrix elements to eigenvalues is made in Chapter 3. Thereby, however, the more mathematical questions concerning the measurability of the eigenvalues of the considered random matrices and concerning the existence of their common density are not strongly dealt with but there are made more or less implicitly respective assumptions.

Other discussed branches of knowledge which are nearly related to random matrices belong not only to physics but also to mathematics as, e.g., the discussion of the distribution of the “nontrivial” zeros of the Riemannian zeta function in Section 1.8. In this and in other sections, the author gives hints to open questions which could stimulate further investigations.

Chapter 16 deals with statistical considerations, in Chapter 17 certain difficult integrals including generalizations of the beta integral are dealt with. Chapter 20 is devoted to bordered matrices and Chapter 21 to certain invariance hypotheses.

The main text is followed by more than 125 pages appendices. In contrast to the first edition the present one is very well illustrated by a large number of figures. The book can be recommended both to physicists and to mathematicians as an excellent introduction into and survey over the area of random matrices and related fields.

Much motivation for considering random matrices is derived from nuclear physics and many results are interpreted within this area via making the hypothesis that the characteristic energies of chaotic systems behave locally as if they were the eigenvalues of a matrix with randomly distributed elements. Chapter 1 introduces into the theory of nuclear spectra from this point of view and Chapter 2 discusses the usual symmetries that a quantum system might possess. The joint density for the various matrix elements of the Hamiltonian is derived. The transition from matrix elements to eigenvalues is made in Chapter 3. Thereby, however, the more mathematical questions concerning the measurability of the eigenvalues of the considered random matrices and concerning the existence of their common density are not strongly dealt with but there are made more or less implicitly respective assumptions.

Other discussed branches of knowledge which are nearly related to random matrices belong not only to physics but also to mathematics as, e.g., the discussion of the distribution of the “nontrivial” zeros of the Riemannian zeta function in Section 1.8. In this and in other sections, the author gives hints to open questions which could stimulate further investigations.

Chapter 16 deals with statistical considerations, in Chapter 17 certain difficult integrals including generalizations of the beta integral are dealt with. Chapter 20 is devoted to bordered matrices and Chapter 21 to certain invariance hypotheses.

The main text is followed by more than 125 pages appendices. In contrast to the first edition the present one is very well illustrated by a large number of figures. The book can be recommended both to physicists and to mathematicians as an excellent introduction into and survey over the area of random matrices and related fields.

Reviewer: W.-D.Richter (Rostock)

### MSC:

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

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

81Q99 | General mathematical topics and methods in quantum theory |

15B52 | Random matrices (algebraic aspects) |