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Eigenvalue distribution of large random matrices. (English) Zbl 1244.15002

Mathematical Surveys and Monographs 171. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5285-9/hbk). xiv, 632 p. (2011).
This monograph, written by well-known experts, begins with a collection of basic material on classical ensembles. The second part of the text covers many recent results and makes it a valuable source for researchers working in the field of random matrix theory. A substantial part of the theory presented was developed by the authors themselves.
The text covers three main aspects of the theory of random matrices: existence and properties of the nonrandom limiting Normalized Counting Measure of eigenvalues, the fluctuations laws of linear eigenvalue statistics and the local regimes. The first two topics can be seen as an analogy to the LLN and the central limit theorem for independent or weakly dependent sequences of random variables. The main part of the book starts by a detailed discussion of the Gaussian ensembles. The second part is on matrix models or invariant ensembles of hermitian and real symmetric matrices. The third part treats ensembles with independent but not necessarily Gaussian random variables. The reader is assumed to be familiar with basic concepts of calculus, linear algebra and probability theory.
This lucidly written book is an important reference for researchers in probability theory, statistics and mathematical physics.
Reviewer: H. M. Mai (Berlin)

MSC:

15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
60F05 Central limit and other weak theorems
15B57 Hermitian, skew-Hermitian, and related matrices
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