An Introduction to Random Matrices

Greg W. Anderson; Alice Guionnet; Ofer Zeitouni

doi:10.1017/CBO9780511801334

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

' … this is a very valuable new reference for the subject, incorporating many modern results and perspectives that are not present in earlier texts on this topic … this book would serve as an excellent foundation with which to begin studying other aspects of random matrix theory.'

Terence Tao Source: Mathematical Reviews

'… the book aims to introduce some of the modern techniques of random matrix theory in a comprehensive and rigorous way. It has a broad range of topics and most of them are fairly accessible. The focus is on introducing and explaining the main techniques, rather than obtaining the most general results. Additional references are given for the reader who wants to continue the study of a certain topic. The writing style is careful and the book is mostly self-contained with complete proofs. This is an excellent new contribution to random matrix theory.'

Source: Journal of Approximation Theory

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An Introduction to Random Matrices

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Contents

Frontmatter
pp i-vi

Contents
pp vii-xii

Preface
pp xiii-xiv

1 - Introduction
pp 1-5

2 - Real and complex Wigner matrices
pp 6-89

3 - Hermite polynomials, spacings and limit distributions for the Gaussian ensembles
pp 90-185

4 - Some generalities
pp 186-321

5 - Free probability
pp 322-413

Appendices
pp 414-464

References
pp 465-480

General conventions and notation
pp 481-483

Index
pp 484-492

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