Random Matrix Theory: Invariant Ensembles and Universality

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This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles—orthogonal, unitary, and symplectic. The authors follow the approach of Tracy and Widom, but the exposition here contains a substantial amount of additional material, in particular, facts from functional analysis and the theory of Pfaffians. The main result in the book is a proof of universality for orthogonal and symplectic ensembles corresponding to generalized Gaussian type weights following the authors' prior work. New, quantitative error estimates are derived. The book is based in part on a graduate course given by the first author at the Courant Institute in fall 2005. Subsequently, the second author gave a modified version of this course at the University of Rochester in spring 2007. Anyone with some background in complex analysis, probability theory, and linear algebra and an interest in the mathematical foundations of random matrix theory will benefit from studying this valuable reference.

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Product Details

  • ISBN-13: 9780821847374
  • Publisher: American Mathematical Society
  • Publication date: 6/15/2009
  • Series: Courant Lecture Notes Series, #18
  • Pages: 217
  • Product dimensions: 6.90 (w) x 9.90 (h) x 0.50 (d)

Table of Contents

Pt. 1 Invariant Random Matrix Ensembles: Unified Derivation of Eigenvalue Cluster and Correlation Functions 1

Ch. 1 Introduction and Examples 3

Ch. 2 Three Classes of Invariant Ensembles 9

Ch. 3 Auxiliary Facts from Functional Analysis, Pfaffians, and Three Integral Identities 37

Ch. 4 Eigenvalue Statistics for the Three Types of Ensembles 65

Pt. 2 Universality for Orthogonal and Symplectic Ensembles 113

Ch. 5 Widom's Formulae for the [beta] = 1 and 4 Correlation Kernels 115

Ch. 6 Large N Eigenvalue Statistics for the [beta] = 1, 4 Ensembles with Monomial Potentials: Universality 139

Bibliography 211

Index 217

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