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Book DetailsCourant Lecture NotesVolume: 18; 2009; 217 ppMSC: Primary 15; 60; 05; 62;
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.ReadershipGraduate students and research mathematicians interested in mathematical foundations of random matrix theory.

Table of Contents

Invariant random matrix ensembles: unified derivation of eigenvalue cluster and correlation functions

Chapter 1. Introduction and examples

Chapter 2. Three classes of invariant ensembles

Chapter 3. Auxiliary facts from functional analysis, Pfaffians, and three integral identities

Chapter 4. Eigenvalue statistics for the three types of ensembles

Universality for orthogonal and symplectic ensembles

Chapter 5. Widom’s formulae for the $\beta =1$ and $4$ correlation kernels

Chapter 6. Large $N$ eigenvalue statistics for the $\beta =1,4$ ensembles with monomial potentials: universality


Additional Material

Reviews

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.
Zentralblatt MATH


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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.
Graduate students and research mathematicians interested in mathematical foundations of random matrix theory.

Invariant random matrix ensembles: unified derivation of eigenvalue cluster and correlation functions

Chapter 1. Introduction and examples

Chapter 2. Three classes of invariant ensembles

Chapter 3. Auxiliary facts from functional analysis, Pfaffians, and three integral identities

Chapter 4. Eigenvalue statistics for the three types of ensembles

Universality for orthogonal and symplectic ensembles

Chapter 5. Widom’s formulae for the $\beta =1$ and $4$ correlation kernels

Chapter 6. Large $N$ eigenvalue statistics for the $\beta =1,4$ ensembles with monomial potentials: universality

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.
Zentralblatt MATH