Orthogonal Polynomials and Random Matrices

by Deift, Percy

ISBN13:
9780821826959
ISBN10:
0821826956
Format: Paperback
Copyright: 2000-11-01
Publisher: New York Univ Courant Inst
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Summary

This volume expands on a set of lectures held at the Courant Institute on Riemann-Hilbert problems, orthogonal polynomials, and random matrix theory. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models. The central question was the following: Why do very general ensembles of random n times n matrices exhibit universal behavior as n > infinity? The main ingredient in the proof is the steepest descent method for oscillatory Riemann-Hilbert problems. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

Preface

Riemann-Hilbert Problems

(12)

What Is a Riemann-Hilbert Problem?

(3)

Examples

(9)

Jacobi Operators

(24)

Jacobi Matrices

(10)

The Spectrum of Jacobi Matrices

(2)

The Toda Flow

(1)

Unbounded Jacobi Operators

(9)

Appendix: Support of a Measure

(2)

Orthogonal Polynomials

(20)

Construction of Orthogonal Polynomials

(6)

A Riemann-Hilbert Problem

(6)

Some Symmetry Considerations

(3)

Zeros of Orthogonal Polynomials

(5)

Continued Fractions

(32)

Continued Fraction Expansion of a Number

(7)

Measure Theory and Ergodic Theory

(12)

Application to Jacobi Operators

(9)

Remarks on the Continued Fraction Expansion of a Number

(4)

Random Matrix Theory

(40)

Introduction

(2)

Unitary Ensembles

(3)

Spectral Variables for Hermitian Matrices

(7)

Distribution of Eigenvalues

101

(12)

Distribution of Spacings of Eigenvalues

113

(7)

Further Remarks on the Nearest-Neighbor Spacing Distribution and Universality

120

(9)

Equilibrium Measures

129

(52)

Scaling

129

(5)

Existence of the Equilibrium Measure μV

134

(11)

Convergence of λχ*

145

(4)

Convergence of 1/n R1 (x1)dx1

149

(10)

Convergence of nx*

159

(8)

Variational Problem for the Equilibrium Measure

167

(2)

Equilibrium Measure for V (x) = tx2m

169

(10)

Appendix: The Transfinite Diameter and Fekete Sets

179

(2)

Asymptotics for Orthogonal Polynomials

181

(56)

Riemann-Hilbert Problem: The Precise Sense

181

(8)

Riemann-Hilbert Problem for Orthogonal Polynomials

189

(2)

Deformation of a Riemann-Hilbert Problem

191

(10)

Asymptotics of Orthogonal Polynomials

201

(7)

Some Analytic Considerations of Riemann-Hilbert Problems

208

(5)

Construction of the Parametrix

213

(17)

Asymptotics of Orthogonal Polynomials on the Real Axis

230

(7)

Universality

237

(22)

Universality

237

(14)

Asymptotics of Ps

251

(8)

Bibliography

259

Supplemental Materials

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