Orthogonal polynomials and random matrices : a Riemann-Hilbert approach Percy Deift.

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Orthogonal polynomials and random matrices : a Riemann-Hilbert approach Percy Deift.



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Book's title: Orthogonal polynomials and random matrices : a Riemann-Hilbert approach Percy Deift.
Library of Congress Control Number: 00061834
International Standard Book Number (ISBN):0821826956 (alk. paper)
Cataloging Source:DLC, DLC DLC
Library of Congress Call Number:QA404.5 .D37 2000
Dewey Decimal Classification Number:515/.55$221
Personal Name:Deift, Percy, 1945-
Publication, Distribution, etc.:Providence, R.I. . American Mathematical Society, (c)2000.
Physical Description:ix, 261 p. : ill. ;, 26 cm.
Title:Courant lecture notes ;$v3
General Note:Originally published: New York : Courant Institute of Mathematical Sciences, New York University, c1999.
Bibliography, etc. Note:Includes bibliographical references (p. 259-261).
Formatted Contents Note:Machine generated contents note: Chapter 1. Riemann-Hilbert Problems 1
1.1. What Is a Riemann-Hilbert Problem? 1
1.2. Examples 4
Chapter 2. Jacobi Operators 13
2.1. Jacobi Matrices 13
2.2. The Spectrum of Jacobi Matrices 23
2.3. The Toda Flow 25
2.4. Unbounded Jacobi Operators 26
2.5. Appendix: Support of a Measure 35
Chapter 3. Orthogonal Polynomials 37
3.1. Construction of Orthogonal Polynomials 37
3.2. A Riemann-Hilbert Problem 43
3.3. Some Symmetry Considerations 49
3.4. Zeros of Orthogonal Polynomials 52
Chapter 4. Continued Fractions 57
4.1. Continued Fraction Expansion of a Number 57
4.2. Measure Theory and Ergodic Theory 64
4.3. Application to Jacobi Operators 76
4.4. Remarks on the Continued Fraction Expansion of a Number 85
Chapter 5. Random Matrix Theory 89
5.1. Introduction 89
5.2. Unitary Ensembles 91
5.3. Spectral Variables for Hermitian Matrices 94
5.4. Distribution of Eigenvalues 101
5.5. Distribution of Spacings of Eigenvalues 113
5.6. Further Remarks on the Nearest-Neighbor Spacing Distribution and
Universality 120
Chapter 6. Equilibrium Measures 129
6.1. Scaling 129
6.2. Existence of the Equilibrium Measure LLV 134
6.3. Convergence of X,* 145
6.4. Convergence of RlI(xl)dxl 149
6.5. Convergence of rlx* 159
6.6. Variational Problem for the Equilibrium Measure 167
6.7. Equilibrium Measure for V(x) = tx2m 169
6.8. Appendix: The Transfinite Diameter and Fekete Sets 179
Chapter 7. Asymptotics for Orthogonal Polynomials 181
7.1. Riemann-Hilbert Problem: The Precise Sense 181
7.2. Riemann-Hilbert Problem for Orthogonal Polynomials 189
7.3. Deformation of a Riemann-Hilbert Problem 191
7.4. Asymptotics of Orthogonal Polynomials 201
7.5. Some Analytic Considerations of Riemann-Hilbert Problems 208
7.6. Construction of the Parametrix 213
7.7. Asymptotics of Orthogonal Polynomials on the Real Axis 230
Chapter 8. Universality 237
8.1. Universality 237
8.2. Asymptotics of Ps 251.
Rubrics: Orthogonal polynomials
Random matrices

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