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Nonlinear analysi s o n manifolds : Sobole v space s an d inequalitie s

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Orthogonal polynomial s an d rando m matrices : A Riemann-Hilber t approac h

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Courant Lecture Notes in Mathematics

Executive Editor Jalal Shatah

Managing Editor Paul D. Monsour

Production Editor Reeva Goldsmith

Copy Editor Melissa Macasieb

Percy Deif t Courant Institute of Mathematical Sciences

3 Orthogona l Polynomial s and Random Matrices: A Riemann-Hilbert Approach

Courant Institute of Mathematical Sciences New York University New York, New York

American Mathematical Society Providence, Rhode Island

http://dx.doi.org/10.1090/cln/003

2000 Mathematics Subject Classification. Pr imar y 30-XX , 33 -XX , 6 0 - X X , 15A90 , 26Cxx.

L i b r a r y o f C o n g r e s s Ca ta log ing - in -Pub i i ca t io n D a t a

Deift, Percy , 1945 -Orthogonal polynomial s an d rando m matrice s : a Riemann-Hiiber t approac h / Perc y Deift .

p. cm . — (Couran t lectur e note s ; 3) Originally published : Ne w Yor k : Couran t Institut e o f Mathematica l Sciences , Ne w Yor k

University, cl999 . Includes bibliographica l references . ISBN 0-8218-2695- 6 1. Orthogona l polynomials . 2 . Rando m matrices . I . Title . II . Series .

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Contents

Preface

Chapter 1.1. 1.2.

Chapter 2.1. 2.2. 2.3. 2.4. 2.5.

Chapter 3.1. 3.2. 3.3. 3.4.

Chapter 4.1. 4.2. 4.3. 4.4.

Chapter 5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

Chapter 6.1. 6.2. 6.3.

• 1. Riemann-Hilber t Problem s What Is a Riemann-Hilbert Problem? Examples

• 2. Jacob i Operators Jacobi Matrices The Spectrum of Jacobi Matrices The Toda Flow Unbounded Jacobi Operators Appendix: Suppor t of a Measure

• 3. Orthogona l Polynomial s Construction of Orthogonal Polynomial s A Riemann-Hilbert Proble m Some Symmetry Consideration s Zeros of Orthogonal Polynomial s

• 4. Continue d Fractions Continued Fraction Expansion of a Number Measure Theory and Ergodic Theory Application to Jacobi Operators Remarks on the Continued Fraction Expansion of a Number

• 5. Rando m Matrix Theory Introduction Unitary Ensembles Spectral Variables for Hermitian Matrices Distribution of Eigenvalues Distribution of Spacings of Eigenvalues Further Remarks on the Nearest-Neighbor Spacin g Distribution an d Universality

• 6. Equilibriu m Measures Scaling Existence of the Equilibrium Measure fi v

Convergence of kx*

be

1 1 4

13 13 23 25 26 35

37 37 43 49 52

57 57 64 76 85

89 89 91 94

101 113

120

129 129 134 145

vii

Vlll CONTENTS

6.4. 6.5. 6.6. 6.7. 6.8.

Chapter 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7.

Chaptei 8.1. 8.2.

Convergence of ^3li(x\)dxi Convergence of r\x* Variational Problem for the Equilibrium Measure Equilibrium Measure for V(x) — tx 2m

Appendix: The Transfinite Diamete r and Fekete Sets

• 7. Asymptotic s for Orthogonal Polynomial s Riemann-Hilbert Problem: The Precise Sense Riemann-Hilbert Problem for Orthogonal Polynomials Deformation o f a Riemann-Hilbert Proble m Asymptotics of Orthogonal Polynomial s Some Analytic Considerations o f Riemann-Hilbert Problem s Construction of the Parametrix Asymptotics of Orthogonal Polynomials on the Real Axis

• 8. Universalit y Universality Asymptotics of Ps

149 159 167 169 179

181 181 189 191 201 208 213 230

237 237 251

Bibliography 259

Preface

In the academic yea r 1996-1997 , I gave a course a t the Courant Institut e on Riemann-Hilbert problems , orthogona l polynomials , an d rando m matri x theory . The lectures for the course were taken down and organized into note form by Ran-dall Pyke , Joh n Podesta , Jos e Ramirez , an d Wen-qin g Xu . Ove r th e las t year , Jinho Baik, Thomas Kriecherbauer, an d Ken McLaughlin have helped me furthe r to bring these notes into their present form. Withou t their help, these notes would never hav e been published , an d I am truly thankfu l t o al l thes e peopl e fo r thei r efforts.

I gave the course in 1996-1997 in an attempt to understand from a more rigor-ous mathematical point of view various results and formulae in Mehta's wonderfu l book Random Matrices [43]. At the same time, I was stimulated and challenged by a set of questions fro m Pete r Sarnak , wh o himself wa s trying to understand [43] . These notes are in many ways a response to his questions, and I deeply appreciat e his clear insights and ready help.

The central question i s the following: Wh y do very general ensembles of ran-dom n x n matrices exhibit universal behavior as n —> oc ?

My work and that of my collaborators Thomas Kriecherbauer, Ke n McLaugh-lin, Stephanos Venakides, and Xin Zhou on this question is reported in [15,16,17]. Apart from certain additional preparatory material , these notes are a pedagogic il-lustration of the general methods and results in [15,16,17], in a special case (see Sections 7 and 8) in which the technical difficulties ar e at a minimum. I thank my colleagues for allowing me to reproduce these results here. Pioneerin g mathemat -ical work o n universality fo r rando m matri x ensemble s wa s don e b y Pastu r an d Scherbina in [51], and Its and Bleher in [5].

In additio n t o the student s an d colleague s mentione d above , I would lik e t o thank Dais y Caldero n fo r he r skil l an d patienc e i n typin g th e final manuscript . Special thanks are also due to Melissa Macasieb for her expert copy-editing o f the text, and to Melissa and Reeva Goldsmith for their care in correcting the TgK file. The final figures were drawn by Daisy Calderon . Th e entire project o f preparin g the manuscrip t fo r publicatio n wa s overseen b y Paul Monsour , an d many , man y thanks are due to him for his great expertise and all his help.

This work was supported in part by NSF Grant DMS-9500867.

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