Selected Title s i n Thi s Serie s Volume 5 Emmanue l Hebe y Nonlinear analysi s o n manifolds : Sobole v space s an d inequalitie s 2000 3 Perc y Deif t Orthogonal polynomial s an d rando m matrices : A Riemann-Hilbert approac h 2000 2 Jala l Shata h an d Michae l Struw e Geometric wav e equation s 2000 1 Qin g Ha n an d Fanghu a Li n Elliptic partial differentia l equation s 2000 Courant Lecture Notes in Mathematics Executive Editor Jalal Shatah Managing Editor Paul D. Monsour Production Editor Reeva Goldsmith Copy Editor Melissa Macasieb http://dx.doi.org/10.1090/cln/003 Percy Deif t Courant Institute of Mathematical Sciences 3 Orthogona l Polynomials 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 2000 Mathematics Subject Classification. Primar y 30-XX , 33-XX , 60-XX , 15A90 , 26Cxx. Library o f Congres s Cataloging-in-Pubiicatio n Dat 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. Orthogonal polynomials . 2 . Random matrices . I . Title. II . Series . QA404.5 .D37 200 0 515'.55—<lc21 00-06183 4 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o mak e fai r us e o f the material , such a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the sourc e i s given . 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Visit the AM S hom e page at URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 0 5 04 03 02 To Rebecca and Abby for your patience and support This page intentionally left blank Contents Preface be Chapter • 1. Riemann-Hilber t Problem s 1 1.1. What Is a Riemann-Hilbert Problem ? 1 1.2. Examples 4 Chapter • 2. Jacob i 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: Suppor t of a Measure 35 Chapter • 3. Orthogona l Polynomials 37 3.1. Construction o f Orthogonal Polynomial s 37 3.2. A Riemann-Hilbert Proble m 43 3.3. Some Symmetry Consideration s 49 3.4. Zeros of Orthogonal Polynomials 52 Chapter • 4. Continue d 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 o f a Number 85 Chapter • 5. Rando m Matrix Theory 89 5.1. Introduction 89 5.2. Unitary Ensembles 91 5.3. Spectral Variables for Hermitian Matrices 94 5.4. Distribution o f Eigenvalues 101 5.5. Distribution o f Spacings of Eigenvalues 113 5.6. Further Remarks on the Nearest-Neighbor Spacin g Distribution an d Universality 120 Chapter • 6. Equilibriu m Measures 129 6.1. Scaling 129 6.2. Existence of the Equilibrium Measure fi v 134 6.3. Convergence of kx* 145 vii Vlll CONTENTS 6.4. Convergence of ^3li(x\)dxi 149 6.5. Convergence of r\x* 159 6.6. Variational Problem for the Equilibrium Measure 167 6.7. Equilibrium Measure for V(x) — tx 2m 169 6.8. Appendix: The Transfinite Diameter and Fekete Sets 179 Chapter • 7. Asymptotic s 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 o f a Riemann-Hilbert Proble m 191 7.4. Asymptotics o f Orthogonal Polynomials 201 7.5. Some Analytic Considerations o f Riemann-Hilbert Problem s 208 7.6. Construction o f the Parametrix 213 7.7. Asymptotics o f Orthogonal Polynomials on the Real Axis 230 Chaptei• 8. Universalit y 237 8.1. Universality 237 8.2. Asymptotics of Ps 251 Bibliography 259 Preface In the academic yea r 1996-1997 , I gave a course a t the Courant Institut e o n 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 m e 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 a m trul y 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 is the following: Wh y do very general ensembles o f 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 o f the general methods an d 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 wor k 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 manuscript 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. This page intentionally left blank This page intentionally left blank Bibliography [1] Abramowitz , M., an d Stegun, I. A., eds. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York, 1972. [2] Akhiezer , N. I. The classical moment problem and some related questions in analysis. Trans- lated by N. Kemmer. Hafner, Ne w York, 1965. [3] Beals , R., an d Coifman, R . Scatterin g an d inverse scatterin g fo r first order operators. Comm. PureAppl. Math. 37: 39-90, 1984 . [4] Beals , R. , Deift , P. , and Tomei , C . Direct and inverse scattering on the line. Mathematica l Surveys and Monographs, 28. American Mathematical Society, Providence, R.I., 1988. [5] Bleher , P. , an d Its , A . R . Semiclassica l asymptotic s o f orthogona l polynomials , Riemann - Hilbert problems, and universality in the matrix model. Ann. of Math. (2) 150: 185-266, 1999. [6] Clancey , K., and Gohberg, I. Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, 3, Birkhauser, Basel-Boston, 1981. [7] Coddington , E. A., and Levinson, N. Theory of ordinary differential equations.