Selected Title s i n Thi s Serie s

Size: px

Start display at page:

Download "Selected Title s i n Thi s Serie s"

Calvin Walters
5 years ago
Views:

2 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 Perc y Deif t Orthogonal polynomial s an d rando m matrices : A Riemann-Hilbert approac h Jala l Shata h an d Michae l Struw e Geometric wav e equation s Qin g Ha n an d Fanghu a Li n Elliptic partial differentia l equation s 2000

3 Courant Lecture Notes in Mathematics Executive Editor Jalal Shatah Managing Editor Paul D. Monsour Production Editor Reeva Goldsmith Copy Editor Melissa Macasieb

4 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

5 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, 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 Orthogonal polynomials. 2. Random matrices. I. Title. II. Series. QA404.5.D '.55 <lc 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 customar y acknowledgmen t o f the sourc e i s given. Republication, systemati c copying, or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society. Request s fo r suc h permission should be addressed to the Assistant to the Publisher, America n Mathematical Society, P. O. Bo x 6248, Providence, Rhod e Islan d Request s ca n als o b e mad e b y e-mai l t o reprint-permissionqams.org held b y the author. Al l rights reserved. Printed i n the Unite d State s o f America. Reprinted b y the America n Mathematica l Society, The America n Mathematica l Societ y retains al l right s except thos e granted t o the Unite d State s Th e pape r use d i n this boo k i s acid-free an d fall s withi n the guideline s established t o ensure permanenc e and durability. Visit the AM S hom e page at URL : /

6 To Rebecca and Abby for your patience and support

7 This page intentionally left blank

8 Contents Preface Chapter 1. Riemann-Hilber t Problem s 1.1. What Is a Riemann-Hilbert Problem? 1.2. Examples Chapter 2. Jacob i Operators 2.1. Jacobi Matrices 2.2. The Spectrum of Jacobi Matrices 2.3. The Toda Flow 2.4. Unbounded Jacobi Operators 2.5. Appendix: Suppor t of a Measure Chapter 3. Orthogona l Polynomials 3.1. Construction o f Orthogonal Polynomial s 3.2. A Riemann-Hilbert Proble m 3.3. Some Symmetry Consideration s 3.4. Zeros of Orthogonal Polynomials Chapter 4. Continue d Fractions 4.1. Continued Fraction Expansion of a Number 4.2. Measure Theory and Ergodic Theory 4.3. Application to Jacobi Operators 4.4. Remarks on the Continued Fraction Expansion o f a Number be Chapter 5. Rando m Matrix Theory Introduction Unitary Ensembles Spectral Variables for Hermitian Matrices Distribution o f Eigenvalues Distribution o f Spacings of Eigenvalues Further Remarks on the Nearest-Neighbor Spacin g Distribution an d Universality Chapter 6. Equilibriu m Measures 6.1. Scaling 6.2. Existence of the Equilibrium Measure fi v 6.3. Convergence of k x * vii

9 Vlll CONTENTS 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 Chapter 7. Asymptotic s for Orthogonal Polynomials 7.1. Riemann-Hilbert Problem: The Precise Sense 7.2. Riemann-Hilbert Problem for Orthogonal Polynomials 7.3. Deformation o f a Riemann-Hilbert Proble m 7.4. Asymptotics of Orthogonal Polynomials 7.5. Some Analytic Considerations o f Riemann-Hilbert Problem s 7.6. Construction of the Parametrix 7.7. Asymptotics of Orthogonal Polynomials on the Real Axis Chaptei 8. Universalit y 8.1. Universality 8.2. Asymptotics of P s Bibliography 259

Preface In the academic yea r 1996-1997, I gave a course at the Courant Institut e o n Riemann-Hilbert problems, orthogona l polynomials, an d rando m matri x theory.

10 Preface In the academic yea r , I gave a course at 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 Randall 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 in an attempt to understand from a more rigorous 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 o f random n x n matrices exhibit universal behavior as n > oc? My work and that of my collaborators Thomas Kriecherbauer, Ke n McLaughlin, 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 illustration of 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

11 This page intentionally left blank

12 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.

13 Bibliography [1] Abramowitz, M., an d Stegun, I. A., eds. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York, [2] Akhiezer, N. I. The classical moment problem and some related questions in analysis. Translated by N. Kemmer. Hafner, Ne w York, [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, [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., [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: , [6] Clancey, K., and Gohberg, I. Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, 3, Birkhauser, Basel-Boston, [7] Coddington, E. A., and Levinson, N. Theory of ordinary differential equations. McGraw-Hill, New York-Toronto-London, [8] Cohen, J., Kesten, H., and Newman, C, eds. Random matrices and their applications. Contemporary Mathematics, 50. American Mathematical Society, Providence, R.I., [9] Deift, P., Integrable Hamiltonian systems. Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994), Lectures in Applied Mathematics, 31. American Mathematical Society, Providence, R.I., [10] Deift, P. A., Its, A. R., an d Zhou, X. A Riemann-Hilber t approac h t o asymptoti c problem s arising i n the theory o f random matri x models an d als o in the theory o f integrabl e statistica l mechanics. Ann. of Math. (2) 146: , [11] Deift, P., Kamvissis, S., Kriecherbauer, T., and Zhou, X. The Toda rarefaction problem. Comm. PureAppl. Math. 49: 35-83, [12] Deift, P., Li, L. C, an d Tomei, C. Toda flows with infinitely many variables. J. Funct. Anal. 64: , [13] Deift, P., and McLaughlin, K. T-R. A continuum limi t o f the Toda lattice. Mem. Amer. Math. Soc. 131(624), [14] Deift, P., McLaughlin, K. T-R., and Kriecherbauer, T. New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95: , [15] Deift, P., McLaughlin, K. T-R., Kriecherbauer, T., Venakides, S., an d Zhou, X. Asymptotic s for polynomials orthogona l wit h respect to varying exponentia l weights. Internal Math. Res. Notices 16 : , [16] Stron g asymptotic s o f orthogona l polynomial s wit h respect t o exponential weights. Comm. PureAppl Math. 52: , [17] Unifor m asymptotic s for polynomials orthogonal with respect to varying exponentia l weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52: , [18] Deift, P. A., Nanda, T., and Tomei, C. Ordinary differential equation s and the symmetric eigenvalue problem. SI AM J. Num. Anal. 20(1): 1-22, [19] Deift, P., Venakides, S., and Zhou, X. The couisionless shock region for the long-time behavior of solutions of the KdV equation. Comm. PureAppl Math. 47(2): ,

260 BIBLIOGRAPHY New results in small dispersion KdV by an extension o f the steepest descent method for Riemann-Hilbert problems. Internal Math. Res. Notices 6: 286-299, 1997. Deift, P., and Zhou, X.

14 260 BIBLIOGRAPHY New results in small dispersion KdV by an extension o f the steepest descent method for Riemann-Hilbert problems. Internal Math. Res. Notices 6: , Deift, P., and Zhou, X. A steepes t descen t metho d fo r oscillator y Riemann-Hilber t problem, Asymptotics for the MKdV equation. Ann. ofmath.(2) 137 : , Asymptotics for the Painleve II equation. Comm. PureAppl. Math. 48: , des Cloizeaux, J., and Mehta, M. L. Asymptotic behavior of spacing distributions for the eigenvalues of random matrices. J. Math. Phys. 14 : , Durrett, R. Probability. Theory and examples. Th e Wadswort h & Brooks/Col e Statis - tics/probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, Calif., Dym, H., an d McKean, H. P. Gaussian processes, function theory, and the inverse spectral problem. Probability and Mathematical Statistics, 31. Academic, New York-London, Dyson, F. Fredholm determinant s an d invers e scatterin g problems. Comm. Math. Phys. 47: , Flaschka, M. The Toda lattice. I. Existence of integrals. Phys. Rev. B (3) 9: , Fokas, A. S. A unified transfor m metho d for solvin g linear and certain nonlinear PDEs. Proc. Roy. Soc. London Ser. A 453: , Fokas, A. S., Its, A. R., an d Kitaev, A. V. An isomonodromy approac h t o the theory o f twodimensional quantum gravity. (Russian) Uspekhi Mat. Nauk 45(6(276)): , 1990 ; translation in Russian Math. Surveys 45(6): , Discrete Painleve equations an d their appearance in quantum gravity. Comm. Math. Phys. 142(2) : , Gohberg, I., and Krein, M. Systems of integral equations on a half-line with kernels depending on the difference o f arguments. Amer. Math. Soc. Transl. (2) 14: , Hellinger, E. Zur Stichtjesschen Kettenbruchtheorie. Ann. of Math. 86: 18-29, Johansson, K. O n fluctuations o f eigenvalue s o f rando m Hermitia n matrices. Duke Math. J. 91(1): , Kamvissis, S. On the long-time behavior o f the double infinite Tod a lattice under initia l data decaying at infinity. Comm. Math. Phys. 153(3): , Katznelson, Y. An introduction to harmonic analysis. Secon d correcte d edition. Dover, New York, Katz, N., and Sarnak, P. Katz, N. M.; Sarnak, P. Zeroes of zeta functions an d symmetry. Bull. Amer. Math. Soc. (N.S.) 36(1): 1-26, Khinchin, A. Ya. Continued fractions. Universit y of Chicago Press, Chicago, Khovanskii, A. N. The application of continued fractions and their generalizations to problems in approximation theory. Translate d by Peter Wynn. Noordhoff, Groningen, Kline, M. Mathematical thought from ancient to modern times. Oxford Universit y Press, New York, Landkof, N. S. Foundations of modern potential theory. Translated from th e Russian by A. P. Doohovskoy. Di e Grundlehren de r mathematischen Wissenschaften, Ban d 180. Springer, New York-Heidelberg, Lax, P. D., and Levermore, CD. The small dispersion limit of the Korteweg-de Vries equation. I. II. III. Comm. PureAppl. Math. 36: , 1983 ; ; Manakov, S. V. Complete integrability and stochastization of discrete dynamical systems. Soviet Phys. JETP 40(2): , Mehta, M. L. Random matrices. Second edition. Academic, Boston, Moser, J. Finitely many mass points on the line under the influence of an exponential potential an integrabl e system. Dynamical systems, theory and applications (Rencontres, BattelleRes. Inst., Seattle, Wash., 1974), Lecture Notes in Physics, 38. Springer, Berlin, [45] Muskhelishvili, N. I. Singular integral equations. Boundary problems of function theory and their application to mathematical physics. Translation by J. R. M. Radok. Noordhoff, Gronin - gen, 1953.

15 BIBLIOGRAPHY 261 [46] Noble, B. Methods based on the Wiener-Ilopf technique for the solution of partial differential equations. Internationa l Scrie s of Monograph s on Pur e and Applie d Mathematics, 7. Pergammon Press, New York-London-Paris-Los Angeles, [47] Odlyzko, A. On the distribution o f spacings betwee n zero s of the zeta function. Math. Comp. 48(177): , [48] Parlett, B. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, N.J., [49] Porter, C. E, ed. Statistical theories of spectra: Fluctuations, A Collection of Reprints, Original Papers, with an Introductory Review. Academic Press, New York, [50] Pastur, L., and Figotin, A. Spectra of random and almost periodic operators. Springer-Verlag, Berlin, [51] Pastur, L., and Schcrbina, L. Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. Preprint, [52] Reed, M. f and Simon, B. Methods of modern mathematical physics, I-IV. Academic Press, New York, [53] Royden, H. Real analysis. Third edition. Macmillan, New York, j 54] Rudnick, Z., and Sarnak, P. Zeros of principal L-functions and random matrix theory. Princeton University, preprint, [55] Ryll-Nardzewski, C, On the ergodic theorems II: Ergodic theory of continued fractions. Studia Math. 12:74-79, [56] Sabat, A. B. One-dimensional perturbations of a differential operator, and the inverse scattering problem, , 298. Problems in meclumics and mathematical physics. Izdat. "Nauka", Moscow, [57] Saff, E. B., and Totik, V. Logarithmic potentials with external fields. Springer, Ne w York - Berlin, [58] Simon, B. Trace ideals and their applications. Cambridge Universit y Press, Cambridge-New York, [59] Th e classica l momen t proble m a s a self-adjoin t finit e differenc e operator. Adv. in Math., in press. [60] Stein, E. M. Singular integrals and differentiability properties of functions. Princeto n Mathe - matical Series, 30. Princeton University Press, Princeton, N.J., [61] Szcgo, G. Orthogonal polynomials. Fourt h edition. America n Mathematica l Society, Collo - quium Publications, 23. American Mathematical Society, Providence, [62] Torchinsky, A. Real variables. Addison-Wesley, Redwood City, Calif., [63] Venakides, S. The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with nontrivial reflection coefficient. Comm. PureAppl. Math. 38: , [64] Venakides, S., Deift, P., and Oba, R. The Toda shock problem. Comm. Pure Appl. Math. 44: , [65] Wall, H. S. Analytic theory of continued fractions. Va n Nostrand, New York, [66] Warner, E W. Foundations of differentiable manifolds and Lie groups. Scott, Forcsma n an d Company, Glenview, [67] Weyl, H. Uber gewohnlike Differentia l gleichunge n mi t Singularitate n un d di e zugehorige n Entwicklungen willkurlicher Funktionen. Ann. of Math. 68: , [68] Widom, H. The asymptotics of a continuous analogu e of orthogona l polynomials. J. Approx. Theory 77(1): 51-64, [69].. Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals. Comm. Math. Phys. 171(1) : , [70] Zakharov, V. E., and Shabat, A. B. Exact theor y o f two-dimensiona l self-focusin g an d onedimensional self-modulatio n o f wave s i n nonlinea r media. Soviet Phys. JETP 34(1) : 62-69, 1972.

Courant Lecture Notes in Mathematics

Stochastic Processes Courant Lecture Notes in Mathematics Executive Editor Jalal Shatah Managing Editor Paul D. Monsour Assistant Editor Reeva Goldsmith http://dx.doi.org/10.1090/cln/016 S. R. S. Varadhan