GRADUATE STUDIES IN MATHEMATICS 172 Combinatorics and Random Matrix Theory Jinho Baik Percy Deift 8SY½G7YMHER American Mathematical Society Combinatorics and Random Matrix Theory https://doi.org/10.1090//gsm/172 GRADUATE STUDIES IN MATHEMATICS 172 Combinatorics and Random Matrix Theory Jinho Baik Percy Deift To Y½c 7Yidan %merican MathematicaP 7ociety Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 05A15, 15B52, 33E17, 35Q15, 41A60, 47B35, 52C20, 60B20, 60K35, 82C23. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-172 Library of Congress Cataloging-in-Publication Data Names: Baik, Jinho, 1973- | Deift, Percy, 1945- | Suidan, Toufic Mubadda, 1975- Title: Combinatorics and random matrix theory / Jinho Baik, Percy Deift, Toufic Suidan. Description: Providence, Rhode Island : American Mathematical Society, 2016. | Series: Graduate studies in mathematics ; volume 172 | Includes bibliographical references and index. Identifiers: LCCN 2015051274 | ISBN 9780821848418 (alk. paper) Subjects: LCSH: Random matrices. | Combinatorial analysis. | AMS: Combinatorics – Enumera- tive combinatorics – Exact enumeration problems, generating functions. msc | Linear and mul- tilinear algebra: matrix theory – Special matrices – Random matrices. msc | Special functions (33-XX deals with the properties of functions as functions) – Other special functions – Painlev´e- type functions. msc | Partial differential equations – Equations of mathematical physics and other areas of application – Riemann-Hilbert problems. msc | Approximations and expansions – Approximations and expansions – Asymptotic approximations, asymptotic expansions (steepest descent, etc.). msc | Operator theory – Special classes of linear operators – Toeplitz operators, Hankel operators, Wiener-Hopf operators. msc | Convex and discrete geometry – Discrete geom- etry – Tilings in 2 dimensions. msc | Probability theory and stochastic processes – Probability theory on algebraic and topological structures – Random matrices (probabilistic aspects; for algebraic aspects see 15B52). msc | Probability theory and stochastic processes - Special pro- cesses – Interacting random processes; statistical mechanics type models; percolation theory. msc | Statistical mechanics, structure of matter – Time-dependent statistical mechanics (dy- namic and nonequilibrium) – Exactly solvable dynamic models. msc Classification: LCC QA188.B3345 2016 | DDC 511/.6–DC23 LC record available at http://lccn.loc. gov/2015051274 Copying and reprinting. 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Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2016 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 212019181716 To my wife Hyunsuk and my daughter Haesue To my wife Rebecca and my daughter Abby To my parents Mubadda and Aida Suidan Contents Preface xi Chapter 1. Introduction 1 §1.1. Ulam’s Problem: Random Permutations 2 §1.2. Random Tilings of the Aztec Diamond 12 §1.3. General Remarks 14 Chapter 2. Poissonization and De-Poissonization 19 §2.1. Hammersley’s Poissonization of Ulam’s Problem 19 §2.2. De-Poissonization Lemmas 21 Chapter 3. Permutations and Young Tableaux 27 §3.1. The Robinson-Schensted Correspondence 28 §3.2. The Number of Standard Young Tableaux 49 §3.3. Applications and Equivalent Models 63 Chapter 4. Bounds on the Expected Value of N 77 §4.1. Lower Bound 77 §4.2. Existence of c 78 §4.3. Young Diagrams in a Markov Chain and an Optimal Upper Bound 82 §4.4. Asymptotics of the Conjugacy Classes of the Symmetric Group 87 vii viii Contents Chapter 5. Orthogonal Polynomials, Riemann-Hilbert Problems, and Toeplitz Matrices 95 §5.1. Orthogonal Polynomials on the Real Line (OPRL) 95 §5.2. Some Classical Orthogonal Polynomials 99 §5.3. The Riemann-Hilbert Problem (RHP) for Orthogonal Polynomials 100 §5.4. Orthogonal Polynomials on the Unit Circle (OPUC) and Toeplitz Matrices 106 §5.5. RHP: Precise Description 111 §5.6. Integrable Operators 118 §5.7. The Strong Szeg˝o Limit Theorem 121 §5.8. Inverses of Large Toeplitz Matrices 130 Chapter 6. Random Matrix Theory 139 §6.1. Unitary Ensembles and the Eigenvalue Density Function 139 §6.2. Andr´eief’s Formula and the Computation of Basic Statisitcs 142 §6.3. Gap Probabilities and Correlation Functions 146 §6.4. Scaling Limits and Universality 152 §6.5. The Tracy-Widom Distribution Function 157 Chapter 7. Toeplitz Determinant Formula 165 §7.1. First Proof 167 §7.2. Second Proof 169 §7.3. Recurrence Formulae and Differential Equations 170 §7.4. Heuristic Argument for Convergence of the Scaled Distribution for L(t) to the Tracy-Widom Distribution 184 Chapter 8. Fredholm Determinant Formula 187 §8.1. First Proof: Borodin-Okounkov-Geronimo-Case Identity 190 §8.2. Second Proof 200 Chapter 9. Asymptotic Results 207 §9.1. Exponential Upper Tail Estimate 208 § 9.2. Exponential Lower Tail Estimate√ 214 §9.3. Convergence of L(t)/t and N / N 224 §9.4. Central Limit Theorem 226 §9.5. Uniform Tail Estimates and Convergence of Moments 239 §9.6. Transversal Fluctuations 240 Contents ix Chapter 10. Schur Measure and Directed Last Passage Percolation 253 §10.1. Schur Functions 253 §10.2. RSK and Directed Last Passage Percolation 273 §10.3. Special Cases of Directed Last Passage Percolation 280 §10.4. Gessel’s Formula for Schur Measure 290 §10.5. Fredholm Determinant Formula 294 §10.6. Asymptotics of Directed Last Passage Percolation 298 §10.7. Equivalent Models 301 Chapter 11. Determinantal Point Processes 305 Chapter 12. Tiling of the Aztec Diamond 317 §12.1. Nonintersecting Lattice Paths 318 §12.2. Density Function 334 §12.3. Asymptotics 347 Chapter 13. The Dyson Process and the Brownian Dyson Process 377 §13.1. Dyson Process 379 §13.2. Brownian Dyson Process 380 §13.3. Derivation of the Dyson Process and the Brownian Dyson Process 381 §13.4. Noncolliding Property of the Eigenvalues of Matrix Brownian Motion 389 §13.5. Noncolliding Property of the Eigenvalues of the Matrix Ornstein-Uhlenbeck Process 395 §13.6. Nonintersecting Processes 402 Appendix A. Theory of Trace Class Operators and Fredholm Determinants 421 Appendix B. Steepest-descent Method for the Asymptotic Evaluation of Integrals in the Complex Plane 431 Appendix C. Basic Results of Stochastic Calculus 437 Bibliography 445 Index 459 Preface As a consequence of certain independent developments in mathematics in recent years, a wide variety of problems in combinatorics, some of long stand- ing, can now be solved in terms of random matrix theory (RMT). The goal of this book is to describe in detail these developments and some of their applications to problems in combinatorics. The book is based on courses on two key examples from combinatorial theory, viz., Ulam’s increasing sub- sequence problem, and the Aztec diamond. These courses were given at the Courant Institute and the University of Michigan by two of the authors (P.D. and J.B., respectively) some ten years ago. The authors are pleased to acknowledge the suggestions, help, and infor- mation they obtained from many colleagues: Eitan Bachmat, Gerard Ben Arous, Alexei Borodin, Thomas Kriecherbauer, Eric Nordenstam, Andrew Odlyzko, Eric Rains, Raghu Varadhan, and Ofer Zeitouni. In particular, Eitan Bachmat and Thomas Kriecherbauer took on the task of reading the manuscript in full, catching typos, and suggesting many very helpful changes to the text. The authors would also like to acknowledge the support of NSF over the years when this book was written in the form of Grants DMS- 0457335, DMS-0757709, DMS-1068646, and DMS-1361782 for J.B., DMS- 0500923, DMS-1001886, and DMS-1300965 for P.D., and DMS-0553403 and DMS-0202530 for T.S. The first author (J.B.) and the third author (T.S.) would, in addition, like to acknowledge the support of an AMS Centen- nial Fellowship (2004–2005) and a Sloan Research Fellowship (2008-2010), respectively. November 2015 xi Bibliography [ABDF11] G. Akemann, J. Baik, and P. Di Francesco (eds.), The Oxford handbook of random matrix theory, Oxford University Press, Oxford, 2011. MR2920518 (2012m:60007) [AD95] D. Aldous and P. Diaconis, Hammersley’s interacting