Conference Board of the Mathematical Sciences CBMS Regional Conference Series in

Number 130

Asymptotics of Random Matrices and Related Models The Uses of Dyson-Schwinger Equations

Alice Guionnet

with support from the 10.1090/cbms/130

Asymptotics of Random Matrices and Related Models The Uses of Dyson-Schwinger Equations

Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics

Number 130

Asymptotics of Random Matrices and Related Models The Uses of Dyson-Schwinger Equations

Alice Guionnet

Published for the Conference Board of the Mathematical Sciences by the

with support from the NSF-CBMS Regional Conference in the Mathematical Sciences on Dyson-Schwinger Equations, Topological Expansions, and Random Matrices held at Columbia University, New York, August 28–September 1, 2017

Partially supported by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

2010 Mathematics Subject Classification. Primary 60B20, 60F05, 60F10, 46L54.

For additional information and updates on this book, visit www.ams.org/bookpages/cbms-130

Library of Congress Cataloging-in-Publication Data Names: Guionnet, Alice, author. | Conference Board of the Mathematical Sciences. | National Science Foundation (U.S.) Title: Asymptotics of random matrices and related models : the uses of Dyson-Schwinger equa- tions / Alice Guionnet. Description: Providence, Rhode Island : Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2019] | Series: CBMS regional conference series in mathematics ; number 130 | “Support from the National Science Foundation.” | Includes bibliographical references and index. Identifiers: LCCN 2018056787 | ISBN 9781470450274 (alk. paper) Subjects: LCSH: Random matrices. | Matrices. | Green’s functions. | Lagrange equations. | AMS: Probability theory and stochastic processes – Probability theory on algebraic and topological structures – Random matrices (probabilistic aspects; for algebraic aspects see 15B52). msc | Functional analysis – Selfadjoint operator algebras (-algebras, von Neumann algebras, etc.) – Free probability and free operator algebras. msc — Probability theory and stochastic processes – Limit theorems – Central limit and other weak theorems. msc — Probability theory and stochastic processes – Limit theorems – Large deviations. msc Classification: LCC QA196.5 .G85 2019 | DDC 519.2/3–dc23 LC record available at https://lccn.loc.gov/2018056787

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Preface vii Chapter 1. Introduction 1 Chapter 2. The example of the GUE 9 Chapter 3. Wigner random matrices 19 3.1. Law of large numbers: Light tails 19 3.2. Law of large numbers: Heavy tails 26 3.3. CLT 30 Chapter 4. Beta-ensembles 35 4.1. Law of large numbers and large deviation principles 35 4.2. Concentration of measure 42 4.3. The Dyson-Schwinger equations 46 4.4. Expansion of the partition function 54 4.5. The Stieltjes transforms approach 57

Chapter 5. Discrete beta-ensembles 63 5.1. Large deviations, law of large numbers 64 5.2. Concentration of measure 66 5.3. Nekrasov’s equations 68 5.4. Second order expansion of linear statistics 75 5.5. Expansion of the partition function 76 Chapter 6. Continuous beta-models: The several cut case 79 6.1. The fixed filling fractions model 80 6.2. Central limit theorem for the full model 88

Chapter 7. Several matrix-ensembles 91 7.1. Non-commutative derivatives 92 7.2. Non-commutative Dyson-Schwinger equations 93 7.3. Independent GUE matrices 93 7.4. Several interacting matrices models 95 7.5. Second order expansion for the free energy 106 Chapter 8. Universality for beta-models 117 Bibliography 137 Index 143

v

Preface

Probability theory is based on the notion of independence. The law of large numbers and the central limit theorem describe the asymptotics of independent variables. However, in many instances one needs to deal with correlated variables, for instance in statistical mechanics. A tremendous effort was carried out to deal with such questions, for instance by developing large deviations theory. However, such general techniques often concern systems whose interaction is of the same order as entropy. These lecture notes are concerned with strongly interacting systems where the interaction overcomes the entropy. Examples of such situations are given by the eigenvalues of random matrices or the uniform tiling of a given domain. We will discuss a technique to deal with such systems: the asymptotic analysis of Dyson- Schwinger (or loop) equations. More specifically, we shall show how to use these techniques to derive the law of large numbers and the central limit theorems. The Dyson-Schwinger equations first showed up in physics. In random matrix theory, they were used to formally compute matrix integrals by solving the topological recursion in the work of Ambjorn, Eynard, and many others. Johansson was the first to use them to rigorously derive the central limit theorem for the empirical measure of the eigenvalues of Gaussian random matrices. Since this seminal work, Dyson-Schwinger equations have been used to derive central limit theorems in many more cases. When I was asked to give the Minerva Lecture Series at Columbia, I thought it was the right time to collect a few of them to highlight the general scheme of this approach. I unfortunately mostly took the time to discuss my own work in this direction, even though I had origianlly planned to cover more related topics such as the local laws derived by Erd¨os-Yau et al. or more general Coulomb gases as studied by Lebl´e and Serfaty. I hope, however, that these lecture notes will motivate the reader to read and find more applications to the asymptotic analysis of Dyson-Schwinger equations. I would like to thank Columbia University, and in particular Ivan Corwin and Andrei Okounkov, for giving me the opportunity to give these lectures. I also thank Jonathan Husson and Felix Parraud for carefully reading these lecture notes and giving me constructive feedback.

vii

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Index

Approximate transport map, 117 Topological recursion, 13 Azuma-Hoeffding’s inequality, 23 Trace, 91 binomial Jack measure, 76 Unitary Ensemble, 113 Universality, 117 Central limit theorem, 15, 52, 60, 73, 86, 94 Concentration of measure, 22, 42, 49, 66 Vandermonde determinant, 35 Continuous Beta-ensembles, 35 Vertices, 10, 96 Correlators, 4 Wigner’s theorem, 19 Discrete Beta-ensembles, 63 Dyson-Schwinger equation, 4, 11, 47, 58, 93, 98, 106, 115

Effective potential, 36, 66, 81 Empirical measure, 4

Free semi-circular variables, 93

GUE, 9, 10

Haar measure, 113 Heavy tails matrices, 26

Large deviation principle, 36, 65 Large deviations from the support, 40 Light tails entries, 19 Lozenge tilings, 63

Maps, 10, 93, 96, 101 Master operator, 47, 107, 122

Nekrasov’s equations, 69 Non-commutative derivatives, 92 Non-commutative law, 91

Off-critical potential, 49 Orthogonal Ensemble, 113

Planar Maps, 10 Planar maps, 101

Schur formula, 22 Semi-circle law, 17 Stieltjes transform, 20 Stirling formula, 66

Topological Expansion, 9

143 SELECTED PUBLISHED TITLES IN THIS SERIES

130 Alice Guionnet, Asymptotics of Random Matrices and Related Models, 2019 129 Wen-Ching , Zeta and L-functions in Number Theory and , 2019 128 Palle E.T. Jorgensen, Harmonic Analysis, 2018 127 Avner Friedman, Mathematical Biology, 2018 126 Semyon Alesker, Introduction to the Theory of Valuations, 2018 125 Steve Zelditch, Eigenfunctions of the Laplacian on a Riemannian Manifold, 2017 124 Huaxin Lin, From the Basic Homotopy Lemma to the Classification of C∗-algebras, 2017 123 Ron Graham and Steve Butler, Rudiments of Ramsey Theory, Second Edition, 2015 122 Carlos E. Kenig, Lectures on the Energy Critical Nonlinear Wave Equation, 2015 121 Alexei Poltoratski, Toeplitz Approach to Problems of the Uncertainty Principle, 2015 120 Hillel Furstenberg, Ergodic Theory and Fractal Geometry, 2014 119 Davar Khoshnevisan, Analysis of Stochastic Partial Differential Equations, 2014 118 Mark Green, Phillip Griffiths, and Matt Kerr, Hodge Theory, Complex Geometry, and Representation Theory, 2013 117 Daniel T. Wise, From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry, 2012 116 Martin Markl, Deformation Theory of Algebras and Their Diagrams, 2012 115 Richard A. Brualdi, The Mutually Beneficial Relationship of Graphs and Matrices, 2011 114 Mark Gross, Tropical Geometry and Mirror Symmetry, 2011 113 Scott A. Wolpert, Families of Riemann Surfaces and Weil-Petersson Geometry, 2010 112 Zhenghan Wang, Topological Quantum Computation, 2010 111 Jonathan Rosenberg, Topology, C∗-Algebras, and String Duality, 2009 110 David Nualart, Malliavin Calculus and Its Applications, 2009 109 Robert J. Zimmer and Dave Witte Morris, Ergodic Theory, Groups, and Geometry, 2008 108 Alexander Koldobsky and Vladyslav Yaskin, The Interface between Convex Geometry and Harmonic Analysis, 2008 107 FanChungandLinyuanLu, Complex Graphs and Networks, 2006 106 Terence Tao, Nonlinear Dispersive Equations, 2006 105 Christoph Thiele, Wave Packet Analysis, 2006 104 Donald G. Saari, Collisions, Rings, and Other Newtonian N-Body Problems, 2005 103 Iain Raeburn, Graph Algebras, 2005 102 Ken Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, 2004 101 Henri Darmon, Rational Points on Modular Elliptic Curves, 2004 100 Alexander Volberg, Calder´on-Zygmund Capacities and Operators on Nonhomogeneous Spaces, 2003 99 Alain Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, 2003 98 Alexander Varchenko, Special Functions, KZ Type Equations, and Representation Theory, 2003 97 Bernd Sturmfels, Solving Systems of Polynomial Equations, 2002 96 Niky Kamran, Selected Topics in the Geometrical Study of Differential Equations, 2002 95 Benjamin Weiss, Single Orbit Dynamics, 2000 94 David J. Saltman, Lectures on Division Algebras, 1999 93 Goro Shimura, Euler Products and Eisenstein Series, 1997

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/cbmsseries/. Probability theory is based on the notion of independence. The celebrated law of large numbers and the central limit theorem describe the asymptotics of the sum of indepen- dent variables. However, there are many models of strongly correlated random variables: for instance, the eigenvalues of random matrices or the tiles in random tilings. Classical tools of probability theory are useless to study such models. These lecture notes describe a general strategy to study

the fluctuations of strongly interacting random variables. Photography Frédéric Bellay/courtesy Galerie Le Réverbère - Lyon/France This strategy is based on the asymptotic analysis of Dyson- Schwinger (or loop) equations: the author will show how these equations are derived, how to obtain the concentration of measure estimates required to study these equations asymptotically, and how to deduce from this analysis the global fluctuations of the model. The author will apply this strategy in different settings: eigenvalues of random matrices, matrix models with one or several cuts, random tilings, and several matrices models.

For additional information and updates on this book, visit www.ams.org/bookpages/cbms-130

CBMS/130

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