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 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 Copying and reprinting. 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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 https://www.ams.org/ 10987654321 242322212019 Contents 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 Bibliography [1] S. Albeverio, L. Pastur, and M. 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