Conference Board of the Mathematical Sciences C B M S 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 Asymptotics of Random Matrices and Related Models The Uses of Dyson-Schwinger Equations Conference Board of the Mathematical Sciences C B M S 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 Partiallysupported 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 informationand 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 ScienceFoundation(U.S.) Title: Asymptotics ofrandom matricesand relatedmodels : the uses of Dyson-Schwinger equa- tions/AliceGuionnet. Description: Providence,RhodeIsland: PublishedfortheConferenceBoardoftheMathematical Sciences by the American Mathematical Society, [2019] | Series: CBMS regional conference series in mathematics ; number 130 | “Support from the National Science Foundation.” | Includesbibliographicalreferencesandindex. Identifiers: LCCN2018056787|ISBN9781470450274(alk. paper) Subjects: LCSH:Randommatrices. |Matrices. |Green’sfunctions. |Lagrangeequations. |AMS: Probabilitytheoryandstochasticprocesses–Probabilitytheoryonalgebraicandtopological structures – Random matrices (probabilistic aspects; for algebraic aspects see 15B52). msc | Functionalanalysis–Selfadjointoperatoralgebras(-algebras,vonNeumannalgebras,etc.) – Freeprobabilityandfreeoperatoralgebras. msc—Probabilitytheoryandstochasticprocesses – Limit theorems – Central limit and other weak theorems. msc — Probability theory and stochasticprocesses–Limittheorems–Largedeviations. msc Classification: LCCQA196.5.G852019|DDC519.2/3–dc23 LCrecordavailableathttps://lccn.loc.gov/2018056787 Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For moreinformation,pleasevisitwww.ams.org/publications/pubpermissions. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. (cid:2)c 2019bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://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, suchgeneraltechniquesoftenconcernsystemswhoseinteractionisofthesameorder as entropy. These lecture notes are concerned with strongly interacting systems wheretheinteractionovercomestheentropy. Examplesofsuchsituationsaregiven by the eigenvalues of random matrices or the uniform tiling of a given domain. We willdiscussatechniquetodealwithsuchsystems: theasymptoticanalysisofDyson- 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-Schwingerequationshavebeenusedtoderivecentrallimittheoremsinmany 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 Erdo¨s-Yau et al. or more general Coulomb gasesasstudiedbyLebl´eandSerfaty. Ihope,however,thattheselecturenoteswill 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 AndreiOkounkov,forgivingmetheopportunitytogivetheselectures. Ialsothank Jonathan Husson and Felix Parraud for carefully reading these lecture notes and giving me constructive feedback. vii CHAPTER 1 Introduction 1.0.1. Some historical references. These lecture notes concern the study of the asymptotics of large systems of particles in very strong mean field interac- tion and in particular the study of their fluctuations. Examples are given by the distributions of eigenvalues of Gaussian random matrices, β-ensembles, random tilingsanddiscreteβ-ensembles, orseveralrandommatrices. Thesemodelsdisplay a much stronger interaction between the particles than the underlying randomness so that classical tools from probability theory fail. Fortunately, these models have in common that their correlators (basically moments of a large class of test func- tions) obey an infinite system of equations that we will call the Dyson-Schwinger equations. They are also called loop equations, Master equations or Ward identi- ties. Dyson-Schwinger equations are usually derived from some invariance or some symmetry of the model, for instance by some integration by parts formula. We shallargueinthesenotesthateventhoughtheseequationsarenotclosed, theyare often asymptotically closed (in the limit where the dimension goes to infinity) so that we can asymptotically solve them and deduce asymptotic expansions for the correlators. This in turn allows to retrieve the global fluctuations of the system, and eventually even more local information such as rigidity. This strategy has been developed at the formal level in physics [2] for a long time. In particular in the work of Eynard and collaborators [17,49–51], it was shown that if one assumes that correlators expand formally in the dimension N, then the coefficients of these expansions obey the so-called topological recursion. For instance, in [29,30], it was shown that assuming a formal expansion holds, Dyson-Schwinger equations induce recurrence relations on the terms in the expan- sion which can be solved by algebraic geometry means. These recurrence relations can even be interpreted as topological recursion, so that the coefficients of these expansions can be given combinatorial interpretations. In fact, it was realized in the seminal works of t’Hooft [88] and Br´ezin-Parisi-Itzykson-Zuber [43] that mo- mentsofGaussianmatricesandmatrixmodelscanbeinterpretedasthegenerating functions for maps. One way to retrieve this result is by using Dyson-Schwinger equations and checking that asymptotically they are similar to the topological re- cursion formulas obeyed by the enumeration of maps, as found by Tutte [92]. In this case, one first needs to analyze the limiting behavior of the system, given by theso-called equilibriummeasure or spectralcurve, andthentheDyson-Schwinger equations,thatisthetopologicalrecursion,willprovidethelargedimensionexpan- sion of the observables. The study of the asymptotics of our large system of particles also starts with the analysis of its limiting behaviour. I usually derive this limiting behaviour as theminimizerofanenergyfunctionalappearingasalargedeviationratefunctional [7], or in concentration of measure estimates [73], but, according to fields, people 1