Asymptotics of Random Matrices and Related Models

154 Pages·2019·1.198 MB·English

by  Alice Guionnet

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
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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