**CBMS Regional Conference Series in Mathematics**

Volume: 130;
2019;
143 pp;
Softcover

MSC: Primary 60; 46;

Print ISBN: 978-1-4704-5027-4

Product Code: CBMS/130

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

**Electronic ISBN: 978-1-4704-5194-3
Product Code: CBMS/130.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $49.50

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#### Supplemental Materials

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

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*Alice Guionnet*

A co-publication of the AMS and CBMS

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 independent
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. 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.

#### Readership

Graduate students and researchers interested in applications to the asymptotic analysis of Dyson-Schwinger equations.

#### Table of Contents

# Table of Contents

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

- Cover Cover11
- Title page iii4
- Preface vii8
- Chapter 1. Introduction 110
- Chapter 2. The example of the GUE 918
- Chapter 3. Wigner random matrices 1928
- Chapter 4. Beta-ensembles 3544
- Chapter 5. Discrete beta-ensembles 6372
- Chapter 6. Continuous beta-models: The several cut case 7988
- Chapter 7. Several matrix-ensembles 91100
- Chapter 8. Universality for beta-models 117126
- Bibliography 137146
- Index 143152
- Back Cover Back Cover1154