**IAS/Park City Mathematics Series**

Volume: 26;
2019;
498 pp;
Hardcover

MSC: Primary 15; 60; 82; 35; 46;

**Print ISBN: 978-1-4704-5280-3
Product Code: PCMS/26**

List Price: $104.00

AMS Member Price: $83.20

MAA Member Price: $93.60

**Electronic ISBN: 978-1-4704-5439-5
Product Code: PCMS/26.E**

List Price: $104.00

AMS Member Price: $83.20

MAA Member Price: $93.60

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# Random Matrices

Share this page *Edited by *
*Alexei Borodin; Ivan Corwin; Alice Guionnet*

A co-publication of the AMS and IAS/Park City Mathematics Institute

Random matrix theory has many roots and many branches in mathematics,
statistics, physics, computer science, data science, numerical
analysis, biology, ecology, engineering, and operations research. This
book provides a snippet of this vast domain of study, with a
particular focus on the notations of universality and
integrability. Universality shows that many systems behave the same
way in their large scale limit, while integrability provides a route
to describe the nature of those universal limits. Many of the ten
contributed chapters address these themes, while others touch on
applications of tools and results from random matrix theory.

This book is appropriate for graduate students and researchers
interested in learning techniques and results in random matrix theory
from different perspectives and viewpoints. It also captures a moment
in the evolution of the theory, when the previous decade brought major
break-throughs, prompting exciting new directions of research.

Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute.

#### Readership

Graduate students and researchers interested in random matrix theory and its many applications.

# Table of Contents

## Random Matrices

Table of Contents pages: 1 2

- Cover Cover11
- Title page i2
- Preface v6
- Introduction vii8
- Riemann–Hilbert Problems 114
- The Semicircle Law and Beyond: The Shape of Spectra of Wigner Matrices 4154
- The Matrix Dyson Equation and its Applications for Random Matrices 7588
- Counting equilibria in complex systems via random matrices 159172
- May model of a complex system: an introduction 159172
- Large-𝑁 asymptotics and large deviations for the Ginibre ensemble 169182
- Counting multiple equilibria via Kac-Rice formulas 177190
- Mean number of equilibria: asymptotic analysis for large deviations 183196
- Appendix: Supersymmetry and characteristic polynomials of real Ginibre matrices. 192205
- Exercises with hints 195208

- A Short Introduction to Operator Limits of Random Matrices 213226
- From the totally asymmetric simple exclusion process to the KPZ fixed point 251264
- The totally asymmetric simple exclusion process 251264
- Distribution function of TASEP 253266
- Determinantal point processes 258271
- Biorthogonal representation of the correlation kernel 263276
- Explicit formulas for the correlation kernel 274287
- The KPZ fixed point 283296
- The 1:2:3 scaling limit of TASEP 293306

- Delocalization of eigenvectors of random matrices 303316
- Microscopic description of Log and Coulomb gases 341354
- Introduction and motivations 342355
- Equilibrium measure and leading order behavior 348361
- Splitting of the Hamiltonian and electric approach 354367
- CLT for fluctuations in the logarithmic cases 363376
- Reexpressing the fluctuations as a ratio of partition functions 364377
- Transport and change of variables 364377
- Energy comparison 366379
- Computing the ratio of partition functions 367380
- Conclusion in the one-dimensional one-cut regular case 368381
- Conclusion in the two-dimensional case or in the general one-cut case 369382

- The renormalized energy 371384
- Large Deviations Principle for empirical fields 376389

- Random matrices and free probability 389402

Table of Contents pages: 1 2