A Dynamical Approach to Random Matrix Theory

Front Cover
American Mathematical Soc., Aug 30, 2017 - Mathematics - 226 pages

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University

This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality.

This manuscript has been developed and continuously improved over the last five years. The authors have taught this material in several regular graduate courses at Harvard, Munich, and Vienna, in addition to various summer schools and short courses.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

 

Contents

Wigner Matrices and Their Generalizations
7
Invariant Ensembles
17
Universality for Generalized Wigner Matrices
29
Weak Local Semicircle Law
45
Proof of the Local Semicircle Law
61
Sketch of the Proof of the Local Semicircle
79
Fluctuation Averaging Mechanism
83
The Rigidity Phenomenon
99
Entropy and the Logarithmic Sobolev Inequality LSI
123
Universality of the Dyson Brownian Motion
151
Continuity of Local Correlation Functions
171
Universality of Wigner Matrices
183
Edge Universality
191
Further Results and Historical Notes
203
References
217
Index
225

Universality for Matrices with Gaussian Convolutions
109

Common terms and phrases

About the author (2017)

László Erdős: Institute of Science and Technology Austria, Klosterneuburg, Austria,
Horng-Tzer Yau: Harvard University, Cambridge, MA

Bibliographic information