A Dynamical Approach to Random Matrix TheoryA 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. 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 |
217 | |
225 | |