Erdős L., Yau H. A Dynamical Approach to Random Matrix Theory

Providence, American Mathematical Society, USA: 2017. — 239 p. — (Courant lecture notes in mathematics 28). — ISBN: 1470436485.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.Wigner Matrices and Their Generalizations
Eigenvalue Density
Invariant Ensembles
Universality for Generalized Wigner Matrices
Local Semicircle Law for Universal Wigner Matrices
Weak Local Semicircle Law
Proof of the Local Semicircle Law
Sketch of the Proof of the Local Semicircle Law Using the Spectral Gap
Fluctuation Averaging Mechanism
Eigenvalue Location: The Rigidity Phenomenon
Universality for Matrices with Gaussian Convolutions
Entropy and the Logarithmic Sobolev Inequality (LSI)
Universality of the Dyson Brownian Motion
Continuity of Local Correlation Functions Under the Matrix OU Process
Universality of Wigner Matrices in Small Energy Windows: GFT
Edge Universality
Further Results and Historical Notes