A Dynamical Approach to Random Matrix Theory

ISBN-13: 9781470436483 / Angielski

Laszlo Erdos;Horng-Tzer Yau

Opis

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.

Spis treści

• Introduction
• 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
• References
• Index