Random matrixes: Eigenvalues distributions and Universality

随机矩阵理论的主要目标之一是研究特征值的分布。近年来，我们在这一领域的核心问题上取得了显著进展。私家侦探有幸参与了其中的一些发展，他希望继续在这些主题上进行研究。在这个提案中，他将讨论其中一些问题的现状，并提出研究几个研究问题，这些问题将导致对这一主题更全面和更深入的理解，特别是与普遍性现象有关的问题。除了研究关于极限分布的传统问题外，我们还将讨论理论的非渐近方面，例如关于大偏差的问题。这些问题中的许多在其他领域的应用中都很重要，例如理论计算机科学和数据挖掘。随机矩阵理论是数学中一个内容丰富的课题。除了本身有趣之外，随机矩阵在统计学、数学物理、组合学、理论计算机科学等各个领域都发挥着重要作用。一个著名的例子是物理学家维格纳的研究，他使用随机矩阵的谱作为核物理学的模型，从而发现了基本的半圆定律。本提案的主要项目是在最精细的尺度上研究随机矩阵谱的极限分布，这是由数学物理和概率论长期存在的问题所激发的。我们还相信，项目中开发的方法将对其他目的有用，例如研究大型（随机）网络和来自随机样本的大数据。

One of the main goals of the theory of random matrices is to study distributions concerning the eigenvalues. In recent years, we have witnessed notable progresses on central problems in this area. The PI has been fortunate to participate in some these developments and he would like to propose to continue his research on these topics. In this proposal, he is going to discuss the current state of some of these problems, and propose to study several research problems that would lead to a more complete and deeper understanding of the subject, especially problems related to the universality phenomenon. Beside studying traditional questions on limiting distributions, we will also discuss non-asymptotic aspects of the theory, e.g. problems concerning large deviations. Many of these problems are important in applications in other fields, such as theoretical computer science and data mining.The theory of random matrices is a rich topic in mathematics. Beside being interesting on their own right, random matrices play fundamental role in various areas such as statistics, mathematical physics, combinatorics, theoretical computer science, etc. A famous example here is the study of physicist Wigner, who used the spectrum of random matrices as a model in nuclear physics, and consequently discovered the fundamental semi-circle law. The main project in this proposal is to study limiting distribution the spectrum of random matrices, at the finest scale, motivated by long standing problems from mathematical physics and probability. We also believe that the methods being developed in the project will be useful for other purposes, such as the study of large (random) networks and large data from random samples.