Random matrixes: Eigenvalues distributions and Universality

随机矩阵理论的主要目标之一是研究关于特征值的分布。近年来，我们在这一领域的中心问题上取得了显著进展。国际和平协会有幸参与了其中的一些发展，他建议继续就这些主题进行研究。在这份提案中，他将讨论其中一些问题的现状，并建议研究几个研究问题，以使人们对这一主题有更完整和更深入的理解，特别是与普遍性现象有关的问题。除了研究传统的极限分布问题外，我们还将讨论该理论的非渐近方面，例如关于大偏差的问题。其中许多问题在理论计算机科学和数据挖掘等其他领域的应用中都是重要的。随机矩阵理论是一个丰富的数学主题。除了本身的趣味性，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基础性的作用。其中一个著名的例子是物理学家维格纳的研究，他利用随机矩阵的谱作为核物理的模型，从而发现了基本的半圆律。这项提案中的主要项目是研究随机矩阵谱的极限分布，在最精细的尺度上，受数学物理和概率学中长期存在的问题的推动。我们还相信，该项目中正在开发的方法将用于其他目的，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.