Singularity, Universality, and Smoothness of Random Walks

随机矩阵在数学物理、数据挖掘、随机噪声扰动、组合学、统计学和理论计算机科学等各个科学领域发挥着核心和基础作用。例如，Wigner半圆分布是在核物理研究中首次发现的，它是随机对称矩阵在大尺寸极限下的特征值的极限分布。本研究项目是对随机矩阵中奇异性和普适性现象的严谨的数学研究。该研究有望使人们对随机矩阵有更全面、更深入的认识，并对相关科学领域产生重大影响。从技术上讲，将研究随机系统的两个方面：奇点和普适性。这包括对维格纳矩阵特征值和随机多项式根的斥力的分析，特别是对矩阵特征值的某些维格纳型估计的研究，以及对随机多项式根在随机系数最小条件下的斥力现象。通用性方面的目标集中在Wigner矩阵的对数行列式的一个中心极限定理型结果和特征向量的某些等分布性质。

Random matrices play a central and fundamental role in various areas of science, including mathematical physics, data mining, random noise perturbation, combinatorics, statistics, and theoretical computer science. For example, the Wigner semicircle distribution, which arises as a limiting distribution of eigenvalues of random symmetric matrices in the large-size limit, was first observed in the study of nuclear physics. This research project is a rigorous mathematical study of the phenomena of singularity and universality in random matrices. This research is expected to lead to a more complete and deeper understanding of random matrices, with considerable impact on related areas of science. Technically speaking, two aspects of random systems will be studied: singularity and universality. This includes the analysis of the repulsion of Wigner matrix eigenvalues and of random polynomial roots, and specifically the study of certain Wegner-type estimates for matrix eigenvalues and, for random polynomial roots, the repulsion phenomenon under minimal conditions on the random coefficients. The goal under the universality aspect focuses on a central limit theorem-type result for the logarithmic determinant of Wigner matrices and certain equidistribution properties of the eigenvectors.