Random Matrix Theory

本课题主要研究元素基本独立的随机矩阵的基本性质。主要目的是识别许多不同类别的随机矩阵系综的特征值密度，局部特征值统计和特征向量分布。在这个项目中考虑的系综包括Wigner矩阵，广义Wigner矩阵，带矩阵，Erdos-Renyi邻接矩阵，非Hermitian（i.i.d.）矩阵）、协方差矩阵、广义协方差矩阵、Wigner或协方差矩阵的有限秩扰动、随机矩阵与确定性矩阵的和或/和积。本项目将研究协方差矩阵的普适性、局部循环律、能带矩阵本征态的定域-离域跃迁、相关矩阵最大本征值的分布等问题，对随机矩阵局部统计的严格分析将有助于揭示高关联多体系统的本质。协方差矩阵的工作将带来数学家和统计学家之间的合作，以更好地理解无线通信，经济现象和人口遗传学。能带矩阵的工作旨在建立半导体和其他无序系统的导电性质。

This project aims to study the basic properties of the random matrices whose entries are basically independent. The main objective is the identification of the eigenvalue densities, the local eigenvalue statistics and the eigenvector distributions of many different classes of random matrix ensembles. The ensembles considered in this project include Wigner matrix, generalized Wigner matrix, band matrix, Erdos-Renyi adjacency matrix, non-Hermitian (i.i.d. matrix), covariance matrix, generalized covariance matrix, finite rank perturbation of Wigner or covariance matrix, the sum or/and product of random matrix and deterministic matrix. Some open problems will be studied in this project, e.g. Universality of covariance matrix, local circular law, localization-delocalization transition of the eigenstates of band matrix, distribution of the largest eigenvalue of correlation matrix and etc.The rigorous analysis on the local statistics of random matrix will shed some light on the nature of the highly correlated many body system. The works on covariance matrix will bring collaboration between mathematicians and statistician to better understand wireless communication, economic phenomenon and population genetics. The works on band matrix aims to establish the conducting properties of semiconductors and other disordered systems.