Random Matrix Theory

随机矩阵理论

• 批准号: 1207961
• 负责人: Jun Yin
• 金额: $ 14万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207961&HistoricalAwards=false
• 关键词: Random; Matrix; Theory

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.

本课题旨在研究条目基本独立的随机矩阵的基本性质。主要目的是识别不同类型的随机矩阵集合的特征值密度、局部特征值统计量和特征向量分布。本文研究的系综包括Wigner矩阵、广义Wigner矩阵、带矩阵、Erdos-Renyi邻接矩阵、非厄米矩阵、协方差矩阵、广义协方差矩阵、Wigner或协方差矩阵的有限秩摄动、随机矩阵与确定性矩阵的和或积。本课题将研究协方差矩阵的普适性、局部循环律、带矩阵特征态的局域-离域跃迁、相关矩阵最大特征值的分布等开放性问题。对随机矩阵的局域统计量的严谨分析将有助于揭示高度相关多体系统的性质。协方差矩阵的研究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。对带矩阵的研究旨在建立半导体和其他无序体系的导电特性。