Singularity, Universality, and Smoothness of Random Walks

随机游走的奇异性、普适性和平滑性

• 批准号: 1600782
• 负责人: Hoi Nguyen
• 金额: $ 13.69万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600782&HistoricalAwards=false
• 关键词: Singularity; Universality; Smoothness; Random; Walks

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.

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中发挥着核心和基础作用。例如，维格纳半圆分布是在核物理研究中首次观察到的，它是随机对称矩阵特征值在大尺寸极限下的极限分布。该研究项目是对随机矩阵中奇异性和普遍性现象的严格数学研究。 这项研究有望带来对随机矩阵更完整、更深入的理解，对相关科学领域产生相当大的影响。从技术上讲，将研究随机系统的两个方面：奇异性和普遍性。这包括对维格纳矩阵特征值和随机多项式根的排斥的分析，特别是对矩阵特征值的某些韦格纳型估计的研究，以及对于随机多项式根的随机系数最小条件下的排斥现象的研究。普适性方面的目标集中于维格纳矩阵的对数行列式的中心极限定理类型结果和特征向量的某些等分布性质。

项目成果

Normal vector of a random hyperplane

随机超平面的法向量

• 发表时间: 2018
• 期刊: International mathematics research notices
• 影响因子: 1
• 作者: Nguyen, Hoi;Vu, Van
• 通讯作者: Vu, Van

Random matrices: overcrowding estimates for the spectrum

随机矩阵：频谱的过度拥挤估计

• 发表时间: 2018
• 期刊: Journal of functional analysis
• 影响因子: 1.7
• 作者: Nguyen, Hoi