Random Matrix Theory and Applications

古典概率论主要建立在没有或弱相关性的系统模型上;随机矩阵统计学则为高度相关的系统提供了唯一已知的一般规律。随机矩阵理论的核心问题是确定随机矩阵统计在随机系统中的普遍程度。在数学上，这个问题通常被称为普适性猜想，它断言随机矩阵统计量与矩阵元素的分布无关。研究者和合作者以前研究过这个猜想的特殊情况。该项目将进一步发展这一方向，并将研究普适性猜想在统计学，组合学，数学物理学和计算机科学中的应用，将概率论和其他领域的研究人员聚集在一起。该项目可能会导致与统计学家和计算机科学家在大型随机矩阵分析方面的富有成效的合作。该项目还将通过数学研究生参与概率论前沿课题的研究来支持培训。Wigner的愿景是用随机矩阵统计来模拟大关联量子系统的局部谱统计，这在概率论和统计物理学中是一个突破性的想法。研究人员和合作者先前证明了维格纳和β系综的局部谱统计的普遍性。这些结果是重要的第一步，在理解维格纳的愿景。该项目的目标是将这种理解扩展到具有重要物理特性的随机矩阵。将涉及以下四个项目：1。Dyson Brown运动的最优松弛时间。2. d-正则图的谱统计量在小d下的普适性，是统计学、计算机科学和大数据分析中的一个非常重要的课题。3.带矩阵的量子唯一遍历性和普适性。这个项目试图证明量子唯一遍历性和普适性本质上同时出现。4.库仑气体的局域密度涨落和线性统计。

Classical probability theory was built largely upon modeling systems with no or weak correlations; random matrix statistics, on the other hand, provide essentially the only known general laws for highly correlated systems. The central question for random matrix theory is to determine to what extent random matrix statistics prevail in random systems. Mathematically, this question is generally referred to as the universality conjecture, which asserts that random matrix statistics are independent of the distributions of the matrix elements. The investigator and collaborators previously studied special cases of this conjecture. This project will develop this direction further and will study applications of the universality conjecture to statistics, combinatorics, mathematical physics, and computer science, bringing together researchers from probability theory and these other fields. The project is likely to result in fruitful collaborations with statisticians and computer scientists on the analysis of large random matrices. The project will also support the training through research involvement of mathematics graduate students working on cutting-edge topics in probability theory.Wigner's vision that the local spectral statistics of large correlated quantum systems are modeled by random matrix statistics is a groundbreaking idea in probability theory and statistical physics. The investigator and collaborators previously proved the universality of the local spectral statistics for both Wigner and beta ensembles. These results are important first steps in understanding Wigner's vision. The project's goal is to extend this understanding to random matrices that have significant physical properties. The following four projects will be addressed: 1. Optimal relaxation time for Dyson Brownian motion. 2. Universality of spectral statistics of d-regular graphs for small d, which is a highly important topic in statistics, computer science, and large data analysis. 3. Quantum unique ergodicity and universality of band matrices. This project seeks to show that quantum unique ergodicity and universality appear essentially concurrently. 4. Local density fluctuation and linear statistics of Coulomb gases.