Random Matrix Theory and Applications

经典概率论在很大程度上建立在没有或弱相关的系统建模上；另一方面，随机矩阵统计基本上为高度相关系统提供了唯一已知的一般规律。随机矩阵理论的中心问题是确定随机矩阵统计在多大程度上适用于随机系统。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。研究者和合作者先前研究了这一猜想的特殊情况。该项目将进一步发展这一方向，并将研究普惠猜想在统计学、组合学、数学物理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。该项目很可能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。该项目还将通过参与研究概率论前沿课题的数学研究生来支持培训。维格纳提出的将大关联量子系统的局部谱统计用随机矩阵统计建模的观点是概率论和统计物理学的一个突破性思想。研究者和合作者先前证明了Wigner和beta系综的局部谱统计的普遍性。这些结果是理解维格纳设想的重要的第一步。该项目的目标是将这种理解扩展到具有重要物理性质的随机矩阵。将处理以下四个项目：戴森布朗运动的最佳松弛时间。2. 小d正则图谱统计的普适性，是统计学、计算机科学和大数据分析中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。这个项目试图表明量子独特的遍历性和普适性本质上同时出现。4. 库仑气体的局部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.