Random Matrices and Applications

在随机矩阵理论中，出现了各种各样的极限分布函数，其中许多是通用的。本课题研究了随机矩阵分布函数的普适性和渐近性。该项目将探索三种特定的情况，其中这些函数有望出现：随机匹配，非平衡相互作用粒子系统，和厄米矩阵模型与外部源。它们说明了随机矩阵理论出现的多样性。研究随机矩阵分布函数的渐近性质。对随机矩阵分布函数的内在性质的研究有望揭示随机矩阵理论的普遍性。随机矩阵理论中的分布函数确实描述了数学和其他科学领域中各种各样的对象。在统计学、物理学、经济学、金融学和电气工程中，一个复杂的系统通常用随机矩阵来建模。更重要的是，当系统的大小趋于无穷大时，一些没有用随机矩阵建模的系统确实表现出类似随机矩阵的行为，这是一种被称为随机矩阵的“普适性”的奇怪现象。这个项目将研究随机矩阵理论中出现的分布函数的一些基本性质，并进一步研究这些函数出现的实例，所有这些都是为了理解是什么使随机矩阵如此普遍。该项目将本科生和研究生纳入研究活动。

In random matrix theory, various limiting distribution functions arise and many of them are universal. This project studies the extent of the universality of random matrix distribution functions and also of asymptotic properties of these functions. The project will explore three specific situations in which such functions are expected to appear: random matchings, nonequilibrium interacting particle systems, and Hermitian matrix models with external sources. They illustrate the diversity of the appearance of random matrix theory. The principal investigator also intends to study the asymptotic properties of random matrix distribution functions. The study of the intrinsic properties of random matrix distribution functions is expected to shed light on the universal nature of random matrix theory. The distribution functions from random matrix theory indeed describe a wide variety of objects from both mathematics and other fields of science. In statistics, physics, economics, finance, and electrical engineering, a complicated system is often modeled in terms of random matrices. More importantly, some systems that are not modeled in terms of random matrices do exhibit random-matrix-like behavior when the size of the systems tends to infinity, a curious phenomenon that is known as the "universality" of random matrices. This project will study some of the basic properties of the distribution functions that arise in random matrix theory and also investigate further instances in which such functions arise, all in an effort to understand what it is that makes random matrices so universal. The project will incorporate undergraduates and graduate students into the research activities.