Random Matrices, Integrable Systems and Related Stochastic Processes

该项目的主要目标和活动是分析出现在各种随机过程中的新极限定律及其相关微分方程。这些随机过程包括Airy过程、Pearcey过程及其高阶推广、不相交的布朗漂移路径模型和一类称为非对称排斥过程的相互作用粒子系统。所采用的主要方法是来自随机矩阵理论，可积系统和算子理论的思想和技术的组合。随机矩阵理论位于数学的几个分支的交叉点，例如概率论，数论，组合学;应用范围从物理学（量子混沌，凝聚态物理学）到电子工程（无线通信）到统计学（高维数据分析）。对这些应用特别重要的是发现了新的普遍极限定律。这些极限定律是由分布函数描述的，现在可以用特殊的“可积”函数来计算。本计画将进一步分析这些不同的极限定律，描述其他定律出现的情况，并提供实现这些定律的计算方法。

The main objective and activity of this project is the analysis of new limit laws and their associated differential equations that appear in a variety of stochastic processes. These stochastic processes include the Airy process, the Pearcey process and its higher order generalizations, a nonintersecting Brownian excursion path model, and a class of interacting particle systems called the asymmetric exclusion process. The main methods to be employed are a combination of the ideas and techniques coming from random matrix theory, integrable systems, and operator theory. Random matrix theory lies at the intersection of several branches of mathematics, e.g. probability theory, number theory, combinatorics; with applications ranging from physics (quantum chaos, condensed matter physics) to electrical engineering (wireless communications) to statistics (high dimensional data analysis). Particularly important for these applications have been the discovery of new universal limit laws. These limit laws are described by distribution functions which are now computable in terms of special "integrable" functions. This project will further analyze these various limit laws, describe circumstances in which other laws arise, and provide computational methods to implement these laws.