Research in Random Matrices and Integrable Systems

这项提案有三个主要项目。第一个是与哈罗德·维多姆一起研究“艾里过程”和可积微分方程之间的联系。艾里过程是一种随机过程，它被认为描述了一大类增长过程。一次艾里过程的分布函数是GUE-Tracy-Widom分布函数。在这种情况下，分布函数被表示为某一算子的Fredholm型行列式(“扩展艾利核”)，或表示为称为Painleve II的某一非线性常微分方程解。艾里过程的有限维分布函数(在许多不同时刻)也可表示为积分算子的Fredholm型行列式(“扩展艾利核”)。目标是找到相应的可积微分方程组，并用这些微分方程组来分析艾里过程。第二个项目，也是与Widom一起完成的，是通过确定所谓的“临界曲线”上的渐近性来完成关于周期Toda方程解的渐近性的早期工作。与研究生Momar Dieng合作的第三个主要项目是为随机矩阵模型GOE和GSE中的次大、次大等特征值的分布函数找到显式的Painleve类型表示。这些分布函数将应用于统计学。如果时间允许，将分析某些涉及Hall-Littlewood对称函数的组合和。著名的钟形曲线，更正式地称为高斯分布函数，因其在社会科学、物理和生物科学以及工程中的广泛应用而广为人知。二十世纪早期的数学家给出了精确的条件，在这些条件下人们可以期望找到高斯分布。现在，在这些学科中，应用这些条件(“独立随机变量之和”)来预测高斯分布的出现是很常见的。当这些条件失效并且我们处理的是强相依随机变量时，我们不能期望看到高斯分布。非常值得注意的是，近年来人们已经意识到，各种随机矩阵模型中的最大特征值的分布函数描述了出现在组合学、增长过程、随机平铺、排队理论、大数据集分析(主成分分析)以及量子点物理应用中的各种问题的新的普遍规律。这些分布函数称为Tracy-Widom分布函数，现在通过称为艾里过程的依赖时间的过程来实现。(艾里过程与布朗运动对高斯分布的作用相同。)这个项目的目标之一是找到表征艾里过程的微分方程式。这些微分方程将有助于分析艾里过程，就像常微分方程式(Painleve II)帮助描述Tracy-Widom分布函数一样。第二个项目是超越最大特征值分布函数，例如，考虑一类称为GOE和GSE的模型的次大特征值分布函数。GOE案例与多元统计特别相关，这些附加的分布函数有望应用于涉及大数据集的问题。

This proposal has three main projects. The first, with Harold Widom, is to study the connection between the ``Airy process'' and integrable differentialequations. The Airy process is a stochastic process that is expected to describe a wide class of growth processes. The distribution function for theAiry process at one single time is the GUE Tracy-Widom distribution function. In this case the distribution function is represented as either a Fredholm determinant of a certain operator (the ``Airy kernel'') or in terms of a solution to a certain nonlinear ordinary differential equation called Painleve II. The finite-dimensional distribution functions for the Airy process (at many different times) are also expressible as Fredholm determinants of an integral operator (the ``extended Airy kernel''). The goal is to find the corresponding integrable differential equations and to use these differential equations to analyze the Airy process. The second project,again with Widom, is to complete earlier work on the asymptotics of solutions to the periodic Toda equations by determining the asymptotics on what are called the ``critical curves.'' The third major project, with graduate student Momar Dieng, is to find explicit Painleve type representations for the distribution function for the next-largest, next-next largest, etc. eigenvalues in the random matrix models GOE and GSE. These distribution functions will have applications to statistics. If time permits certain combinatorial sums involving Hall-Littlewood symmetric functions will be analyzed.The famous bell-shaped curve, known more formally as the Gaussian distribution function, is well-known due to its many applications in thesocial sciences, the physical and biological sciences, and engineering. Mathematicians in the early part of the twentieth century gave precise conditions under which one can expect to find the Gaussian distribution. It is now common in these disciplines to apply these conditions (``sums of independent random variables'') to predict the appearance of the Gaussian distribution. When these conditions fail and we are dealing with strongly dependent random variables, we cannot expect to see the Gaussian. Quite remarkably it has been realized in recent years that the distribution functions of the largest eigenvalues in various random matrix models describe new universal laws for a wide variety of problems appearing incombinatorics, growth processes, random tilings, queuing theory, the analysis of large data sets (``principal component analysis'') as well as applications to the physics of quantum dots. These distribution functions, known as the Tracy-Widom distribution functions, are now realized in terms of a time dependent process called the Airy process. (The Airy process plays the same role as Brownian motion does to the Gaussian distribution.) One of the goals of this project is to find differential equations that characterize the Airy process. These differential equationswill facilitate analysis of the Airy process much in the same way that the ordinary differential equation (Painleve II) has aided in the description of the Tracy-Widom distribution functions. A second project is to go beyond the largest eigenvalue distribution functions and to consider, for example, the next-largest eigenvalue distribution functions for a class of models called GOE and GSE. The GOE case is particularly relevant to multivariate statistics and these additional distribution functions can be expected to find applications to problems involving large data sets.