This proposal has three main projects. The first, with Harold Widom, is to study the connection between the ``Airy process'' and integrable differential equations. The Airy process is a stochastic process that is expected to describe a wide class of growth processes. The distribution function for the Airy 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 the social 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 in combinatorics, 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 equations will 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.
