Random Matrices, Integrable Systems and Related Stochastic Processes

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

The main activity of the project is the analysis of new limit laws and their associated integrable 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 ideas and techniques coming from random matrix theory, integrable systems and operator theory. An operator theory application to integrable systems would establish new limit theorems for a class of operator determinants providing solutions to the cylindrical Toda equations in so-called critical cases. The methods are related to those used in the study of truncated Wiener-Hopf and Toeplitz operators. Many physical systems possess such complicated behavior that exact predictions become impossible. Random matrix theory provides mathematical models that allow a simulation of such behavior and predictions that allow comparison with experiment. This was its original motivation, but it has since had significant applications in other areas of mathematics, science and technology, in such diverse subjects as communications, probability, statistics, number theory, condensed matter physics, and engineering. One can anticipate that ideas from random matrix theory and techniques developed in part by the PI and his collaborators will be instrumental in the study of the stochastic processes describe above. Further, the project should provide, through the integrable differential equations, implementable numerical algorithms to compute properties of these processes. This last aspect is most important for applications.