Random Matrices, Integrable Systems and Related Stochastic Processes

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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.