UNIVERSITY OF WISCONSIN SYSTEM
This project explores problems connected to large scale properties of random matrices and interacting particle systems. The beta-ensemble is a one parameter family of distributions of random points on a line; for specific parameter values it describes the eigenvalues of some of the most famous classical random matrix ensembles. Understanding the scaling limit of these classical ensembles has been an important problem of random matrix theory. This project aims to build on the recent results of the PI and a collaborator in which the point process scaling limits of the beta-ensembles are described. The plan is to extract more information and to reach a better understanding of the limit process while exploring connections to other fields.
Interacting particle systems arise from various applications in several fields. Examples include growth and deposition models, traffic models, chemotaxis, the spreading of a disease. In recent decades a concerted effort has been made to provide mathematically rigorous analysis of such systems. One of the goals of this proposal is to analyze the equilibrium fluctuations in a certain family of interacting particle systems. The plan is to develop robust, model-independent methods to compare various particle systems and to work towards proving the universality of the scaling exponent in a broad class of models.
The project deals with problems related to large random systems with lots of components and non-trivial interactions. The treatment of these problems requires a wide range of tools which connects them to various other fields of mathematics besides probability, e.g.
combinatorics, complex analysis, operator theory, partial differential equations. Such systems occur in many other fields in addition to mathematics, for instance in physics, biology, engineering and economics.