Random Matrices and Interacting Systems

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

In the 1950's the physicist Eugene Wigner proposed the study of large random symmetric matrices in order to approximate the behavior of complicated self-adjoint operators. His goal was to understand the scaling properties of the eigenvalues of random matrices to get insight about the spectrum of certain operators arising from the study of heavy nuclei. In the last half century, random matrices found applications in a wide range of areas both within mathematics (e.g., combinatorics, number theory, and the study of interacting stochastic systems) and various other fields as well (e.g., information theory, financial mathematics, and RNA-folding). In the recent years, a new characterization has been established for limit laws of random matrices via the spectra of certain random self-adjoint differential operators. This provides a new connection between random matrices and self-adjoint operators more than half a century after Wigner's original idea. This research project explores this new area. This project builds on recent results of the investigator and collaborators on random operator representations of scaling limits of random matrices. The aim is to study the resultant random operators in order to learn more about the limit point processes. Questions under study include the application of the classical theory of differential operators to random matrices, the study of operator level convergence results for random matrix models with quantitative error bounds, and the investigation of the connection between random matrix limits and diffusions in the hyperbolic plane. The project also includes investigations related to the study of the large-scale behavior of interacting stochastic systems. It is conjectured that a wide family of one-dimensional interacting stochastic systems share an unusual scaling behavior with limit distributions related to random matrix theory; this is the Kardar-Parisi-Zhang (KPZ) universality class. The investigator intends to explore various models belonging (or conjectured to belong) to the KPZ universality class, with a special focus on directed polymers, a model describing random paths in a space-time environment that is also random.