Exact Solvability in Random Matrices and Data Sciences

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The project focuses on random matrices - rectangular arrays of random numbers. While the primary focus is on theoretical properties of such matrices, the questions are motivated by applications in other sciences: economics, statistics, and physics. In data sciences, rectangular arrays appear whenever the observations are naturally arranged in two dimensions, for instance, in time and space. In quantum mechanics, matrices and related operators appear in modelling of a physical system. When the amount of available information about such a system is limited, a proper modelling is by taking the matrix to be random. This project provides research training opportunities for graduate students. Random matrices and their eigenvalues play a central role in many research areas, including high-energy physics, growth models, number theory, and high-dimensional statistics. The project revolves around exactly solvable or integrable families of random matrices, for which the eigenvalues are accessible through explicit formulas, actions of differential operations, orthogonal polynomials, and other essentially algebraic techniques. The goal of this project is three-fold: to search for these families, develop delicate asymptotic results about them (which usually go far beyond theorems available for generic systems), and use them for obtaining asymptotic predictions for much wider classes of random matrices and related objects of applied interest. The central objects gluing together different parts of the project are beta-ensembles, which are N-dimensional distributions uniting and generalizing the laws of eigenvalues of various random matrices. While in classical contexts the parameter beta takes values 1, 2, or 4, depending on whether the matrices under consideration are real, complex, or are quaternion matrices, this project emphasize a point of view in which beta is allowed to take arbitrary positive real values and should be interpreted as the inverse temperature in the terminology of statistical mechanics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.