Orthogonal Polynomials and Random Matrices

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

It was the physicist Eugene Wigner who in the 1950's first used eigenvalues of random matrices to model the interactions of neutrons for heavy nuclei. Random matrices have since become a major research area with connections to mathematical physics, probability theory, number theory, numerical analysis, and orthogonal polynomials. Indeed, there is a well known anecdote about an interaction in the early 1970's between the physicist Freeman Dyson, and number theorist Hugh Montgomery, at Princeton, where their discussions led to the realization that there is a link between random matrices and the Riemann Zeta function of number theory. The PI's focus is on "universal" behavior of these random matrices: certain features seem to be independent of almost any underlying assumption, and consequently hold very generally. This has been known for a long time, and has been explored by both mathematical physicists and pure mathematicians. The techniques that have been developed to study this "universality" have been useful in many other areas of mathematics. This proposal will develop appropriate tools from orthogonal polynomials and classical analysis, and use these to establish "universal" features in as great a generality as possible. There will be also be an educational component to the project, involving collaboration with other researchers, organization of conferences, editorial duties, and supervision of undergraduate and/or graduate students. The specific goals of the project include investigating the ramifications and generalizations of a variational property recently established by the PI. It is hoped that this will enable one to establish universality limits for Hermitian ensembles under minimal conditions on measures with compact support, and also for varying measures. Monotonicity in the measure, and techniques of orthogonal polynomials are key tools in this approach. Somewhat more ambitious is the goal of extending the variational principle to beta-ensembles. This will include asymptotics of generalized Christoffel functions involving alternating polynomials in several variables. Discrete analogues will also be studied. Another major goal is developing the theory of Dirichlet orthogonal polynomials, and investigating their application to the Lindelof hypothesis for the Riemann Zeta function. Additional goals include investigating biorthogonal polynomials, and related numerical quadratures; orthogonal polynomials on curves in the plane, and Pade approximations.