Universality Limits, Orthogonal Polynomials and Spaces of Entire Functions

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

The main goal of the project is to investigate new techniques for establishing universality limits for random matrices, in the unitary case. A probability distribution is placed on the space of n by n Hermitian matrices, leading to possibly dependent entries - as distinct from the case of independently distributed entries. One then studies the correlation of spacing of m-tuples of eigenvalues, especially when m is fixed, and the size n of the matrix approaches infinity. The conjectured limit, which in the bulk of the spectrum involves the sinc kernel, is called a universality limit, because it is independent of the point and the underlying measure. One main goal is to prove this universality in the bulk under minimal conditions on the underlying distribution, using new techniques developed by the investigator. Another more challenging goal involves analogous questions at the endpoints of the spectrum, or at points inside the spectrum, where the underlying distribution exhibits irregularities. Universality limits first rose to prominence in the work of the physicist Eugene Wigner. He was trying to model scattering of neutrons off heavy nuclei. Because of the large number of interactions, it seemed natural to try a probabilistic model. Remarkably, eigenvalues of random Hermitian matrices turned out to be an appropriate setting, and orthogonal polynomials were a key tool in the analysis. Subsequently random matrices have been connected with number theory, and the zeros of the Riemann zeta function, and also with other topics. The project's outcomes should also be of interest in other disciplines where random matrices arise. It is also hoped that graduate and undergraduate students will become involved in the research.