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.
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 the 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’s goal was to develop appropriate tools from orthogonal polynomials and classical analysis, and use these to establish "universal" features in as great a generality as possible. There was also 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 included establishing universality limits in as general a context as possible. This was achieved by showing that for measures on the real line with arbitrary compact support, universality does hold in some sense, namely in the sense of convergence in measure. In addition, a variational property was established for the m point correlation function arising from ensembles associated with Hermitian matrices. Some progress was made for beta ensembles associated with certain discrete measures. In addition, universality limits were established in a number of settings, including a multivariate setting, on subarcs of the unit circle, and for rational orthogonal functions. Other foci of the project included orthogonal and biorthogonal polynomials, and aspects of approximation theory. Asymptotics were obtained as the degree of the polynomials increase to infinity. Some progress was also made on shape preserving approximation for weighted polynomials. The results were disseminated in papers and at conferences. Over 35 papers were published or accepted during the period of support, and several plenary conference talks were given. In addition, two undergraduate students were supervised for summer research projects, and Ulfar Stefansson completed his Ph.D. The PI also co-organized a major international conference Constructive Functions 2014, held at Vanderbilt University in May 2014.
