Applications of multivariate orthogonal polynomials

Random Matrix Theory (RMT) arose in the 1950s out of a need for a model to predict the spectral behaviour of systems that were too complex to describe precisely. Investigations of large matrices whose entries were random variables were able to reproduce features of complex quantum systems that were universal---that is to say, do not depend on the precise details of the system under consideration, but more general features such as whether or not time-reversal symmetry is present. Although RMT was introduced to study complex quantum systems, for example excitation levels of large atomic nuclei, it was observed that spectral statistics of a single particle could be predicted using RMT, provided that the classical dynamics are chaotic.

This leads to the Random Matrix Conjecture, still not fully resolved to this day, that the spectral statistics of a chaotic system with time-reversal symmetry, correspond exactly to the large matrix size limit of Gaussian symmetric matrix statistics; if time-reversal symmetry is absent then the spectral statistics instead are Gaussian Hermitian. RMT has also been observed in the distribution of zeros of the famous{Riemann zeta function. This correspondence was given theoretical evidence by Montgomery and established beyond reasonable doubt by a virtuoso effort of numerical computations performed by Odlyzko.

The objective of this project will be to develop new tools and methods for RMT, based on the theory of Jack polynomials. These are multivariate homogeneous symmetric polynomials that are eigenfunctions of certain partial differential operators. Their importance in RMT comes from an integration formula due to Kadell for a Jack polynomial against measures naturally occurring in RMT that can be explicitly evaluated.

For other ensembles of random matrices the multivariate orthogonal polynomials are expected to appear in analogous ways. These multivariate polynomials are expanded in the basis of Jack polynomials. The utility of Jack polynomials and multivariate orthogonal polynomials has only recently started to be realised, and there are numerous open problems that are ripe for a Ph.D. thesis.

This project will aim to solve the following problems about multivariate orthogonal polynomials: There are potentially many useful applications to knowing answers to these problems. For example a good solution to problem 1) would lead directly to better evaluations of certain integrals used in RMT. 94% of Loughborough’s research impact is rated world-leading or internationally excellent.

REF 2021