The proposed research aims to expand our knowledge of random matrix ensembles with two species of eigenvalues. Among the ensembles to be studied are real asymmetric ensembles and ensembles of cosquare matrices. The latter ensembles consist of matrices whose eigenvalues are complex and satisfy an invariance property with respect to the unit circle. This invariance property is analogous to that of complex conjugation in real asymmetric ensembles, and the unit circle replaces the real line as an exceptional set in the study of eigenvalue statistics. Of particular interest is the study of the integrable structure of cosquare ensembles. Recent work by the PI on the integrable structure of real asymmetric ensembles suggests that cosquare ensembles have an analogous integrable structure. This discovery will further our understanding of ensembles with two (or more) different species of eigenvalues. Specific topics to be investigated include the joint eigenvalue density, a closed form for ensemble averages and correlation functions, as well as the limit law for the spectral radius (as the size of the matrices increases) and asymptotic results for the proportion of eigenvalues of unit modulus. The proposed research also aims to apply these results to Diophantine problems via the investigation of ensembles which arise naturally from heights of polynomials. The connection between polynomials and cosquare ensembles stems from the fact that the set characteristic polynomials of cosquare matrices is identical to the set of monic self-inversive polynomials. The latter arise naturally in the study of integer polynomials of small height.
An additional goal is the study of the integrable structure of Hermitian ensembles whose inverse temperature parameter (beta) is a square integer. Random matrix theory is a branch of mathematics which has application to fields as varied as physics, neuroscience, wireless communication and the study of prime numbers. This ubiquity arises via ?universal? mathematical models which are naturally applicable in many diverse contexts. The study of these universal models not only leads to new discoveries in their areas of application, but also leads to surprising connections between seemingly disparate areas of science and engineering. Random matrix theory is but one example of the pursuit of abstract mathematical knowledge leading to advances in applied sciences. The goal of the proposed research is the investigation of two particular universal models and their application to the study of the integers. One of these universal models also has applications in the study of random networks, dynamics and cortical electric activity; the proposed research will potentially further our understanding of these fields. The other proposed model is new and has application to the study of integers; past precedents suggest that this universal model will find eventual application in a wider variety of contexts.
