Moments of characteristic polynomials and their derivatives in the classical compact ensembles

Abstract

The study of statistical properties of characteristic polynomials of random matrices from the classical compact ensembles is rich and diverse in applications. In this thesis, we study a variety of moments of characteristic polynomials, their derivatives and logarithmic derivatives, motivated by how our results can inform number theoretic analogues. We strive to make the moment parameter as general as possible, and the point where we are evaluating our moments as general as possible too.
The first obstacle of evaluating moments of the derivative of characteristic polynomials is that when we differentiate the characteristic polynomial, we lose the nice factorisation. The methods developed in this thesis allow us to get around the obstacle of expanding powers of sums. We note that in all the classical compact ensembles, the moments of the derivative of characteristic polynomials are proportional to the moments of the characteristic polynomials.
Our first results show how the asymptotics of integer moments of the logarithmic derivative of characteristic polynomials evaluated at a point approaching 1 from the USp(2N) and SO(N) are governed by the likelihood of matrices in those ensembles having an eigenvalue at or near the point 1.
Next, we have a handful of exact results of complex moments of the derivative of characteristic polynomials from USp(2N) and SO(2N) at the point 1, as well as exact small integer moments from U(N) at a real point s.
In the final chapter, we consider exact integer moments of the derivative of characteristic polynomials over U(N) evaluated on the unit circle. We give an exact representation involving a sum of determinants, prove a factorisation theorem and provide an exact form up to the 12th moment.

Date of Award	27 Sept 2022
Original language	English
Awarding Institution	University of Bristol
Supervisor	Nina C Snaith (Supervisor)