# Moments and correlations of random matrices and symmetric function theory

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)

### Abstract

The central theme of this thesis is random matrices and their connections to combinatorics and probability theory. We present the results on correlations of eigenvalues for unitary invariant Hermitian ensembles, also called the $\beta=2$ Hermitian ensembles, using symmetric functions.

Classical compact groups such as the unitary group, the orthogonal group and the symplectic group have always been the representatives of $\beta = 2$ ensembles. These groups are computationally simple compared to other ensembles due to the compactness of support of the eigenvalues and the underlying representation theory. The group characters are symmetric functions in the eigenvalues. Many quantities relating to the correlations of eigenvalues, the notable ones being the joint moments of traces and joint moments of characteristic polynomials, can be effectively studied using the symmetric function theory and the representation theory of compact groups. Such a combinatorial approach to computing correlations is highly successful as it enables calculating the exact formulae and provides a route to compute large matrix asymptotics.

We develop a parallel theory for Hermitian ensembles, in particular for the Gaussian, Laguerre and Jacobi ensembles. We provide exact formulae for joint moments of traces and joint moments of characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces, we derive explicit asymptotic formulae for the rate of convergence of the moments of polynomial functions of Gaussian unitary matrices to those of a standard normal distribution when the matrix size tends to infinity.

We also calculate the asymptotics of the moments of characteristic polynomials of Hermitian ensembles, specifically the Gaussian unitary ensemble, as the matrix size tends to infinity. Our approach reveals that the even and odd dimensional Gaussian unitary matrices contribute differently to the moments and combine in a unique way to produce the semi-circle law.
Date of Award 28 Sep 2021 English The University of Bristol Francesco Mezzadri (Supervisor) & Jon Keating (Supervisor)

'