Hermitian, Non-Hermitian and Multivariate Finite Free Probability

Abstract

Finite free probability is a relatively new field that uses expected characteristic polynomials to study sums and products of unitarily invariant random matrices. In the first half of this dissertation we derive additive and multiplicative central limit theorems (CLTs) in finite free probability. We establish direct connections between these CLTs and CLTs in classical probability theory, random matrix theory, and free probability. We also define additive and multiplicative Brownian motions in finite free probability and relate them to point processes in statistical/quantum physics as well as to Brownian motions in classical probability theory, random matrix theory, and free probability. In the second half of this dissertation we introduce/define a new branch of finite free probability, which we call Multivariate/non-Hermitian finite free probability. We derive additive and multiplicative CLTs in Multivariate/non-Hermitian finite free probability and establish direct connections between these CLTs and CLTs in multivariate/complex classical probability theory, non-Hermitian random matrix theory, and multivariate/non-Hermitian free probability.