Matrix liberation process I: Large deviation upper bound and almost sure convergence

TL;DR

This paper introduces the matrix liberation process, establishes a large deviation upper bound for its empirical distribution, and proves its almost sure convergence to the free probability liberation process as matrix size grows.

Contribution

It defines the matrix liberation process and provides the first large deviation upper bound, demonstrating convergence to free probability limits in large matrix regimes.

Findings

Established a large deviation upper bound for the empirical distribution

Proved almost sure convergence to the liberation process in large N limit

Characterized properties of the rate function in the large deviation principle

Abstract

We introduce the concept of matrix liberation process, a random matrix counterpart of the liberation process in free probability, and prove a large deviation upper bound for its empirical distribution with several properties on its rate function. As a simple consequence we obtain the almost sure convergence of the empirical distribution of the matrix liberation process to that of the corresponding liberation process as continuous processes in large $N$ limit.