Spacings - An Example for Universality in Random Matrix Theory

TL;DR

This paper explores the universality of local eigenvalue statistics in Random Matrix Theory, focusing on empirical spacing distributions and their convergence to universal limits through analytical and numerical results.

Contribution

It provides new analytical and numerical results on the empirical spacing distribution and its convergence, including a detailed explanation of foundational concepts in Random Matrix Theory.

Findings

Empirical spacing distribution converges to the universal limit.

Numerical simulations support the analytical results.

A sketch of the proof for weak convergence is presented.

Abstract

Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the empirical spacing distribution and its Kolmogorov distance from the universal limit. We describe new results, some analytical, some numerical, that are contained in [27]. A large part of the paper is devoted to explain basic definitions and facts of Random Matrix Theory, culminating in a sketch of the proof of a weak version of convergence for the empirical spacing distribution $\sigma_N$.