Empirical Spacings of Unfolded Eigenvalues

TL;DR

This paper proves the convergence of empirical eigenvalue spacings in random matrix ensembles, emphasizing practical relevance and using unfolding techniques to analyze all spacings with optimal convergence rates.

Contribution

It extends previous results by focusing on empirical distributions, employing unfolding for uniform density, and establishing strong bulk universality with optimal convergence rates.

Findings

Empirical eigenvalue spacings converge as the number of points increases.

Unfolding transforms ensembles for uniform density, enabling analysis of all spacings.

Bounds on convergence rates are established for the empirical distribution.

Abstract

We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the empirical distribution of nearest neighbor spacings. We extend existing results for the spacing distribution in two ways. On the one hand, we believe the empirical distribution to be of more practical relevance than the so far considered expected distribution. On the other hand, we use the unfolding, a non-linear rescaling, which transforms the ensemble such that the density of particles is asymptotically constant. This allows to consider all empirical spacings, where previous results were restricted to a tiny fraction of the particles. Moreover, we prove bounds on the rates of convergence. The main ingredient for the proof, a strong bulk universality result for correlation functions in the unfolded setting including optimal rates, should be of independent interest.