Empirical spacings of unfolded eigenvalues

Martin Venker,Kristina Schubert

doi:10.1214/ejp.v20-4436

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.

Highlights

The universal behaviour of eigenvalue statistics of random matrices has attracted much interest over the last decades
The belief has emerged that limit laws obtained in random matrix theory are ubiquitous in large systems of strongly correlated particles on the real line
The ubiquity of certain limit laws has been established within Random Matrix Theory (RMT) as the universality phenomenon, which means that for large matrix sizes many eigenvalue statistics exhibit the same limit distributions for essentially different matrix models, provided these models share the same symmetry

Summary

Introduction

The universal behaviour of eigenvalue statistics of random matrices has attracted much interest over the last decades. S2 provides a good approximation to Before we can further review results on the spacing distribution for invariant ensembles, we first need to introduce the notion of the equilibrium measure in order to rescale the particles It is well-known that under very mild assumptions on V and J, there is a measure μV on R with compact support that is the weak limit of the expected empirical spectral distribution of PN,V , i.e. for all continuous and bounded g : R → R. This shows that for expected spacings, an average over an increasing number of spacings as in (1.10) is not necessary to obtain the limiting distribution Tao proved this for Hermitian Wigner matrices and the result was later extended in [EY12] and [BFG15] to all symmetry classes and β-ensembles. This natural non-linear rescaling produces an ensemble with constant equilibrium density

Statement of Results

The convergence of EN

Proofs of remaining statements