Particle systems with repulsion exponent $\beta$ and random matrices

Martin Venker

doi:10.1214/ecp.v18-2864

Abstract

We consider a class of particle systems generalizing the $\beta$-Ensembles from random matrix theory. In these new ensembles, particles experience repulsion of power $\beta>0$ when getting close, which is the same as in the $\beta$-Ensembles. For distances larger than zero, the interaction is allowed to differ from those present for random eigenvalues. We show that the local bulk correlations of the $\beta$-Ensembles, universal in random matrix theory, also appear in these new ensembles.

Highlights

Introduction and Main ResultsA central theme in random matrix theory is the universality phenomenon, which means that many essentially different matrix distributions lead in the limit of growing dimension to the same spectral statistics.In the past 15 years or so, much progress has been made in proving universality of local spectral distributions, especially correlations between neighboring eigenvalues in the bulk of the spectrum and of the largest eigenvalues
We consider a class of particle systems generalizing the β-Ensembles from random matrix theory
We show that the local bulk correlations of the β-Ensembles, universal in random matrix theory, appear in these new ensembles

Summary

Introduction

Introduction and Main ResultsA central theme in random matrix theory is the universality phenomenon, which means that many essentially different matrix distributions lead in the limit of growing dimension to the same spectral statistics.In the past 15 years or so, much progress has been made in proving universality of local spectral distributions, especially correlations between neighboring eigenvalues in the bulk of the spectrum and of the largest eigenvalues. There exists a constant αh ≥ 0 such that for all real analytic, strongly convex and even Q with αQ > αh, the following holds: The first correlation measure ρhN,,1Q,β converges weakly to a compactly supported probability measure μhQ,β which has a non-zero and continuous density on the interior of its support. Let G be the Gaussian potential G(t) := x2 and recall that the corresponding limiting measure μQ,β is the semicircle distribution (with a certain variance depending ecp.ejpecp.org on β).

Results

Conclusion