Particle Systems with Repulsion Exponent $\beta$ and Random Matrices

TL;DR

This paper introduces a generalized class of particle systems with variable repulsion exponents, extending the $eta$-Ensembles from random matrix theory, and demonstrates that their local bulk correlations are universal.

Contribution

It generalizes $eta$-Ensembles by allowing different interactions at larger distances while preserving local bulk correlation universality.

Findings

Local bulk correlations are universal in the new ensembles.

Particles exhibit $eta$-like repulsion at close distances.

Interaction differences at larger distances do not affect local universality.

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.