Local universality of repulsive particle systems and random matrices

TL;DR

This paper demonstrates that certain repulsive particle systems exhibit local correlation behaviors in the bulk that match those of random Hermitian matrices, specifically following the sine kernel, despite lacking a spectral representation.

Contribution

It establishes the universality of local correlations in repulsive particle systems, linking them to random matrix theory without requiring a spectral structure.

Findings

Local correlations match sine kernel predictions in the bulk.

Results hold for systems without spectral or determinantal representations.

Provides a potential explanation for sine kernel appearance beyond classical random matrices.

Abstract

We study local correlations of certain interacting particle systems on the real line which show repulsion similar to eigenvalues of random Hermitian matrices. Although the new particle system does not seem to have a natural spectral or determinantal representation, the local correlations in the bulk coincide in the limit of infinitely many particles with those known from random Hermitian matrices; in particular they can be expressed as determinants of the so-called sine kernel. These results may provide an explanation for the appearance of sine kernel correlation statistics in a number of situations which do not have an obvious interpretation in terms of random matrices.