Bulk Universality of General $\beta$-Ensembles with Non-convex Potential

TL;DR

This paper proves the bulk universality of general beta-ensembles with non-convex potentials, extending previous results by removing the convexity assumption through a novel convexification technique.

Contribution

It introduces a convexified measure that preserves local statistics, enabling universality results for non-convex potentials in beta-ensembles.

Findings

Bulk universality established for non-convex potentials

Convexification technique preserves local statistics

Extends previous results beyond convex potentials

Abstract

We prove the bulk universality of the $\beta$-ensembles with non-convex regular analytic potentials for any $\beta>0$. This removes the convexity assumption appeared in our earlier work. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a "convexified measure" so that the local statistics are preserved under this convexification.