Universality of General $\beta$-Ensembles

TL;DR

This paper proves that the spacing distributions of log-gases with convex analytic potentials are universal across all inverse temperatures, matching those of Gaussian $eta$-ensembles, for any $eta>0$.

Contribution

It establishes the universality of $eta$-ensembles with convex analytic potentials for all $eta>0$, extending previous results to a broader class of potentials.

Findings

Spacing distributions match those of Gaussian $eta$-ensembles.

Universality holds for all $eta>0$ and convex analytic potentials.

Results apply to any inverse temperature $eta$, confirming broad universality.

Abstract

We prove the universality of the $\beta$-ensembles with convex analytic potentials and for any $\beta>0$, i.e. we show that the spacing distributions of log-gases at any inverse temperature $\beta$ coincide with those of the Gaussian $\beta$-ensembles.