Edge Universality of Beta Ensembles

TL;DR

This paper establishes the universality of spectral edge behavior in beta ensembles and generalized Wigner matrices under certain regularity conditions, extending previous results to broader classes of potentials.

Contribution

It proves edge universality for beta ensembles with $eta \,\geq 1$ and regular potentials, and extends bulk universality to $\,\mathscr{C}^4$ potentials.

Findings

Edge universality holds for beta ensembles with $eta \,\geq 1$.

Edge universality applies to generalized Wigner matrices across all symmetry classes.

Bulk universality extends to $\,\mathscr{C}^4$ potentials.

Abstract

We prove the edge universality of the beta ensembles for any $\beta\ge 1$, provided that the limiting spectrum is supported on a single interval, and the external potential is $\mathscr{C}^4$ and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class $\mathscr{C}^4$.