Edge Universality of Beta Ensembles

Understand

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 $\mathscr{C}^4$.