Gap universality of generalized Wigner and $\beta$-ensembles

Abstract

We consider generalized Wigner ensembles and general $\beta$-ensembles with analytic potentials for any $\beta \ge 1$ . The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian $\beta$-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any potential $C^4(\mathbb{R})$.