Local semicircle law in the bulk for Gaussian β-ensemble

Abstract

We use the tridiagonal matrix representation to derive a local semicircle law for Gaussian beta ensembles at the optimal level of n-1+δ for any δ > 0. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of n-1/2+δ; then an induction argument allows us to reach the optimal level. This result was obtained in a different setting, using different methods, by Bourgade, Erd\"os, and Yau and in Bao and Su. Our approach is new and could be extended to other tridiagonal models.