Local semicircle law in the bulk for Gaussian $\beta$-ensemble
  We use the tridiagonal matrix representation to derive a local semicircle law for Gaussian beta ensembles at the optimal level of $n^{-1+\delta}$ for any $\delta > 0$. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of $n^{-1/2+\delta}$; 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. 
Philippe Sosoe, Percy Wong
math.PR