Local semicircle law in the bulk for Gaussian $β$-ensemble

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, Erdös, and Yau and in Bao and Su. Our approach is new and could be extended to other tridiagonal models.

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