Local semicircle law in the bulk for Gaussian $\beta$-ensemble

TL;DR

This paper establishes a local semicircle law at the optimal scale for Gaussian beta ensembles using a novel approach based on tridiagonal matrix representation and resolvent expansion.

Contribution

The authors introduce a new method for deriving the local semicircle law in Gaussian beta ensembles at the optimal scale, extending previous results with a different technique.

Findings

Established local semicircle law at scale n^{-1+δ}

Derived semicircle law at intermediate scale n^{-1/2+δ}

Method can be extended to other tridiagonal models

Abstract

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.