Local semicircle law at the spectral edge for Gaussian $\beta$-ensembles

TL;DR

This paper establishes a near-optimal local semicircle law at the spectral edge for Gaussian beta-ensembles, extending previous results for Wigner matrices with new combinatorial techniques.

Contribution

It proves the local semicircle law at the spectral edge for Gaussian beta-ensembles at a nearly optimal scale, using moment calculations of the tridiagonal model.

Findings

Law holds at scale n^{-2/3+ε} for all β ≥ 1

Moment calculations up to p_n = O(n^{2/3-ε}) are used

Extension of Sinai and Soshnikov's results to beta-ensembles

Abstract

We study the local semicircle law for Gaussian $\beta$-ensembles at the edge of the spectrum. We prove that at the almost optimal level of $n^{-2/3+\epsilon}$, the local semicircle law holds for all $\beta \geq 1$ at the edge. The proof of the main theorem relies on the calculation of the moments of the tridiagonal model of Gaussian $\beta$-ensembles up to the $p_n$-moment where $p_n = O(n^{2/3-\epsilon})$. The result is the analogous to the result of Sinai and Soshnikov for Wigner matrices, but the combinatorics involved in the calculations are different.