Local Semicircle Law at the Spectral Edge for Gaussian β-Ensembles

Abstract

We study the local semicircle law for Gaussian β-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 β ≥ 1 at the edge. The proof of the main theorem relies on the calculation of the moments of the tridiagonal model of Gaussian β-ensembles up to the p n -moment, where \({p_n = O(n^{2/3-\epsilon})}\). The result is analogous to the result of Sinai and Soshnikov (Funct Anal Appl 32(2), 1998) for Wigner matrices, but the combinatorics involved in the calculations are different.