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

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+is an element of), the local semicircle law holds for all beta >= 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-is an element of)). 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.