Local semicircle law at the spectral edge for Gaussian $\beta$-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 $n^{-2\mathrm{/}3+ϵ}$, the local semicircle law holds for all $\beta \ge 1$ 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 $p_n=O(n^{2\mathrm{/}3-ϵ})$. The result is the analogous to the result of Sinai and Soshnikov for Wigner matrices, but the combinatorics involved in the calculations are different.