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

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