Local semicircle law for the Gaussian beta ensembles

Abstract

In this thesis, we present two results, related to the mesoscopic behaviour of the Gaussian \beta-ensembles, at the edge and in the bulk respectively. For the first result, 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 analogous to the result of Sinai and Soshnikov for Wigner matrices, but the combinatorics involved in the calculations are different.

For the second result, we use the tridiagonal matrix representation to derive a local semicircle law for Gaussian \beta-ensembles at the optimal level of n^{-1+\delta} for any \delta > 0. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of n^{-1/2+\delta}; then an induction argument allows us to reach the optimal level. This result was obtained in a different setting, using different methods, by Bourgade, Erdos, and Yau and in Bao and Su. The approach is new and could be extended to other tridiagonal models.