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

Abstract

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 $\beta \ge 1$ 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 $p_n=O(n^{2/3-\epsilon })$. The result is the analogous to the result of Sinai and Soshnikov for Wigner matrices, but the combinatorics involved in the calculations are different.

16 pages, 2 figures

References in corpus (4)

• Local semicircle law and complete delocalization for Wigner random matrices
• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• From Random Matrices to Stochastic Operators

Cited by in corpus (1)

• Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles