Local semicircle law in the bulk for Gaussian $\beta$-ensemble

Abstract

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, Erdös, and Yau and in Bao and Su. Our approach is new and could be extended to other tridiagonal models.

31 pages, version 3 corrected typos and expanded sections 4-5

References in corpus (4)

• Local semicircle law and complete delocalization for Wigner random matrices
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• From Random Matrices to Stochastic Operators
• Bulk Universality of General $\beta$-Ensembles with Non-convex Potential

Cited by in corpus (3)

• Optimal local law and central limit theorem for $\beta$-ensembles
• Local density for two-dimensional one-component plasma
• Convergence of the eigenvalue density for beta-Laguerre ensembles on short scales