Bulk eigenvalue statistics for random regular graphs

Abstract

We consider the uniform random d-regular graph on N vertices, with d ∈ [N$^{\alpha}$,N$^{2/3−\alpha}$] for arbitrary α > 0. We prove that in the bulk of the spectrum the local eigenvalue correlation functions and the distribution of the gaps between consecutive eigenvalues coincide with those of the Gaussian Orthogonal Ensemble.