Bulk eigenvalue statistics for random regular graphs

Abstract

We consider the uniform random $d$-regular graph on $N$ vertices, with $d\in[N^{\alpha},N^{2/3-\alpha}]$ for arbitrary $\alpha>0$. 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.