Bulk eigenvalue statistics for random regular graphs

TL;DR

This paper proves that the eigenvalue statistics in the bulk of the spectrum for large random regular graphs match those of the Gaussian Orthogonal Ensemble, extending understanding of spectral universality.

Contribution

It establishes bulk eigenvalue universality for random regular graphs with degrees growing polynomially with the number of vertices.

Findings

Eigenvalue correlation functions match GOE predictions

Distribution of eigenvalue gaps aligns with GOE

Results hold for degrees in a specified polynomial range

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.