Spectral Statistics of Random d-Regular Graphs

Abstract

In this thesis we study the uniform random $d$-regular graphs on $N$ vertices from a random matrix theory point of view. In the first part of this thesis, we focus on uniform random $d$-regular graphs with large but fixed degree. In the bulk of the spectrum down to the optimal spectral scale, we prove that the Green's functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original graphs. This result implies that the Kesten--McKay law holds for the empirical eigenvalue density down to the smallest scale and the bulk eigenvectors are completely delocalized. Our method is based on estimating the Green's function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs. In the second part of this thesis, we prove, for $1\ll d\ll N^{2/3}$, 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. In order to show this, we interpolate between the adjacent matrices of random $d$-regular graphs and the Gaussian orthogonal ensemble using a constrained version of Dyson Brownian motion.