Convergence of the density of states and delocalization of eigenvectors on random regular graphs

TL;DR

This paper investigates the spectral properties of random regular graphs, showing convergence of the density of states to that of an infinite tree and providing bounds on eigenvector delocalization.

Contribution

It proves the convergence of the integrated density of states and establishes uniform bounds on the rate of convergence for random regular graphs.

Findings

Density of states converges to that of the infinite regular tree.

Uniform bounds on the convergence rate are established.

Eigenvector delocalization results are derived from Green function estimates.

Abstract

Consider a random regular graph of fixed degree $d$ with $n$ vertices. We study spectral properties of the adjacency matrix and of random Schr\"odinger operators on such a graph as $n$ tends to infinity. We prove that the integrated density of states on the graph converges to the integrated density of states on the infinite regular tree and we give uniform bounds on the rate of convergence. This allows to estimate the number of eigenvalues in intervals of size comparable to $\log_{d-1}^{-1}(n)$. Based on related estimates for the Green function we derive results about delocalization of eigenvectors.