Sparse regular random graphs: Spectral density and eigenvectors

TL;DR

This paper studies the spectral properties of sparse regular random graphs, showing convergence to the semicircle law and delocalization of eigenvectors as the graph size grows.

Contribution

It provides new theoretical results on spectral density convergence and eigenvector delocalization in sparse regular random graphs.

Findings

Spectral distribution converges to the semicircle law as degree increases.

Eigenvectors are delocalized with high probability.

Concentration estimates on eigenvalues over small intervals.

Abstract

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized.