Local semicircle law for random regular graphs

TL;DR

This paper proves that the spectral properties of large random regular graphs closely follow the semicircle law at very fine scales, leading to eigenvector delocalization and quantum ergodicity implications.

Contribution

It establishes the local semicircle law for random regular graphs with degree at least $( ext{log} N)^4$, extending spectral analysis to optimal scales.

Findings

Green's function approximates semicircle law at optimal scale

Eigenvectors are completely delocalized

Quantum unique ergodicity holds probabilistically

Abstract

We consider random $d$-regular graphs on $N$ vertices, with degree $d$ at least $(\log N)^4$. We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by Wigner's semicircle law, down to the optimal scale given by the typical eigenvalue spacing (up to a logarithmic correction). Aside from well-known consequences for the local eigenvalue distribution, this result implies the complete (isotropic) delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity.