Local Kesten--McKay law for random regular graphs

TL;DR

This paper proves that the spectral density of large random regular graphs follows the Kesten--McKay law down to the smallest scales, showing eigenvector delocalization, by analyzing Green's functions and local graph structures.

Contribution

It establishes the local spectral law for random regular graphs at optimal scales, extending the understanding of eigenvalue distributions and eigenvector behavior.

Findings

Spectral density follows Kesten--McKay law at small scales

Eigenvectors are completely delocalized in the bulk spectrum

Green's functions can be approximated by those of tree-like graphs

Abstract

We study the adjacency matrices of random $d$-regular graphs with large but fixed degree $d$. In the bulk of the spectrum $[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$ 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 spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors. 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.