On quantum percolation in finite regular graphs

TL;DR

This paper investigates the spectral properties of quantum percolation on finite regular graphs, linking eigenvalues and eigenvector localization to infinite graph spectra, and shows that small percolation probabilities preserve absolutely continuous spectra on regular trees.

Contribution

It provides a quantitative relationship between finite and infinite graph spectra and demonstrates the persistence of absolutely continuous spectrum under small percolation on regular trees.

Findings

Eigenvalues and eigenvector delocalization relate to infinite graph spectra.

Small percolation probability preserves absolutely continuous spectrum on regular trees.

Results apply to large-girth regular graphs and Cayley graphs.

Abstract

The aim of this paper is twofold. First, we study eigenvalues and eigenvectors of the adjacency matrix of a bond percolation graph when the base graph is finite and well approximated locally by an infinite regular graph. We relate quantitatively the empirical measure of the eigenvalues and the delocalization of the eigenvectors to the spectrum of the adjacency operator of the percolation on the infinite graph. Secondly, we prove that percolation on an infinite regular tree with degree at least $3$ preserves the existence of an absolutely continuous spectrum if the removal probability is small enough. These two results are notably relevant for bond percolation on a uniformly sampled regular graph or a Cayley graph with large girth.