Mean quantum percolation

TL;DR

This paper investigates the spectral properties of various random graphs, demonstrating the presence of a non-trivial continuous spectral component in supercritical regimes and introducing techniques to analyze the spectral measure.

Contribution

It develops two new methods to lower bound the continuous part of the spectral measure and applies them to prove the existence of continuous spectrum in supercritical random graphs.

Findings

Spectral measure of 2D bond percolation has a non-trivial continuous part in the supercritical phase.

Supercritical Erdős-Rényi graphs exhibit a continuous spectral component.

Examples of random graphs with purely continuous spectrum are provided.

Abstract

We study the spectrum of adjacency matrices of random graphs. We develop two techniques to lower bound the mass of the continuous part of the spectral measure or the density of states. As an application, we prove that the spectral measure of bond percolation in the two dimensional lattice contains a non-trivial continuous part in the supercritical regime. The same result holds for the limiting spectral measure of a supercritical Erdos-Renyi graph and for the spectral measure of a unimodular random tree with at least two ends. We give examples of random graphs with purely continuous spectrum.