Lifshitz tails for spectra of Erdős–Rényi random graphs

Abstract

We consider the discrete Laplace operator Δ^(N) on Erdős–Rényi random graphs with N vertices and edge probability p/N. We are interested in the limiting spectral properties of Δ^(N) as N→∞ in the subcritical regime 0<p<1 where no giant cluster emerges. We prove that in this limit the expectation value of the integrated density of states of Δ^(N) exhibits a Lifshitz-tail behavior at the lower spectral edge E=0.