Lifshitz tails for spectra of Erdos-Rényi random graphs

Abstract

We consider the discrete Laplace operator Δ (N) on Erdos-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. © Institute of Mathematical Statistics, 2006.