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 $N\to\infty$ 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.