Lifshitz tails for spectra of Erdős--Rényi random graphs

Oleksiy Khorunzhiy, Werner Kirsch, Peter Müller

Published 2005-02-27, updated 2006-03-14Version 3

We consider the discrete Laplace operator $\Delta^{(N)}$ on Erd\H{o}s--R\'{e}nyi random graphs with $N$ vertices and edge probability $p/N$. We are interested in the limiting spectral properties of $\Delta^{(N)}$ 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 $\Delta^{(N)}$ exhibits a Lifshitz-tail behavior at the lower spectral edge E=0.