Lifshitz tails for spectra of Erdős--Rényi random graphs

Abstract

We consider the discrete Laplace operator ${\Delta }^{(N)}$ on Erdős--Rényi 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.

Published at http://dx.doi.org/10.1214/1050516000000719 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Spectral Analysis of Percolation Hamiltonians
• Spectral properties of the Laplacian on bond-percolation graphs
• Quantum site percolation on amenable graphs

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