Lifshitz tails for spectra of Erd\H{o}s--R\'{e}nyi random graphs
Oleksiy Khorunzhiy, Werner Kirsch, Peter M\"uller

TL;DR
This paper studies the spectral properties of the Laplace operator on Erdős–Rényi random graphs in the subcritical regime, demonstrating Lifshitz-tail behavior at the lower spectral edge as the number of vertices grows large.
Contribution
It proves the Lifshitz-tail behavior of the expected integrated density of states for the Laplace operator on Erdős–Rényi graphs in the subcritical phase, a novel spectral analysis result.
Findings
Lifshitz-tail behavior at spectral edge E=0
Expected density of states exhibits exponential decay near zero
Results hold in the large N limit for subcritical p
Abstract
We consider the discrete Laplace operator on Erd\H{o}s--R\'{e}nyi random graphs with vertices and edge probability . We are interested in the limiting spectral properties of as in the subcritical regime where no giant cluster emerges. We prove that in this limit the expectation value of the integrated density of states of exhibits a Lifshitz-tail behavior at the lower spectral edge E=0.
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