Lifshitz tails for spectra of Erd\H{o}s--R\'{e}nyi random graphs

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 $\Delta^{(N)}$ 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.