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

Oleksiy Khorunzhiy·Werner Kirsch·Peter Müller
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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.

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Oleksiy Khorunzhiy, Werner Kirsch, Peter Müller. 2005-02-27. Lifshitz tails for spectra of Erd\H{o}s--R\'{e}nyi random graphs. https://doi.org/10.1214/1050516000000719

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