Oleksiy Khorunzhiy, Werner Kirsch, Peter Mueller
Lifshits tails for spectra of Erdoes-Renyi random graphs
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ABSTRACT.  We consider the discrete Laplace operator $\Delta^{(N)}$ on Erdoes-Renyi 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 Lifshits-tail behaviour at the lower spectral edge $E=0$.