Emergence of extended states at zero in the spectrum of sparse random graphs

Abstract

We confirm the long-standing prediction that $\mathit{c}=\mathit{e}\approx 2.718$ is the threshold for the emergence of a nonvanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdős–Rényi random graph with average degree c. This is achieved by a detailed second-order analysis of the resolvent ${(\mathit{A}-\mathit{z})}^{-1}$ near the singular point $\mathit{z}=0$, where A is the adjacency operator of the Poisson–Galton–Watson tree with mean offspring c. More generally, our method applies to arbitrary unimodular Galton–Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution.