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

We confirm the long-standing prediction that $c=e\approx 2.718$ is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erd\H{o}s-Renyi random graph with average degree $c$. This is achieved by a detailed second-order analysis of the resolvent $(A-z)^{-1}$ near the singular point $z=0$, 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.