Delocalization Transition for Critical Erdos-Renyi Graphs

We analyse the eigenvectors of the adjacency matrix of a critical ErdosRenyi graph G(N, d/ N), where d is of order log N. We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent gamma(w) of an eigenvector w, defined through parallel to w parallel to(infinity)/parallel to w parallel to(2) = N-gamma(w). Our results remain valid throughout the optimal regime root log N << d <= O(log N).