Delocalization Transition for Critical Erdős-Rényi Graphs.

We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi 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 γ ( w ) of an eigenvector w , defined through ‖ w ‖ ∞ / ‖ w ‖ 2 = N - γ ( w ) . Our results remain valid throughout the optimal regime log N ≪ d ⩽ O ( log N ) .