Delocalization Transition for Critical Erdős-Rényi Graphs

We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi graph $$ , where d is of order $$ . 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 $$ of an eigenvector $$ , defined through $$ . Our results remain valid throughout the optimal regime $$ .