Delocalization Transition for Critical Erdős–Rényi Graphs

We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph [Formula: see text] , where d is of order [Formula: see text] . 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 [Formula: see text] of an eigenvector [Formula: see text] , defined through [Formula: see text] . Our results remain valid throughout the optimal regime [Formula: see text] .