We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi graph
, 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
of an eigenvector w, defined through
\|\mathbf w\|_\infty /
\|\mathbf w\|_2 = N^{-\gamma(\mathbf w)}
|
. Our results remain valid throughout the optimal regime
\sqrt{\log N} \ll d \leq O(\log N)
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