The completely delocalized region of the Erdős-Rényi graph

We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability d N . We determine the full region of delocalization by determining the critical values of d log N down to which delocalization persists: for d log N > 1 log 4−1 all eigenvectors are completely delocalized, and for d log N > 1 all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [1, 3] that localized eigenvectors exist in the corresponding spectral regions.