The completely delocalized region of the Erdos-Renyi graph

We analyse the eigenvectors of the adjacency matrix of the Erdos-Renyi 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/logN > 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.