The completely delocalized region of the Erdős-Rényi graph

Abstract

We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability $\frac{d}{N}$. We determine the full region of delocalization by determining the critical values of $\frac{d}{logN}$ down to which delocalization persists: for $\frac{d}{logN}>\frac{1}{log4-1}$ all eigenvectors are completely delocalized, and for $\frac{d}{logN}>1$ 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.