# The completely delocalized region of the Erdős-Rényi graph

Johannes Alt, Raphael Ducatez, Antti Knowles

Johannes Alt, Raphael Ducatez, Antti Knowles

**7/9/21**Published in : arXiv:2109.03227

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}{\log N} down to which delocalization persists: for \frac{d}{\log N} > \frac{1}{\log 4 - 1} all eigenvectors are completely delocalized, and for \frac{d}{\log N} > 1 all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [arXiv:2005.14180, arXiv:2106.12519] that localized eigenvectors exist in the corresponding spectral regions.