Critical Erd{\H o}s-R\'enyi digraph: all eigenvectors away from zero are delocalized

We consider the adjacency matrix of the directed Erd{\H o}s-Rényi graph.

As long as the expected degree is larger than the logarithm of the number of vertices, the graph is connected, we show that all eigenvectors are completely delocalized.

Below this critical scale, we prove eigenvector delocalization if the corresponding eigenvalue is away from zero.

This contrasts the \emph{undirected} or Hermitian setting, where large eigenvalues have localized eigenvectors [arXiv:2005.14180].

Our results also hold for sparse random matrices with independent entries, which can be viewed as weighted Erd{\H o}s-Rényi digraphs.