On the skew-spectral distribution of randomly oriented graphs

Abstract

The randomly oriented graph $G_{n,p}^{\sigma }$ is an Erdős-Rényi random graph $G_{n,p}$ with a random orientation $\sigma$, which assigns to each edge a direction so that $G_{n,p}^{\sigma }$ becomes a directed graph. Denote by $S_n$ the skew-adjacency matrix of $G_{n,p}^{\sigma }$. Under some mild assumptions, it is proved in this paper that, the spectral distribution of $S_n$ (under some normalization) converges to the standard semicircular law almost surely as $n\to \infty$. It is worth mentioning that our result does not require finite moments of the entries of the underlying random matrix.