We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with entries weighted by iid signs. Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the graph has degree linear in the number of vertices. Towards establishing the same result for the adjacency matrix without iid weights, we prove that it is invertible with high probability. Along the way we make use of Stein's method of exchangeable pairs to establish some graph discrepancy properties.
Random regular digraphs: singularity and spectrum