Circular law for sparse random regular digraphs

Fix a constant C≥ 1 and let d=d(n) satisfy d≤ln^C n for every large integer n. Denote by A_n the adjacency matrix of a uniform random directed d-regular graph on n vertices. We show that, as long as d→∞ with n, the empirical spectral distribution of appropriately rescaled matrix A_n converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in directed d-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of A_n based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between the matrix entries.