Circular law for sparse random regular digraphs

Fix a constant C ≥ 1 and let d = d(n) satisfy d ≤ lnC n for every large integer n. Denote by An the adjacency matrix of a uniform random directed d-regular graph on n vertices. We show that if d → ∞ as n → ∞, the empirical spectral distribution of the appropriately rescaled matrix An 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 a 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 An based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between matrix entries.