Discrepancy properties for random regular digraphs

For the uniform random regular directed graph we prove concentration inequalities for (1) codegrees and (2) the number of edges passing from one set of vertices to another. As a consequence, we can deduce discrepancy properties for the distribution of edges essentially matching results for \er digraphs obtained from Chernoff-type bounds; in particular, we go beyond what can be obtained by comparison methods. The proofs make use of the method of exchangeable pairs, developed for concentration of measure by Chatterjee. Exchangeable pairs are constructed using two involutions on the set of regular digraphs: a well-known "simple switching" operation, as well as a novel "reflection" operation. The discrepancy properties proved in this paper were an important component in the recent proof by the author that random 0/1 matrices with constant row and column sum $d$ are invertible with high probability if $\min(d, n-d)=\omega(\log^2n)$.