  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
Erd\H{o}s-R\'enyi digraphs obtained from Chernoff-type bounds. 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.
