  The switch chain is a well-known Markov chain for sampling directed graphs
with a given degree sequence. While not ergodic in general, we show that it is
ergodic for regular degree sequences. We then prove that the switch chain is
rapidly mixing for regular directed graphs of degree d, where d is any positive
integer-valued function of the number of vertices. We bound the mixing time by
bounding the eigenvalues of the chain. A new result is presented and applied to
bound the smallest (most negative) eigenvalue. This result is a modification of
a lemma by Diaconis and Stroock, and by using it we avoid working with a lazy
chain. A multicommodity flow argument is used to bound the second-largest
eigenvalue of the chain. This argument is based on the analysis of a related
Markov chain for undirected regular graphs by Cooper, Dyer and Greenhill, but
with significant extension required.
