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

A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs