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{\bf Catherine Greenhill}
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{\bf A Polynomial Bound on the Mixing Time of a Markov Chain for Sampling Regular Directed Graphs}
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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~[\emph{Annals of Applied
  Probability} 1991], 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~[\emph{Combinatorics, Probability and
  Computing} 2007], but with significant extension required.  

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