On the switch Markov chain for perfect matchings

We study a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph. This Markov chain was proposed by Diaconis, Graham and Holmes as a possible approach to a sampling problem arising in Statistics. They considered several classes of graphs, and conjectured that the switch chain would mix rapidly for graphs in these classes. Here we settle their conjecture almost completely. We ask: for which graph classes is the Markov chain ergodic and for which is it rapidly mixing? We provide a precise answer to the ergodicity question and close bounds on the mixing question. We show for the first time that the mixing time of the switch chain is polynomial in the class of monotone graphs. This class was identified by Diaconis, Graham and Holmes as being of particular interest in the statistical setting.