Edge isoperimetry and rapid mixing on matroids and geometric Markov chains

Abstract

We show how to bound the mixing time and log-Sobolev constants of Markov chains by bounding the edge-isoperimetry of their underlying graphs. To do this we use two recent techniques, one involving Average Conductance and the other log-Sobolev constants. We show a sort of strong conductance bound on a family of geometric Markov chains, give improved bounds for the mixing time of a Markov chain on balanced matroids, and in both cases find lower bounds on the log-Sobolev constants of these chains.