Edge isoperimetry and rapid mixing on matroids and geometric Markov chains

Authors:
Ravi Montenegro;Jung-Bae Son
Affiliations:
Department of Mathematics, Yale University, New Haven, CT;Division of Informatics, King's Buildings, University of Edinburgh, Edinburgh EH9 3JZ, UK
Venue:
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Year:
2001

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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.