A semidefinite bound for mixing rates of Markov chains

Abstract

We study the method of bounding the spectral gap of a reversible Markov chain by establishing canonical paths between the states. We provide natural examples where improved bounds can be obtained by allowing variable length functions on the edges. We give a simple heuristic for computing good length functions. Further generalization using multicommodity flow yields a bound which is an invariant of the Markov chain, and which can be computed at an arbitrary precision in polynomial time via semidefinite programming. We show that, for any reversible Markov chain on n states, this bound is off by a factor of order at most log2n, and that this can be tight.