Lifting Markov chains to speed up mixing

Abstract

There are several examples where the mixing time of a Markov chain can be reduced substantially, often to about its square root, by `lifting', i.e., by splitting each state into several states. In several examples of random walks on groups, the lifted chain not only mixes better, but is easier to analyze. We characterize the best mixing time achievable through lifting in terms of multicommodity flows. We show that the reduction to square root is best possible. If the lifted chain is time-reversible, then the gain is smaller, at most a factor of log(1/π0), where π0 is the smallest stationary probability of any state. We give an example showing that a gain of a factor of log(1/π0)/log log(1/π0) is possible.