Factoring graphs to bound mixing rates

This paper develops a new technique for bounding the mixing rate of a Markov chain by decomposing the state space into factors. The first application is an efficient Monte Carlo Markov chain algorithm for generating random three-colorings of 2-dimensional lattice regions. This provides a rigorous tool for studying some properties of the 3-state Potts model and the ice model from statistical mechanics. As a second application, we develop similar techniques to bound the mixing rate of a Metropolis sampling algorithm by a type of "temperature factorization". Both factorization theorems work by using known mixing properties of related Markov chains to establish the efficiency of a new sampling algorithm.