Analyzing Glauber Dynamics by Comparison of Markov Chains

A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable conngurations which are based on local moves, or \Glauber dynamics." Typically these single site update algorithms are diicult to analyze, so often the Markov chain is modiied to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and Saloo-Coste to show that several of the natural single point update algorithms are eecient. The strategy is to relate the mixing rate of these algorithms to the corresponding non-local algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating tilings, colorings and independent sets.