Eecient Exact Sampling from the Ising Model Using Swendsen-wang

In Monte Carlo Markov Chain theory, a particle on a Markov chain is run for a 'long time' until the particle is nearly in the stationary distribution. Coupling from the past is a methodology which allows samples to be taken exactly from the stationary distribution of the chain, but which requires that we have a completely coupling chain. One chain often used for the ferromagnetic Ising model is the Swendsen-Wang chain. We give the rst method for running Swendsen-Wang as a completely coupling chain, thereby giving a procedure for taking exact samples from the ferromagnetic Ising model using Swendsen-Wang. Moreover, this chain completely couples in polynomial time when a parameter of the chain is inversely proportional to the maximum degree of the graph, or when this parameter is close to 1. This has two implications. First, the Swendsen-Wang chain is rapidly mixing over this set of parameter values, a fact recently (and independently) shown by Cooper and Frieze using path coupling. Second, over this set of parameter values the exact sampling procedure will have an expected running time which is polynomial.