Decomposition Methods and Sampling Circuits in the Cartesian Lattice

Decomposition theorems are useful tools for bounding the convergence rates of Markov chains. The theorems relate the mixing rate of a Markov chain to smaller, derivative Markov chains, defined by a partition of the state space, and can be useful when standard, direct methods fail. Not only does this simplify the chain being analyzed, but it allows a hybrid approach whereby different techniques for bounding convergence rates can be used on different pieces. We demonstrate this approach by giving bounds on the mixing time of a chain on circuits of length 2n in ZZ.