Aggregate Path Coupling: Higher Dimensional Theory

Abstract

In this chapter, we extend the aggregate path coupling technique derived in the previous section for the Blume-Capel model to a large class of statistical mechanical models that is disjoint from the mean-field Blume-Capel model. The aggregate path coupling method presented here extends the classical path coupling method for Gibbs ensembles in two directions. First, we consider macroscopic quantities in higher dimensions and find a monotone contraction path by considering a related variational problem in the continuous space. We also do not require the monotone path to be a nearest-neighbor path. In fact, in most situations we consider, a nearest-neighbor path will not work for proving contraction. Second, the aggregation of the mean path distance along a monotone path is shown to contract for some but not all pairs of configurations. Yet, we use measure concentration and large deviation principle to show that proving contraction for pairs of configurations, where at least one of them is close enough to the equilibrium, is sufficient for establishing rapid mixing.