Cluster Algorithms for Discrete Models of Colloids with Bars

Colloids are mixtures of two different types of molecules. The model has a hard-core constraint forcing all the molecules to occupy non-overlapping positions, but there are no additional interactions between molecules; all non-overlapping arrangements are equally likely. It is believed that colloids undergo a phase transition whereby at low density the two types of molecules will be uniformly interspersed, while at high density large clusters will form and the two types of molecules will effectively separate. While local algorithms are not believed to work at or beyond the critical point, an algorithm due to Dress and Krauth [3] offers an alternative approach to sampling potentially beyond the critical point where clusters begin to form. We study the DK algorithm on a colloid model consisting of long bars and small diamonds on the periodic lattice Z2n. We show that if we restrict the model to allow at most one bar in each column of the lattice region, then local algorithms are slow, but the DK algorithm is provably efficient (if the bars are long enough). However, we show that when we allow any number of bars per column, the DK algorithm also requires exponential time to reach equilibrium.