Graphical representations and cluster algorithms I. Discrete spin systems

Graphical representations similar to the FK representation are developed for a variety of spin-systems. In several cases, it is established that these representations have (FKG) monotonicity properties which enables characterization theorems for the uniqueness phase and the low-temperature phase of the spin system. Certain systems with intermediate phases and/or first-order transitions are also described in terms of the percolation properties of the representations. In all cases, these representations lead, in a natural fashion, to Swendsen-Wang-type algorithms. Hence, at least in the above-mentioned instances, these algorithms realize the program described by Kandel and Domany, Phys. Rev. B 43 (1991) 8539–8548. All of the algorithms are shown to satisfy a Li-Sokal bound which (at least for systems with a divergent specific heat) implies critical slowing down. However, the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points.