Cluster algorithms: Beyond suppression of critical slowing down

The cluster algorithm pioneered by Swendsen and Wang is widely acclaimed for its ability to suppress dynamic slowing down near a critical point, However, the cluster approach permits the formulation of Monte Carlo algorithms that yield important additional efficiency gains. For systems with long-range interactions, Luijten and Blote have introduced a method in which the number of operations per spin flip is independent of the number of interactions between a spin and the other spins in the system. Thus, the computational effort for the simulation of an N-particle system is reduced from O(N-2) to O(N), which has helped to resolve several open questions concerning critical behavior in systems with long-range interactions. As a second example of what can be achieved with cluster methods, we discuss some illustrative properties of a newly-developed geometric cluster algorithm for interacting fluids.