Dual Monte Carlo and cluster algorithms

We discuss the development of cluster algorithms from the viewpoint of probability theory and not from the usual viewpoint of a particular model. From the viewpoint of probability theory, we detail the nature of a cluster algorithm, make explicit the assumptions embodied in all clusters of which we are aware, and define the construction of free-cluster algorithms. We also illustrate these procedures by rederiving the Swendsen-Wang algorithm, presenting the details of the loop algorithm for a worldline simulation of a quantum S=1/2 model, and proposing a free-cluster version of the Swendsen-Wang replica method for the random Ising model. How the principle of maximum entropy might be used to aid the construction of cluster algorithms is also discussed