Abstract
Motivated by the Swendsen-Wang algorithm for the Ising model with no external magnetic field the authors consider a particle system on a discrete lattice evolving according to the following parallel algorithm: at each time step connected clusters of particles are removed with probability 1/2 and at each empty site a particle is created with probability p. Due to the presence of arbitrarily large clusters the interaction can have an arbitrary range. In the one-dimensional case they solve the model completely by means of a novel path expansion in spacetime. In the two-dimensional case they show that, due to strong time correlation, the probability of having a large cluster of particles does not rapidly decay exponentially in the cardinality of the cluster even for p very small. They then prove ergodicity for 1-p close to zero.
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