Cellular automata for contour dynamics

The dynamics of deterministic partially permutive cellular automata in two dimensions is considered. In particular we lay out the design principles and analyze the dynamics of generalized voter-type rules. They provide a two parameter model class of cellular automata which exhibit annealing, diffusive and critical domain behavior. We analyze the dynamics rigorously in certain cases and also show that all types of behavior are very close to their ideal counterparts observed in probabilistic models based on strong independence assumptions. This is further evidence that the statistical mechanics extends to cover also classes of purely deterministic dynamics.