Stochastic cellular automata with Gibbsian invariant measures

A geometric characterization is given of a class of stochastic cellular automata, whose invariant probability measures are Gibbsian distributions (i.e. Markovian random fields). This class, whose members may be considered as the Cartesian product of convex polytopes with a finite number of vertices. contains all known automata of this type (e.g., the Metropolis, Gibbs sampler, and heat bath algorithms), and also a new family, which is characterized by having nonreversible dynamic behavior. The fact that there is a complete geometric structure, instead of isolated points, opens the possibility of using classical techniques (such as convex programming) for the design of optimal automata. Some examples of applications are given, oriented towards the solution of image restoration problems.