Statistical Mechanics of Probabilistic Cellular Automata

Abstract

The necessary and sufficient conditions under which fully probabilistic cellular-automata (PCA) rules possess an underlying Hamiltonian (i.e., are "reversible") are established. It is argued that, even for irreversible rules, continuous ferromagnetic transitions in PCA with "up-down" symmetry belong in the universality class of kinetic Ising models. The nonstationary (e.g., periodic) states achieved for asymptotically large times by certain PCA rules in the (mean field) limit of infinite dimension are argued to persist in two and three dimensions, where fluctuations are strong.