Ergodicity of probabilistic cellular automata: A constructive criterion

We give a sequence of criteria (of increasing complexity) for the exponential ergodicity of discrete time interacting particle systems. Each criterion involves estimating the dependence on initial conditions of the process on finite space-time volumes. It generalizes and improves the existing single site condition and is the analog of the Dobrushin-ShlosmanC _v condition in equilibrium statistical mechanics. Our dynamic criterion may also be used to prove the uniqueness of Gibbs state in situations where theC _v condition fails. As a converse we prove that if there is a certain form of convergence to the stationary measure faster thann ^–d , wheren is the time andd is the dimension of the lattice, then our condition holds for some space-time volumes and hence the convergence must be exponentially fast.