Randomly interacting particle systems: The uniqueness regime

We introduce spatial disorder in a large system of interacting particles th at evolve according to a non-reversible dynamical law. We show that if the regions where the components strongly interact are scarce, several general properties of the discrete and continuous time dynamics remain unaffected b y the disorder. For the discrete time dynamics we prove that the unique invariant measure i s Gibbsian, its two-point spatial correlation function decays exponentially fast for increasing distances and, for a restricted class of models (i.e., directed probabilistic cellular automata), we prove almost sure and disord er-averaged upper bounds for the rate of relaxation towards equilibrium. Mo reover we show, by an example, that under our conditions these bounds are ( almost) optimal. For the continuous time dynamics, after showing the existence of the infini te volume limit, we derive approximations by a discrete time updating syste m, valid uniformly in time. (C) Elsevier, Paris.