Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

Pierre-Yves Louis (1)

1 Institut für Mathematik [Potsdam]

For a general attractive Probabilistic Cellular Automata on S Z d , we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the inuence from the boundary for the invariant measures of the system restricted to nite boxes. For a class of reversible PCA dynamics on {−1, +1} Z d , with a naturally associated Gibbsian potential ϕ, we prove that a (spatial-) weak mixing condition (WM) for ϕ implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to ϕ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.

Mots clés: Ergodicity; Interacting Particle Systems; Weak Mixing Con-dition; Probabilistic Cellular Automata; Exponential rate of convergence; Gibbs measure

Domaines: Probabilités [math.PR]; Automates cellulaires et gaz sur réseau [nlin.CG]

Pierre-Yves Louis. Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions. Electronic Communications in Probability, 2004, ⟨10.1214/ECP.v9-1116⟩. ⟨hal-01277059v2⟩