WIAS Preprint No. 823, (2003)

Convergence towards equilibrium of Probabilistic Cellular Automata


  • Louis, Pierre-Yves

2010 Mathematics Subject Classification

  • 60G60 60J10 60K35 82C20 82C26


  • Probabilistic Cellular Automata, Interacting Particle Systems, Coupling, Attractive Dynamics, %Stochastic Ordering, Weak Mixing Condition, Ergodicity, Exponential rate of convergence, Gibbs measure


We first introduce some coupling of a finite number of Probabilistic Cellular Automata dynamics (PCA), preserving the stochastic ordering. Using this tool, and under some assumption ( 𝒜) we establish ergodicity for general attractive probabilistic cellular automata on Sd, where S is finite: this means the convergence towards equilibrium of these Markovian parallel dynamics, in the uniform norm, exponentially fast. For a class of reversible PCA dynamics on {-1,+1}d, with a naturally associated Gibbsian potential 𝜑, we prove that a Weak Mixing condition for 𝜑 implies the validity of the assumption (𝒜), thus the 'exponential ergodicity' of the dynamics towards the unique Gibbs measure associated to 𝜑 holds. On some particular examples of this PCA class, we verify that our assumption (𝒜) is weaker than the Dobrushin-Vasershtein ergodicity condition. For some precise PCA, the 'exponential ergodicity' holds as soon as there is no phase transition.

Appeared in

  • Electronic Communications in Probability, vol. 9, pp 119-131, 7.10.2004

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