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, defined on the whole lattice, with finite spin space: this means the convergence towards equilibrium of these Markovian parallel dynamics, in the uniform norm, exponentially fast. For a class of reversible PCA dynamics with spin space (-1,+1), with a naturally associated Gibbsian potential, we prove that a Weak Mixing condition implies the validity of the assumption (A), thus the `exponential ergodicity' of the dynamics towards the unique Gibbs measure 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.
- Electronic Communications in Probability, vol. 9, pp 119-131, 7.10.2004