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Keywords
- Probabilistic Cellular Automata (4)
- Wahrscheinlichkeitstheorie (2)
- coupling (2)
- monotone coupling (2)
- stochastic ordering (2)
- Attractive Dynamics (1)
- Coupling (1)
- ERgodicity of Markov Chains (1)
- Gibbs measures (1)
- Interacting Particle Systems (1)
- Interacting particle systems (1)
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- Stochastic Ordering (1)
- Stochastik (1)
- Stochastische Zellulare Automaten (1)
- Weak Mixing Condition (1)
- Wechselwirkende Teilchensysteme (1)
- continuous time Markov Chains (1)
- dynamical system representation (1)
- monotone random (1)
- monotonicity (1)
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- Institut für Mathematik (7)
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We first introduce some coupling of a finite number of Probabilistic Cellular Automata dynamics (PCA), preserving the stochastic ordering. Using this tool, for a general attractive probabilistic cellular automata on SZd, where S is finite, we prove that a condition (A) is equivalent to the (time-) convergence towards equilibrium of this Markovian parallel dynamics, in the uniform norm, exponentially fast. This condition (A) means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite ‘box’-volume. For a class of reversible PCA dynamics on {−1, +1}Zd , 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 special PCA, the ‘exponential ergodicity’ holds as soon as there is no phase transition.