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Ergodicity of PCA: equivalence between spatial and temporal mixing conditions. (English) Zbl 1059.60098
Summary: For a general attractive probabilistic cellular automaton on $$S^{\mathbb{Z}^d}$$, we prove that the (time-)convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition $$(\mathcal A)$$. This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on $$\{-1,+1\}^{\mathbb{Z}^d}$$ with a naturally associated Gibbsian potential $$\varphi$$, we prove that a (spatial-)weak mixing condition $$(\mathcal {WM})$$ for $$\varphi$$ implies the validity of the assumption $$(\mathcal A)$$; thus exponential (time-)ergodicity of these dynamics towards the unique Gibbs measure associated to $$\varphi$$ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G60 Random fields 37B15 Dynamical aspects of cellular automata 37H99 Random dynamical systems 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
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