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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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