Louis, Pierre-Yves Ergodicity of PCA: equivalence between spatial and temporal mixing conditions. (English) Zbl 1059.60098 Electron. Commun. Probab. 9, 119-131 (2004). 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. Cited in 11 Documents 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 PDFBibTeX XMLCite \textit{P.-Y. Louis}, Electron. Commun. Probab. 9, 119--131 (2004; Zbl 1059.60098) Full Text: DOI arXiv EuDML