an:02145783
Zbl 1059.60098
Louis, Pierre-Yves
Ergodicity of PCA: equivalence between spatial and temporal mixing conditions
EN
Electron. Commun. Probab. 9, 119-131 (2004).
00114168
2004
j
60K35 60G60 37B15 37H99 60J10 82C26
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.