@article{ECP1116,
author = {Pierre-Yves Louis},
title = {Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {9},
year = {2004},
keywords = {},
abstract = {For a general attractive Probabilistic Cellular Automata 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.},
pages = {no. 13, 119-131},
issn = {1083-589X},
doi = {10.1214/ECP.v9-1116},
url = {http://ecp.ejpecp.org/article/view/1116}}