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2004 Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions
Pierre-Yves Louis
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Electron. Commun. Probab. 9: 119-131 (2004). DOI: 10.1214/ECP.v9-1116

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.

Citation

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Pierre-Yves Louis. "Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions." Electron. Commun. Probab. 9 119 - 131, 2004. https://doi.org/10.1214/ECP.v9-1116

Information

Accepted: 7 October 2004; Published: 2004
First available in Project Euclid: 26 May 2016

zbMATH: 1059.60098
MathSciNet: MR2108858
Digital Object Identifier: 10.1214/ECP.v9-1116

Subjects:
Primary: 60G60
Secondary: 37B15, 37H99, 60J10, 60K35, 68Q80, 82C20, 82C26

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