## Electronic Communications in Probability

### Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

Pierre-Yves Louis

#### 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.

#### Article information

Source
Electron. Commun. Probab. Volume 9 (2004), paper no. 13, 119-131.

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

https://projecteuclid.org/euclid.ecp/1464286693

Digital Object Identifier
doi:10.1214/ECP.v9-1116

Mathematical Reviews number (MathSciNet)
MR2108858

Zentralblatt MATH identifier
1059.60098

Rights

#### Citation

Louis, Pierre-Yves. Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions. Electron. Commun. Probab. 9 (2004), paper no. 13, 119--131. doi:10.1214/ECP.v9-1116. https://projecteuclid.org/euclid.ecp/1464286693

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