Electronic Communications in Probability

Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

Pierre-Yves Louis

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 37B15: Cellular automata [See also 68Q80] 37H99: None of the above, but in this section 60J10 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 68Q80: Cellular automata [See also 37B15] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 82C26: Dynamic and nonequilibrium phase transitions (general)

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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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