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 Electronic Communications in Probability > Vol. 9 (2004) > Paper 13 open journal systems 


Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

Pierre-Yves Louis, Technische Universitat Berlin


Abstract
For a general attractive Probabilistic Cellular Automata on SZ^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 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}Z^d with a naturally associated Gibbsian potential φ, we prove that a (spatial-) weak mixing condition WM   for   φ implies the validity of the assumption A ; thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to φ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.


Full text: PDF

Pages: 119-131

Published on: October 7, 2004
Bibliography
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Electronic Communications in Probability. ISSN: 1083-589X