Ergodicity of PCA: equivalence between spatial and temporal mixing conditions
- For a general attractive Probabilistic Cellular Automata on SZd, 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}Zd, with a naturally associated Gibbsian potential ;, we prove that a (spatial-) weak mixing condition (WM) for ; implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to ; holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.
Author details: | Pierre-Yves Louis |
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URL: | http://www.math.washington.edu/%7Eejpecp/ECP/viewarticle.php?id=1704&layout=abstract |
Publication type: | Article |
Language: | English |
Year of first publication: | 2004 |
Publication year: | 2004 |
Release date: | 2017/03/24 |
Source: | Electronic Communications in Probability. - 9 (2004), S. 119 - 131 |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
Peer review: | Nicht referiert |