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Ergodicity of PCA

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

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Author:Pierre-Yves Louis
Subtitle (English):equivalence between spatial and temporal mixing conditions
Document Type:Postprint
Year of Completion:2004
Publishing Institution:Universität Potsdam
Release Date:2006/03/20
Tag:Stochastische Zellulare Automaten; Wechselwirkende Teilchensysteme
ERgodicity of Markov Chains; Gibbs measures; Interacting particle systems; Probabilistic Cellular Automata
GND Keyword:Wahrscheinlichkeitstheorie
Source:ELECTRONIC COMMUNICATIONS IN PROBABILITY. - ISSN 1083-589X . - 9 (2004), S. 119 - 131
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Notes extern:
AMS 2000 Subject classification: 60G60 , 60J10 , 60K35 , 82C20 , 82C26 , 37B15

first published at:
Electronic Communications in Probability, 9 (2004) paper 13, pages 119-131