## Ergodicity of PCA : equivalence between spatial and temporal mixing conditions

- 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.

Author: | Pierre-Yves Louis |
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URN: | urn:nbn:de:kobv:517-opus-6589 |

Document Type: | Postprint |

Language: | English |

Year of Completion: | 2004 |

Publishing Institution: | Universität Potsdam |

Release Date: | 2006/03/20 |

Tag: | Stochastische Zellulare Automaten; Wechselwirkende TeilchensystemeERgodicity 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 |

RVK - Regensburg Classification: | - |

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 |