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Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

Part of: Special processes Markov processes Theory of computing Ergodic theory Topological dynamics

Published online by Cambridge University Press:  04 January 2016

Ana Bušić ,
Jean Mairesse  and
Irène Marcovici
Ana Bušić*
Affiliation:
INRIA - École Normale Supérieure
Jean Mairesse*
Affiliation:
Université Paris Diderot
Irène Marcovici*
Affiliation:
Université Paris Diderot
*
∗ Postal address: Laboratoire d'Informatique de l'École Normale Supérieure (UMR 8548), INRIA - École Normale Supérieure, 23 avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France. Email address: [email protected]
∗∗ Postal address: CNRS, UMR 7089, LIAFA, Université Paris Diderot, Sorbonne Paris Cité, F-75205 Paris, France.
∗∗ Postal address: CNRS, UMR 7089, LIAFA, Université Paris Diderot, Sorbonne Paris Cité, F-75205 Paris, France.
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Abstract

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A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

Keywords

Probabilistic cellular automataperfect samplingergodicityinvariant measure

MSC classification

Primary: 37B15: Cellular automata 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains
Secondary: 37A25: Ergodicity, mixing, rates of mixing 60K35: Interacting random processes; statistical mechanics type models; percolation theory 68Q80: Cellular automata
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 960 - 980
Copyright
© Applied Probability Trust 

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