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Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

  • Ana Bušić (a1), Jean Mairesse (a2) and Irène Marcovici (a2)

Abstract

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.

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Copyright

Corresponding author

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: ana.busic@inria.fr
∗∗ Postal address: CNRS, UMR 7089, LIAFA, Université Paris Diderot, Sorbonne Paris Cité, F-75205 Paris, France.
∗∗∗∗ Email address: irene.marcovici@liafa.univ-paris-diderot.fr

References

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Keywords

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Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

  • Ana Bušić (a1), Jean Mairesse (a2) and Irène Marcovici (a2)

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