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December 2013 Probabilistic cellular automata, invariant measures, and perfect sampling
Ana Bušić, Jean Mairesse, Irène Marcovici
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Adv. in Appl. Probab. 45(4): 960-980 (December 2013). DOI: 10.1239/aap/1386857853

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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Ana Bušić. Jean Mairesse. Irène Marcovici. "Probabilistic cellular automata, invariant measures, and perfect sampling." Adv. in Appl. Probab. 45 (4) 960 - 980, December 2013. https://doi.org/10.1239/aap/1386857853

Information

Published: December 2013
First available in Project Euclid: 12 December 2013

zbMATH: 1327.37008
MathSciNet: MR2853437
Digital Object Identifier: 10.1239/aap/1386857853

Subjects:
Primary: 37B15, 60J05, 60J22
Secondary: 37A25, 60K35, 68Q80

Rights: Copyright © 2013 Applied Probability Trust

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Vol.45 • No. 4 • December 2013
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