Advances in Applied Probability

Probabilistic cellular automata, invariant measures, and perfect sampling

Ana Bušić, Jean Mairesse, and Irène Marcovici

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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.

Article information

Source
Adv. in Appl. Probab., Volume 45, Number 4 (2013), 960-980.

Dates
First available in Project Euclid: 12 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1386857853

Digital Object Identifier
doi:10.1239/aap/1386857853

Mathematical Reviews number (MathSciNet)
MR2853437

Zentralblatt MATH identifier
1327.37008

Subjects
Primary: 37B15: Cellular automata [See also 68Q80] 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains [See also 65C40]
Secondary: 37A25: Ergodicity, mixing, rates of mixing 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 68Q80: Cellular automata [See also 37B15]

Keywords
Probabilistic cellular automata perfect sampling ergodicity invariant measure

Citation

Bušić, Ana; Mairesse, Jean; Marcovici, Irène. Probabilistic cellular automata, invariant measures, and perfect sampling. Adv. in Appl. Probab. 45 (2013), no. 4, 960--980. doi:10.1239/aap/1386857853. https://projecteuclid.org/euclid.aap/1386857853


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