## Advances in Applied Probability

### Probabilistic cellular automata, invariant measures, and perfect sampling

#### 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

#### 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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