Probabilistic cellular automata, invariant measures, and perfect sampling

Authors Ana Busic, Jean Mairesse, Irene Marcovici

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Ana Busic
Jean Mairesse
Irene Marcovici

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Ana Busic, Jean Mairesse, and Irene Marcovici. Probabilistic cellular automata, invariant measures, and perfect sampling. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 296-307, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


In a probabilistic cellular automaton (PCA), the cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. A PCA can be viewed as a Markov chain whose ergodicity is investigated. A classical cellular automaton (CA) is a particular case of PCA. For a 1-dimensional CA, 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 be also a PCA.
  • probabilistic cellular automata
  • perfect sampling
  • ergodicity


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