Probabilistic cellular automata and random fields with i.i.d. directions
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, p. 455-475

Let us consider the simplest model of one-dimensional probabilistic cellular automata (PCA). The cells are indexed by the integers, the alphabet is {0,1}, and all the cells evolve synchronously. The new content of a cell is randomly chosen, independently of the others, according to a distribution depending only on the content of the cell itself and of its right neighbor. There are necessary and sufficient conditions on the four parameters of such a PCA to have a Bernoulli product invariant measure. We study the properties of the random field given by the space-time diagram obtained when iterating the PCA starting from its Bernoulli product invariant measure. It is a non-trivial random field with very weak dependences and nice combinatorial properties. In particular, not only the horizontal lines but also the lines in any other direction consist of i.i.d. random variables. We study extensions of the results to Markovian invariant measures, and to PCA with larger alphabets and neighborhoods.

Considérons le modèle le plus simple d’automates cellulaires probabilistes (ACP) de dimension 1. Les cellules sont indexées par les entiers relatifs, l’alphabet est {0,1}, et toutes les cellules évoluent de manière synchrone. Le nouveau contenu d’une cellule est choisi aléatoirement, indépendamment des autres, selon une distribution dépendant seulement du contenu de la cellule et de sa voisine de droite. On connaît des conditions nécessaires et suffisantes portant sur les quatre paramètres d’un tel ACP pour qu’il ait la mesure produit de Bernoulli comme mesure invariante. Nous étudions les propriétés du champ aléatoire formé par le diagramme espace-temps obtenu lorsqu’on itère l’ACP à partir de sa mesure invariante de Bernoulli. Il s’agit d’un champ aléatoire non trivial, présentant de très faibles dépendances et de jolies propriétés combinatoires. En particulier, les lignes horizontales mais aussi les lignes selon les autres directions sont constituées de variables aléatoires i.i.d. Nous étudions l’extension de ces résultats à des mesures invariantes de forme markovienne, ainsi qu’aux ACP ayant des alphabets et des voisinages plus grands.

DOI : https://doi.org/10.1214/12-AIHP530
Classification:  37B15,  60J05,  60G60
Keywords: probabilistic cellular automata, product-form invariant measures, random fields
@article{AIHPB_2014__50_2_455_0,
     author = {Mairesse, Jean and Marcovici, Ir\`ene},
     title = {Probabilistic cellular automata and random fields with i.i.d. directions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     pages = {455-475},
     doi = {10.1214/12-AIHP530},
     mrnumber = {3189079},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_2_455_0}
}
Mairesse, Jean; Marcovici, Irène. Probabilistic cellular automata and random fields with i.i.d. directions. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 455-475. doi : 10.1214/12-AIHP530. http://www.numdam.org/item/AIHPB_2014__50_2_455_0/

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