Let us consider the family of one-dimensional probabilistic cellular automata (PCA) with memory two having the following property: the dynamics is such that the value of a given cell at time is drawn according to a distribution which is a function of the states of its two nearest neighbours at time , and of its own state at time . We give conditions for which the invariant measure has a product form or a Markovian form, and prove an ergodicity result holding in that context. The stationary space-time diagrams of these PCA present different forms of reversibility. We describe and study extensively this phenomenon, which provides families of Gibbs random fields on the square lattice having nice geometric and combinatorial properties. Such PCA naturally arise in the study of different models coming from statistical physics. We review from a PCA approach some results on the -vertex model and on the enumeration of directed animals, and we also show that our methods allow to find new results for an extension of the classical TASEP model. As another original result, we describe some families of PCA for which the invariant measure can be explicitly computed, although it does not have a simple product or Markovian form.
Considérons la famille d’automates cellulaires probabilistes (ACP) de dimension un avec mémoire deux ayant la propriété suivante : la dynamique est telle que la valeur d’une cellule au temps est tirée aléatoirement selon une distribution qui est une fonction de l’état de ses deux voisines les plus proches au temps , et de son propre état au temps . Nous donnons des conditions pour lesquelles la loi invariante d’un tel ACP est une mesure de forme produit ou une mesure markovienne, et prouvons un résultat d’ergodicité s’appliquant dans ce contexte. Les diagrammes espace-temps de ces ACP possèdent différentes formes de réversibilité. Nous décrivons et étudions ce phénomène, qui fournit des familles de champs aléatoires de Gibbs sur la grille carrée ayant des propriétés géométriques et combinatoires remarquables. De tels ACP apparaissent de manière naturelle dans l’étude de différents modèles de physique statistique. En utilisant le point de vue des ACP, nous retrouvons des résultats portant sur le modèle à sommets et sur l’énumération des animaux dirigés, et nous montrons aussi que nos méthodes permettent de trouver de nouveaux résultats sur une extension du modèle classique de TASEP. Un autre résultat original de ce travail est la description de familles d’ACP pour lesquels la loi invariante est explicite, mais n’est ni une mesure de forme produit, ni une mesure markovienne.
Revised:
Accepted:
Published online:
Keywords: Probabilistic cellular automata, invariant measures, ergodicity, reversibility
@article{AHL_2020__3__501_0, author = {Casse, J\'er\^ome and Marcovici, Ir\`ene}, title = {Probabilistic cellular automata with memory two: invariant laws and multidirectional reversibility}, journal = {Annales Henri Lebesgue}, pages = {501--559}, publisher = {\'ENS Rennes}, volume = {3}, year = {2020}, doi = {10.5802/ahl.39}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.39/} }
TY - JOUR AU - Casse, Jérôme AU - Marcovici, Irène TI - Probabilistic cellular automata with memory two: invariant laws and multidirectional reversibility JO - Annales Henri Lebesgue PY - 2020 SP - 501 EP - 559 VL - 3 PB - ÉNS Rennes UR - http://archive.numdam.org/articles/10.5802/ahl.39/ DO - 10.5802/ahl.39 LA - en ID - AHL_2020__3__501_0 ER -
%0 Journal Article %A Casse, Jérôme %A Marcovici, Irène %T Probabilistic cellular automata with memory two: invariant laws and multidirectional reversibility %J Annales Henri Lebesgue %D 2020 %P 501-559 %V 3 %I ÉNS Rennes %U http://archive.numdam.org/articles/10.5802/ahl.39/ %R 10.5802/ahl.39 %G en %F AHL_2020__3__501_0
Casse, Jérôme; Marcovici, Irène. Probabilistic cellular automata with memory two: invariant laws and multidirectional reversibility. Annales Henri Lebesgue, Volume 3 (2020), pp. 501-559. doi : 10.5802/ahl.39. http://archive.numdam.org/articles/10.5802/ahl.39/
[Bax82] Exactly solved models in statistical mechanics, 1982 (http://physics.anu.edu.au/theophys/_files/Exactly.pdf) | MR | Zbl
[BC15] Discrete time -TASEPs, Int. Math. Res. Not., Volume 2015 (2015) no. 2, pp. 499-537 | DOI | MR | Zbl
[BCG16] Stochastic six-vertex model, Duke Math. J., Volume 165 (2016) no. 3, pp. 563-624 | DOI | MR | Zbl
[BF05] Invariant measures and convergence properties for cellular automaton 184 and related processes, J. Stat. Phys., Volume 118 (2005) no. 3-4, pp. 589-623 | DOI | MR | Zbl
[BGM69] Invariant random Boolean fields (in Russian)., Mat. Zametki, Volume 6 (1969), pp. 555-566 | Zbl
[BM98] New enumerative results on two-dimensional directed animals, Discrete Math., Volume 180 (1998) no. 1-3, pp. 73-106 | DOI | MR | Zbl
[BMM13] Probabilistic cellular automata, invariant measures, and perfect sampling, Adv. Appl. Probab., Volume 45 (2013) no. 4, pp. 960-980 | DOI | MR | Zbl
[Cas16] Probabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distribution, Adv. Appl. Probab., Volume 48 (2016) no. 2, pp. 369-391 | DOI | MR | Zbl
[Cas18] Edge correlation function of the 8-vertex model when , Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD), Volume 5 (2018) no. 4, pp. 557-619 | DOI | MR | Zbl
[CM15] Markovianity of the invariant distribution of probabilistic cellular automata on the line, Stochastic Processes Appl., Volume 125 (2015) no. 9, pp. 3458-3483 | DOI | MR | Zbl
[DCGH + 18] The Bethe ansatz for the six-vertex and XXZ models: An exposition, Probab. Surv., Volume 15 (2018), pp. 102-130 | DOI | MR | Zbl
[Dha82] Equivalence of the two-dimensional directed-site animal problem to Baxter’s hard-square lattice-gas model, Phys. Rev. Lett., Volume 49 (1982) no. 14, pp. 959-962 | DOI | MR
[Dur10] Probability: theory and examples, Cambridge Series in Statistical and Probabilistic Mathematics, 31, Cambridge University Press, 2010 | MR | Zbl
[Ger61] On dominance and varieties of commuting matrices, Ann. Math., Volume 73 (1961), pp. 324-348 | DOI | MR
[GG01] The ergodic theory of traffic jams, J. Stat. Phys., Volume 105 (2001) no. 3-4, pp. 413-452 | DOI | MR | Zbl
[GKLM89] From PCAs to equilibrium systems and back, Commun. Math. Phys., Volume 125 (1989) no. 1, pp. 71-79 | DOI | MR | Zbl
[Gur92] A note on commuting pairs of matrices, Linear Multilinear Algebra, Volume 31 (1992) no. 1-4, pp. 71-75 | DOI | MR | Zbl
[HMM19] Percolation games, probabilistic cellular automata, and the hard-core model, Probab. Theory Relat. Fields, Volume 174 (2019) no. 3-4, pp. 1187-1217 | DOI | MR | Zbl
[KV80] Reversible Markov chains with local interaction, Multicomponent random systems (Advances in Probability and Related Topics), Volume 6, Marcel Dekker, 1980, pp. 451-469 | MR | Zbl
[LBM07] Directed animals and gas models revisited, Electron. J. Comb., Volume 14 (2007) no. 4, R71, 36 pages | MR | Zbl
[LMS90] Statistical mechanics of probabilistic cellular automata, J. Stat. Phys., Volume 59 (1990) no. 1-2, pp. 117-170 | DOI | MR | Zbl
[Mar16] Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond, 12th Conference on Computability in Europe, CiE 2016, Proceedings (Beckmann, Arnold; Jonoska, Natasša, eds.), Springer (2016), pp. 153-163 | DOI
[Mel18] The free-fermion eight-vertex model: couplings, bipartite dimers and Z-invariance (2018) (https://arxiv.org/abs/1811.02026)
[MM14a] Around probabilistic cellular automata, Theor. Comput. Sci., Volume 559 (2014), pp. 42-72 | DOI | MR | Zbl
[MM14b] Probabilistic cellular automata and random fields with i.i.d. directions, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 50 (2014) no. 2, pp. 455-475 | DOI | Numdam | MR | Zbl
[MT55] Pairs of matrices with property . II, Trans. Am. Math. Soc., Volume 80 (1955) no. 2, pp. 387-401 | MR | Zbl
[TVS + 90] Discrete local Markov systems, Stochastic cellular systems: ergodicity, memory, morphogenesis (Nonlinear Science: Theory & Applications), Manchester University Press, 1990, pp. 1-175
[Vas78] Bernoulli and Markov stationary measures in discrete local interactions, Developments in statistics, Vol. 1 (Krishnaiah, Paruchuri R., ed.), Academic Press Inc., 1978, pp. 99-112
Cited by Sources: