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.
[Bax82] Exactly solved models in statistical mechanics, 1982 (http://physics.anu.edu.au/theophys/_files/Exactly.pdf) | MR | Zbl
[BGM69] Invariant random Boolean fields (in Russian)., Mat. Zametki, Volume 6 (1969), pp. 555-566 | Zbl
[Mar16] Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond, 12th Conference on Computability in Europe, CiE 2016, Proceedings (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)
[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