Probabilistic cellular automata with memory two: invariant laws and multidirectional reversibility
Annales Henri Lebesgue, Volume 3 (2020) , pp. 501-559.

KeywordsProbabilistic cellular automata, invariant measures, ergodicity, reversibility

Abstract

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 $t+1$ is drawn according to a distribution which is a function of the states of its two nearest neighbours at time $t$, and of its own state at time $t-1$. 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 $8$-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.

References

[Bax82] Baxter, Rodney J. Exactly solved models in statistical mechanics, 1982 (http://physics.anu.edu.au/theophys/_files/Exactly.pdf) | MR 690578 | Zbl 0538.60093

[BC15] Borodin, Alexei; Corwin, Ivan Discrete time $q$-TASEPs, Int. Math. Res. Not., Volume 2015 (2015) no. 2, pp. 499-537 | Article | MR 3340328 | Zbl 1310.82030

[BCG16] Borodin, Alexei; Corwin, Ivan; Gorin, Vadim Stochastic six-vertex model, Duke Math. J., Volume 165 (2016) no. 3, pp. 563-624 | Article | MR 3466163 | Zbl 1343.82013

[BF05] Belitsky, Vladimir; Ferrari, Pablo A. Invariant measures and convergence properties for cellular automaton 184 and related processes, J. Stat. Phys., Volume 118 (2005) no. 3-4, pp. 589-623 | Article | MR 2123649 | Zbl 1126.37301

[BGM69] Belyaev, Yuriĭ K.; Gromak, Yu. I.; Malyshev, Vadim A. Invariant random Boolean fields (in Russian)., Mat. Zametki, Volume 6 (1969), pp. 555-566 | Zbl 0201.18605

[BM98] Bousquet-Mélou, Mireille New enumerative results on two-dimensional directed animals, Discrete Math., Volume 180 (1998) no. 1-3, pp. 73-106 | Article | MR 1603701 | Zbl 0974.05002

[BMM13] Bušić, Ana; Mairesse, Jean; Marcovici, Irène Probabilistic cellular automata, invariant measures, and perfect sampling, Adv. Appl. Probab., Volume 45 (2013) no. 4, pp. 960-980 | Article | MR 3161292 | Zbl 1327.37008

[Cas16] Casse, Jérôme 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 | Article | MR 3511766 | Zbl 1366.37024

[Cas18] Casse, Jérôme Edge correlation function of the 8-vertex model when $a+c=b+d$, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD), Volume 5 (2018) no. 4, pp. 557-619 | Article | MR 3900291 | Zbl 1406.82006

[CM15] Casse, Jérôme; Marckert, Jean-François Markovianity of the invariant distribution of probabilistic cellular automata on the line, Stochastic Processes Appl., Volume 125 (2015) no. 9, pp. 3458-3483 | Article | MR 3357616 | Zbl 1371.37016

[DCGH + 18] Duminil-Copin, Hugo; Gagnebin, Maxime; Harel, Matan; Manolescu, Ioan; Tassion, Vincent The Bethe ansatz for the six-vertex and XXZ models: An exposition, Probab. Surv., Volume 15 (2018), pp. 102-130 | Article | MR 3775121 | Zbl 1430.60080

[Dha82] Dhar, Deepak 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 | Article | MR 675054

[Dur10] Durrett, Rick Probability: theory and examples, Cambridge Series in Statistical and Probabilistic Mathematics, Volume 31, Cambridge University Press, 2010 | MR 2722836 | Zbl 1202.60001

[Ger61] Gerstenhaber, Murray On dominance and varieties of commuting matrices, Ann. Math., Volume 73 (1961), pp. 324-348 | Article | MR 132079

[GG01] Gray, Lawrence; Griffeath, David The ergodic theory of traffic jams, J. Stat. Phys., Volume 105 (2001) no. 3-4, pp. 413-452 | Article | MR 1871652 | Zbl 1048.90066

[GKLM89] Goldstein, Sheldon; Kuik, Roelof; Lebowitz, Joel L.; Maes, Christian From PCAs to equilibrium systems and back, Commun. Math. Phys., Volume 125 (1989) no. 1, pp. 71-79 | Article | MR 1017739 | Zbl 0683.68045

[Gur92] Guralnick, Robert Mickael A note on commuting pairs of matrices, Linear Multilinear Algebra, Volume 31 (1992) no. 1-4, pp. 71-75 | Article | MR 1199042 | Zbl 0754.15011

[HMM19] Holroyd, Alexander E.; Marcovici, Irène; Martin, James B. Percolation games, probabilistic cellular automata, and the hard-core model, Probab. Theory Relat. Fields, Volume 174 (2019) no. 3-4, pp. 1187-1217 | Article | MR 3980314 | Zbl 1418.91100

[KV80] Kozlov, O.; Vasilyev, N. B. Reversible Markov chains with local interaction, Multicomponent random systems (Advances in Probability and Related Topics) Volume 6, Marcel Dekker, 1980, pp. 451-469 | MR 599544 | Zbl 0444.60099

[LBM07] Le Borgne, Yvan; Marckert, Jean-François Directed animals and gas models revisited, Electron. J. Comb., Volume 14 (2007) no. 4, R71, 36 pages | MR 2365970 | Zbl 1157.82332

[LMS90] Lebowitz, Joel L.; Maes, Christian; Speer, Eugene R. Statistical mechanics of probabilistic cellular automata, J. Stat. Phys., Volume 59 (1990) no. 1-2, pp. 117-170 | Article | MR 1049965 | Zbl 1083.82522

[Mar16] Marcovici, Irène Ergodicity of Noisy Cellular Automata: The Coupling Method and Beyond, 12th Conference on Computability in Europe, CiE 2016, Proceedings (2016), pp. 153-163 | Article

[Mel18] Melotti, Paul The free-fermion eight-vertex model: couplings, bipartite dimers and Z-invariance (2018) (https://arxiv.org/abs/1811.02026)

[MM14a] Mairesse, Jean; Marcovici, Irène Around probabilistic cellular automata, Theor. Comput. Sci., Volume 559 (2014), pp. 42-72 | Article | MR 3280727 | Zbl 1360.68615

[MM14b] Mairesse, Jean; Marcovici, Irène 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 | Article | Numdam | MR 3189079 | Zbl 1359.37027

[MT55] Motzkin, Theodore Samuel; Taussky, Olga Pairs of matrices with property $L$. II, Trans. Am. Math. Soc., Volume 80 (1955) no. 2, pp. 387-401 | MR 86781 | Zbl 0067.25401

[TVS + 90] Toom, André; Vasilyev, N. B.; Stavskaya, O. N.; Mityushin, L. G.; Kurdyumov, G. L.; Pirogov, S. A. Discrete local Markov systems, Stochastic cellular systems: ergodicity, memory, morphogenesis (Nonlinear Science: Theory & Applications), Manchester University Press, 1990, pp. 1-175

[Vas78] Vasilyev, N. B. 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