## Electronic Journal of Probability

### Ergodicity of some classes of cellular automata subject to noise

#### Abstract

Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise.

We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 41, 44 pp.

Dates
Accepted: 18 March 2019
First available in Project Euclid: 12 April 2019

https://projecteuclid.org/euclid.ejp/1555034440

Digital Object Identifier
doi:10.1214/19-EJP297

#### Citation

Marcovici, Irène; Sablik, Mathieu; Taati, Siamak. Ergodicity of some classes of cellular automata subject to noise. Electron. J. Probab. 24 (2019), paper no. 41, 44 pp. doi:10.1214/19-EJP297. https://projecteuclid.org/euclid.ejp/1555034440

#### References

• [1] A. Adamatzky (ed.), Cellular automata, Encyclopedia of Complexity and Systems Science, Springer, 2009.
• [2] V. Belitsky and P. A. Ferrari, Ballistic annihilation and deterministic surface growth, Journal of Statistical Physics 80 (1995), 517–543.
• [3] V. Belitsky and P. A. Ferrari, Invariant measures and convergence properties for cellular automaton 184 and related processes, Journal of Statistical Physics 118 (2005), no. 3–4, 589–623.
• [4] J. van den Berg and J. E. Steif, On the existence and nonexistence of finitary codings for a class of random fields, The Annals of Probability 27 (1999), no. 3, 1501–1522.
• [5] M. Bramson and C. Neuhauser, Survival of one-dimensional cellular automata under random perturbations, The Annals of Probability 22 (1994), no. 1, 244–263.
• [6] A. Bušić, J. Mairesse, and I. Marcovici, Probabilistic cellular automata, invariant measures, and perfect sampling, Advances in Applied Probability 45 (2013), no. 4, 960–980.
• [7] T. Ceccherini-Silberstein and M. Coornaert, Cellular automata and groups, Springer, 2010.
• [8] P. Chassaing and J. Mairesse, A non-ergodic probabilistic cellular automaton with a unique invariant measure, Stochastic Processes and their Applications 121 (2010), no. 11, 2474–2487.
• [9] B. Chopard and M. Droz, Cellular automata modeling of physical systems, Cambridge University Press, 1998.
• [10] C. F. Coletti and P. Tisseur, Invariant measures and decay of correlations for a class of ergodic probabilistic cellular automata, Journal of Statistical Physics 140 (2010), 103–121.
• [11] T. M. Cover and J. A. Thomas, Elements of information theory, Wiley, 1991.
• [12] P. Dai Pra, P.-Y. Louis, and S. Rœlly, Stationary measures and phase transition for a class of probabilistic cellular automata, ESAIM: Probability and Statistics 6 (2002), 89–104.
• [13] M. Denker, C. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Springer-Verlag, 1976.
• [14] R. Durrett, Oriented percolation in two dimensions, The Annals of Probability 12 (1984), no. 4, 999–1040.
• [15] R. Durrett, Lecture notes on particle systems and percolation, Wadsworth & Brooks/Cole, 1988.
• [16] P. Ferrari, Ergodicity for a class of probabilistic cellular automata, Revista de Matemáticas Aplicadas 12 (1991), 93–102.
• [17] P. A. Ferrari, A. Maass, S. Martinez, and P. Ney, Cesàro mean distribution of group automata starting from measures with summable decay, Ergodic Theory and Dynamical Systems 20 (2000), no. 6, 1657–1670.
• [18] R. Fisch, The one-dimensional cyclic cellular automaton: a system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics, Journal of Theoretical Probabilities 3 (1990), no. 2, 311–338.
• [19] P. Gács, Reliable computation with cellular automata, Journal of Computer and System Sciences 32 (1986), no. 1, 15–78.
• [20] P. Gács, Reliable cellular automata with self-organization, Journal of Statistical Physics 103 (2001), no. 1–2, 45–267.
• [21] M. Garzon, Models of massive parallelism, Springer, 1995.
• [22] H.-O. Georgii, Gibbs measures and phase transitions, De Gruyter, 1988.
• [23] S. Goldstein, R. Kuik, J. L. Lebowitz, and C. Maes, From PCA’s to equilibrium systems and back, Communications in Mathematical Physics 125 (1989), 71–79.
• [24] L. Gray, The behavior of processes with statistical mechanical properties, Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten, ed.), The IMA Volumes in Mathematics and Its Applications, vol. 8, Springer, 1987, pp. 131–167.
• [25] P. Guillon and G. Richard, Nilpotency and limit sets of cellular automata, Proceedings of the 33rd International Symposium (MFCS 2008), LNCS, vol. 5162, Springer, 2008, pp. 375–386.
• [26] B. Hellouin de Menibus and M. Sablik, Self-organisation in cellular automata with coalescent particles: qualitative and quantitative approaches, Journal of Statistical Physics 167 (2017), no. 5, 1180–1220.
• [27] B. Hellouin de Menibus and M. Sablik, Characterization of sets of limit measures of a cellular automaton iterated on a random configuration, Ergodic Theory and Dynamical Systems 38 (2018), no. 2, 601–650.
• [28] B. Hellouin de Menibus, V. Salo, and G. Theyssier, Characterizing asymptotic randomization in abelian cellular automata, Ergodic Theory and Dynamical Systems (To appear).
• [29] R. Holley, Free energy in a Markovian model of a lattice spin system, Communications in Mathematical Physics 23 (1971), no. 2, 87–99.
• [30] A. E. Holroyd, I. Marcovici, and J. B. Martin, Percolation games, probabilistic cellular automata, and the hard-core model, Probability Theory and Related Fields (To appear).
• [31] J.-F. Marckert J. Casse, Markovianity of the invariant distribution of probabilistic cellular automata on the line, Stochastic Processes and their Applications 125 (2015), no. 9, 3458–3483.
• [32] B. Jahnel and C. Külske, A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit, Stochastic Processes and their Applications 125 (2015), no. 6, 2427–2450.
• [33] J. Kari, The nilpotency problem of one-dimensional cellular automata, SIAM Journal on Computing 21 (1992), no. 3, 571–586.
• [34] J. Kari, Theory of cellular automata: A survey, Theoretical Computer Science 334 (2005), 3–33.
• [35] J. Kari and S. Taati, Statistical mechanics of surjective cellular automata, Journal of Statistical Physics 160 (2015), no. 5, 1198–1243.
• [36] O. Kozlov and N. Vasilyev, Reversible Markov chains with local interaction, Multicomponent Random Systems (R. L. Dobrushin and Ya. G. Sinai, eds.), Marcel Dekker, 1980, pp. 451–469.
• [37] P. Kůrka, Cellular automata with vanishing particles, Fundamenta Informaticae 58 (2003), no. 3–4, 203–221.
• [38] P. Kůrka, Topological and symbolic dynamics, Cours Spécialisés, vol. 11, Société Mathématique de France, 2003.
• [39] J. K. Lebowitz, C. Maes, and E. R. Speer, Statistical mechanics of probabilistic cellular automata, Journal of Statistical Physics 59 (1990), no. 1–2, 117–170.
• [40] T. M. Liggett, Interacting particle systems, Springer, 1985.
• [41] D. A. Lind, Applications of ergodic theory and sofic systems to cellular automata, Physica D. Nonlinear Phenomena 10 (1984), no. 1–2, 36–44.
• [42] T. Lindvall, Lectures on the coupling method, Dover, 2002.
• [43] P.-Y. Louis, Ergodicity of PCA: Equivalence between spatial and temporal mixing conditions, Electronic Communications in Probability 9 (2004), 119–131.
• [44] P.-Y. Louis and F. R. Nardi (eds.), Probabilistic cellular automata: Theory, applications and future perspectives, Springer, 2018.
• [45] C. Maes and S. B. Shlosman, Ergodicity of probabilistic cellular automata: A constructive criterion, Communications in Mathematical Physics 135 (1991), no. 2, 233–251.
• [46] J. Mairesse and I. Marcovici, Around probabilistic cellular automata, Theoretical Computer Science 559 (2014), 42–72.
• [47] J. Mairesse and I. Marcovici, Probabilistic cellular automata and random fields with i.i.d. directions, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 50 (2014), no. 2, 455–475.
• [48] C. E. M. Pearce and F. K. Fletcher, Oriented site percolation, phase transitions and probability bounds, Journal of Inequalities in Pure and Applied Mathematics 6 (2005), no. 5, 135.
• [49] O. Penrose, Foundations of statistical mechanics: A deductive treatment, Pergamon, 1970.
• [50] M. Pivato and R. Yassawi, Limit measures for affine cellular automata, Ergodic Theory and Dynamical Systems 22 (2002), no. 4, 1269–1287.
• [51] J. G. Propp and D. B. Wilson, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms 9 (1996), no. 1–2, 223–252.
• [52] G. Rozenberg, T. Bäck, and J. N. Kok (eds.), Handbook of natural computing, vol. 1, Springer, 2012.
• [53] Th. W. Ruijgrok and E. G. D. Cohen, Deterministic lattice gas models, Physics Letters A 133 (1988), no. 7–8, 415–418.
• [54] V. Salo, On nilpotency and asymptotic nilpotency of cellular automata, Electronic Proceedings in Theoretical Computer Science 90 (2012), 86–96.
• [55] J. E. Steif, $\bar{d}$-Convergence to equilibrium and space-time Bernoulicity for spin systems in the $m<\varepsilon$ case, Ergodic Theory and Dynamical Systems 11 (1991), no. 3, 547–575.
• [56] T. Toffoli and N. Margolus, Cellular automata machines, MIT Press, 1987.
• [57] A. Toom, Stable and attractive trajectories in multicomponent systems, Multicomponent Random Systems (R. L. Dobrushin and Ya. G. Sinai, eds.), Marcel Dekker, 1980, pp. 549–575.
• [58] A. L. Toom, N. B. Vasilyev, O. N. Stavskaya, L. G. Mityushin, G. L. Kuryumov, and S. A. Pirogov, Discrete local Markov systems, Stochastic cellular systems: ergodicity, memory, morphogenesis (R. L. Dobrushin, V. I. Kryukov, and A. L. Toom, eds.), Manchester University Press, 1990.
• [59] N. B. Vasilyev, Bernoulli and Markov stationary measures in discrete local interactions, Locally Interacting Systems and Their Application in Biology (R. L. Dobrushin, V. I. Kryukov, and A. L. Toom, eds.), Springer, 1978, pp. 99–112.
• [60] S. Wolfram (ed.), Theory and applications of cellular automata, World Scientific, 1986.
• [61] S. Wolfram (ed.), A new kind of science, Wolfram Media, 2002.
• [62] H. Yaguchi, Application of entropy analysis to discrete-time interacting particle systems on the one-dimensional lattice, Hiroshima Mathematical Journal 30 (2000), no. 1, 137–165.