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Overview: PCA Models and Issues

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Probabilistic Cellular Automata

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 27))

Abstract

Probabilistic cellular automata (PCA) are interacting discrete stochastic dynamical systems used as a modeling tool for a wide range of natural and societal phenomena. Their key features are: (i) a stochastic component that distinguishes them from the well-known cellular automata (CA) algorithms and (ii) an underlying parallelism that sets them apart from purely asynchronous simulation dynamics in statistical mechanics, such as interacting particle systems and Glauber dynamics. On the applied side, these features make PCA an attractive computational framework for high-performance computing, distributed computing, and simulation. Indeed, PCA have been put to good use as part of multiscale simulation frameworks for studying natural systems or large interconnected network structures. On the mathematical side, PCA have a rich mathematical theory that leads to a better understanding of the role of randomness and synchronicity in the evolution of large systems. This book is an attempt to present a wide panorama of the current status of PCA theory and applications. Contributions cover important issues and applications in probability, statistical mechanics, computer science, natural sciences, and dynamical systems. This initial chapter is intended both as a guide and an introduction to the issues discussed in the book. The chapter starts with a general overview of PCA modeling, followed by a presentation of conspicuous applications in different contexts. It closes with a discussion of the links between approaches and perspectives for future developments.

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References

  1. Achlioptas, D., Naor, A., Peres, Y.: Rigorous location of phase transitions in hard optimization problems. Nature 435(June), 759–764 (2005)

    Article  Google Scholar 

  2. Adachi, S., Peper, F., Lee, J.: Computation by asynchronously updating cellular automata. J. Stat. Phys. 114(January), 261–289 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alber, M.S., Kiskowski, M.A., Glazier, J.A., Jiang, Y.I.: On cellular automaton approaches to modeling biological cells. In: J. Rosenthal, D.S. Gilliam (eds.) Mathematical Systems Theory in Biology, Communications, Computation, and Finance. The IMA Volumes in Mathematics and its Applications. The IMA Volumes in Mathematics and its Applications, vol. 7105, p. 12. Springer, New York (2002)

    Google Scholar 

  4. Alippi, C., Polycarpou, M., Panayiotou, C., Ellinas, G. (eds.): Computing with Probabilistic Cellular Automata. Lecture Notes in Computer Science, vol. 1. Springer, Berlin (2009)

    Google Scholar 

  5. Almeida, R.M., Macau, E.E.N.: Stochastic cellular automata model for wildland fire spread dynamics. J. Phys. Conf. Ser. 285(1), 12,038 (2011)

    Google Scholar 

  6. Arrighi, P., Schabanel, N., Theyssier, G.: Stochastic cellular automata: correlations, decidability and simulations. Fundamenta Informaticae 126(2–3), 121–156 (2013)

    Google Scholar 

  7. Aubert, M., Badoual, M., Féreol, S., Christov, C., Grammaticos, B.: A cellular automaton model for the migration of glioma cells. Phys. Biol. 3(2), 93 (2006)

    Article  Google Scholar 

  8. Bahr, D., Passerini, E.: Statistical Mechanics of Collective Behavior: Macro-Sociology. J. Math. Sociol. 23(1), 29–49 (1998)

    Article  MATH  Google Scholar 

  9. Balister, P., Bollobás, B., Kozma, R.: Large deviations for mean field models of probabilistic cellular automata. Random Struct. Algorithms 29(3), 399–415 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bandini, S., Manzoni, S., Vizzari, G.: Agent Based Modeling and Simulation. In: Computational Complexity. Theory, Techniques, and Applications, pp. 105–117 (2012)

    Google Scholar 

  11. Bandini, S., Mauri, G., Serra, R.: Cellular automata: from a theoretical parallel computational model to its application to complex systems. Parallel Comput. 27, 539–553 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bandman, O.: Simulating spatial dynamics by probabilistic cellular automata. Lect. Notes Comput. Sci. 2493, 10–19 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Barrat, A., Barthelemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  14. Bartolozzi, M., Thomas, A.W.: Stochastic cellular automata model for stock market dynamics. Phys. Rev. E 046112(4), 1–17 (2004)

    Google Scholar 

  15. Berezner, S.A., Krutina, M., Malyshev, V.A.: Exponential convergence of Toom’s probabilistic cellular automata. J. Stat. Phys. 73(5–6), 927–944 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Berry, H., Fatès, N.: Robustness of the critical behaviour in the stochastic Greenberg–Hastings cellular automaton model. Int. J. Unconv. Comput. 7(1), 65–85 (2011)

    Google Scholar 

  17. Birkner, M., Depperschmidt, A.: Survival and complete convergence for a spatial branching system with local regulation. Ann. Appl. Probab. 17(5/6), 1777–1807 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Blok, H.J., Bergersen, B.: Synchronous versus asynchronous updating in the game of Life. Phys. Rev. E 59(4), 1–16 (1999)

    Article  Google Scholar 

  19. Boccara, N., Cheong, K.: Critical behaviour of a probabilistic automata network SIS model for the spread of an infectious disease in a population of moving individuals. J. Phys. A 26, 3707–3717 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bouchaud, J.P.: Crises and collective socio-economic phenomena: simple models and challenges. J. Stat. Phys. 151(3–4), 567–606 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bovier, A., den Hollander, F.: Metastability, Grundlehren der mathematischen Wissenschaften, vol. 351. Springer International Publishing, Cham (2015)

    Google Scholar 

  22. Bovier, A., den Hollander, F., Spitoni, C.: Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures. Ann. Probab. 38(2), 661–713 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bovier, A., Manzo, F.: Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics. J. Stat. Phys. 107(3–4), 757–779 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bramson, M., Neuhauser, C.: Survival of one-dimensional cellular automata under random perturbations. Ann. Probab. 22(1), 244–263 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bricmont, J., Bosch, H.V.D.: Intermediate model between majority voter PCA and its mean field model. J. Stat. Phys. 158(5), 1090–1099 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Burkhead, E.G., Hawkins, J.M., Molinek, D.K.: A dynamical study of a cellular automata model of the spread of HIV in a lymph node. Bull. Math. Biol. 71(1), 25–74 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Bušić, A., Fatès, N., Mairesse, J., Marcovici, I.: Density classification on infinite lattices and trees. Electron. J. Probab. 18, 109–120 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Bušić, A., Mairesse, J., Marcovici, I.I.: Probabilistic cellular automata, invariant measures, and perfect sampling. Adv. Appl. Probab. 980(July 2012), 960–980 (2011)

    Google Scholar 

  29. Carvalho, K.C.D., Tomé, T.: Anisotropic probabilistic cellular automaton for a predator-prey system. Braz. J. Phys. 37(2a), 466–471 (2007)

    Article  Google Scholar 

  30. Casse, J.: Probabilistic cellular automata with general alphabets letting a Markov chain invariant. Adv. Appl. Probab. 48(2) (2016)

    Google Scholar 

  31. Casse, J., Marckert, J.F.: Markovianity of the invariant distribution of probabilistic cellular automata on the line. Stochastic Process. Appl. 1(9), 1–29 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Cerf, R., Manzo, F.: Nucleation and growth for the Ising model in d dimensions at very low temperatures. Ann. Probab. 41(6), 3697–3785 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cervelle, J.: Constructing continuous systems from discrete cellular automata. In: P. Bonizzoni, V. Brattka, B. Loewe (eds.) The Nature of Computation. Logic, Algorithms, Applications. Lecture Notes in Computer Science, vol. 7921, pp. 55–64. Springer, Berlin (2013)

    Google Scholar 

  34. Cervelle, J., Formenti, E.: Algorithmic Complexity and Cellular Automata. Computational Complexity. Theory, Techniques, and Applications pp. 132–146 (2012)

    Google Scholar 

  35. Chassaing, P., Gerin, L.: Asynchronous cellular automata and Brownian motion. In: 2007 Conference on Analysis of Algorithms, Discrete Math. Theor. Comput. Sci. Proc., AH, pp. 385–401. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2007)

    Google Scholar 

  36. Chassaing, P., Mairesse, J.: A non-ergodic probabilistic cellular automaton with a unique invariant measure. Stochastic Process. Appl. 121(11), 2474–2487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Chopard, B., Droz, M.: Cellular automata modeling of physical systems. Comput. Compl. Theory Tech. Appl. 122, 407–433 (1998)

    Google Scholar 

  38. Chopard, B., Dupuis, A., Masselot, A., Luthi, P.: Cellular automata and lattice boltzmann techniques: an approach to model and simulate complex systems. Adv. Complex Syst. 05(02n03), 103–246 (2002)

    Google Scholar 

  39. Chopard, B., Ouared, R., Deutsch, A., Hatzikirou, H., Wolf gladrow, D.: Lattice-gas cellular automaton models for biology: from fluids to cells. Acta biotheoretica 58(4), 329–340 (2010)

    Google Scholar 

  40. Cirillo, E.N.M., Nardi, F.R.: Metastability for a stochastic dynamics with a parallel heat bath updating rule. J. Stat. Phys. 110(1–2), 183–217 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Cirillo, E.N.M., Nardi, F.R.: Relaxation height in energy landscapes: an application to multiple metastable states. J. Stat. Phys. 150(6), 1080–1114 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Cirillo, E.N.M., Nardi, F.R., Polosa, A.D.: Magnetic order in the Ising model with parallel dynamics. Phys. Rev. E 64, 057,103 (2001)

    Google Scholar 

  43. Cirillo, E.N.M., Nardi, F.R., Sohier, J.: Metastability for general dynamics with rare transitions: escape time and critical configurations. J. Stat. Phys. 161(2), 365–403 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Cirillo, E.N.M., Nardi, F.R., Spitoni, C.: Competitive nucleation in reversible probabilistic cellular automata. Phys. Rev. E 78 (2008)

    Google Scholar 

  45. Cirillo, E.N.M., Nardi, F.R., Spitoni, C.: Metastability for reversible probabilistic cellular automata with self-interaction. J. Stat. Phys. 132(3), 431–471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Coletti, C.F., Tisseur, P.: Invariant measures and decay of correlations for a class of ergodic probabilistic cellular automata. J. Stat. Phys. 140(1), 103–121 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. Cook, M.: Universality in Elementary Cellular Automata. Complex Syst. 15, 1–40 (2004)

    MathSciNet  MATH  Google Scholar 

  48. Cox, J.T., Durrett, R.: Limit theorems for the spread of epidemics and forest fires. Stochastic Process. Appl. 30, 171–191 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  49. Dai Pra, P., Louis, P.Y., Roelly, S.: Stationary measures and phase transition for a class of probabilistic cellular automata. ESAIM Probab. Stat. 6, 89–104 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  50. Dai Pra, P., Runggaldier, W., Sartori, E., Tolotti, M.: Large portfolio losses; a dynamic contagion model. Ann. Appl. Probab. 19(1), 1–38 (2007)

    MathSciNet  MATH  Google Scholar 

  51. Dai Pra, P., Scoppola, B., Scoppola, E.: Sampling from a Gibbs measure with pair interaction by means of PCA. J. Stat. Phys. 149(4), 722–737 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Dai Pra, P., Scoppola, B., Scoppola, E.: Fast mixing for the low temperature 2d Ising model through irreversible parallel dynamics. J. Stat. Phys. 159(1), 1–20 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Dawson, D.A.: Synchronous and asynchronous reversible Markov systems. Canad. Math. Bull. 17(5), 633–649 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  54. Dawson, D.A.: Stable states of probabilistic cellular automata. Inf. Control 34(2), 93–106 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  55. De Masi, A., Esposito, R., Lebowitz, J.L., Presutti, E.: Hydrodynamics of stochastic cellular automata. Commun. Math. Phys. 125(1), 127–145 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  56. Delorme, M., Mazoyer, J. (eds.): Cellular Automata: A Parallel Model. Springer, Berlin (1999)

    Google Scholar 

  57. Demongeot, J., Goles, E., Tchuente, M. (eds.): Dynamical Systems and Cellular Automata. Academic Press, New York (1985)

    Google Scholar 

  58. Dennunzio, A., Formenti, E., Fatès, N.: Foreword: cellular automata and applications. Nat. Comput. 12(3), 305–305 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  59. Depoorter, J., Maes, C.: Stavskaya’s measure is weakly Gibbsian. Markov Process. Related Fields 12(4), 791–804 (2006)

    MathSciNet  MATH  Google Scholar 

  60. Deroulers, C., Aubert, M., Badoual, M., Grammaticos, B.: Modeling tumor cell migration: from microscopic to macroscopic models. Phys. Rev. E 79(3), 1–14 (2009)

    Article  MathSciNet  Google Scholar 

  61. Derrida, B.: Dynamical phase transitions in spin models and automata. In: Fundamental Problems in Statistical Mechanics VII (Altenberg, 1989), pp. 273–309. North-Holland, Amsterdam (1990)

    Google Scholar 

  62. Deuschel, J.D., Greven, A. (eds.): Interacting Stochastic Systems. Springer, Berlin (2005)

    Google Scholar 

  63. Deutsch, A., Dormann, S., Maini, P.K.: Cellular Automaton Modeling of Biological Pattern Formation: Characterization, Applications, and Analysis. Modeling and Simulation in Science Engineering and Technology. Birkhäuser (2005)

    Google Scholar 

  64. Dhar, D.: Equivalence of the two-dimensional directed-site animal problem to Baxter’s hard-square lattice-gas model. Phys. Rev. Lett. 49(14), 959–962 (1982)

    Article  MathSciNet  Google Scholar 

  65. Diakonova, M., Mackay, R.S.: Mathematical examples of space-time phases. Int. J. Bifurc. Chaos 21(08), 1–8 (2011)

    Article  Google Scholar 

  66. Dobrushin, R.L.: Markov processes with a large number of locally interacting components: existence of a limit process and its ergodicity. Problemy Peredači Informacii 7(2), 70–87 (1971)

    MathSciNet  Google Scholar 

  67. Domany, E., Kinzel, W.: Equivalence of cellular automata to Ising models and directed percolation. Phys. Rev. Lett. 53(4), 311–314 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  68. Dommers, S., Giardinà, C., van der Hofstad, R.: Ising models on power-law random graphs. J. Stat. Phys. 141(4), 638–660 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  69. Drossel, B., Clar, S., Schwabl, F.: Exact results for the one-dimensional self-organized critical forest-fire model. Phys. Rev. Lett. 71(23), 3739 (1993)

    Article  Google Scholar 

  70. Drossel, B., Schwabl, F.: Self-organized critical forest-fire model. Phys. Rev. Lett. 69(11), 1629–1632 (1992)

    Article  Google Scholar 

  71. Durrett, R.: Oriented percolation in two dimension. Ann. Probab. 12(4), 929–1227 (1984)

    Article  MathSciNet  Google Scholar 

  72. Durrett, R.: Stochastic spatial models. SIAM Rev. 41(4), 677–718 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  73. Durrett, R., Levin, S.A.: Stochastic spatial models: a user’s guide to ecological applications. Philos. Trans. B 343, 329–350 (1994)

    Article  Google Scholar 

  74. Earl, D.J., Deem, M.W.: Parallel tempering: theory, applications, and new perspectives. Phys. Chem. Chem. Phys. PCCP 7(23), 3910–3916 (2005)

    Article  Google Scholar 

  75. Elsayed, W.M., El Bassiouny, A.H., Radwan, E.F.: Applying inhomogeneous probabilistic cellular automata rules on epidemic model. Int. J. Adv. Res. Artif. Intell. 2(4) (2013)

    Google Scholar 

  76. Farmer, J.D., Foley, D.: The economy needs agent-based modelling. Nature 460(7256), 685–686 (2009)

    Article  Google Scholar 

  77. Fatès, N.: FiatLux: a simulation program in Java for cellular automata and discrete dynamical systems. http://fiatlux.loria.fr (Cecill licence) APP IDDN.FR.001.300004.000.S.P.2013.000.10000

  78. Fatès, N.: Solving the decentralised gathering problem with a reaction-diffusion-chemotaxis scheme. Swarm Intell. 4(2), 91–115 (2010)

    Article  Google Scholar 

  79. Fatès, N.: A guided tour of asynchronous cellular automata. Lect. Notes Comput. Sci. 8155, 15–30 (2013)

    Article  MATH  Google Scholar 

  80. Fatès, N.: Stochastic cellular automata solutions to the density classification problem. Theory Comput. Syst. 53(2), 223–242 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  81. Fatès, N.: Aesthetics and Randomness in Cellular Automata. pp. 137–139. Springer International Publishing, Berlin (2016)

    Google Scholar 

  82. Fatès, N., Gerin, L.: Examples of fast and slow convergence of 2D asynchronous cellular systems. In: Umeo, H., Morishita, S., Nishinari, K., Komatsuzaki, T., Bandini, S. (eds.) Cellular Automata. Lecture Notes in Computer Science, vol. 5191, pp. 184–191. Springer, Berlin (2008)

    Chapter  Google Scholar 

  83. Fatès, N., Morvan, M.: An experimental study of robustness to asynchronism for elementary cellular automata. Complex Syst. 16(1), 1–27 (2005)

    MathSciNet  MATH  Google Scholar 

  84. Fernández, R., Toom, A.: Non-gibbsianness of the invariant measure of non-reversible cellular automata with totally asymmetric noise. Astérisque 287, 71–87 (2003)

    MathSciNet  MATH  Google Scholar 

  85. Ferrari, P.: Ergodicity for a class of probabilistic cellular automata. Rev. Mat. Apl. 12(2), 93–102 (1991)

    MathSciNet  MATH  Google Scholar 

  86. Ferrari, P., Frigessi, A., Schonmann, R.H.: Convergence of some partially parallel gibbs samplers with annealing. Ann. Appl. Probab. 3(1), 137–153 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  87. Soares Filho, B.S., Coutinho Cerqueira, G., Lopes Pennachin, C.: Dinamica–a stochastic cellular automata model designed to simulate the landscape dynamics in an Amazonian colonization frontier. Ecol. Model. 154(3), 217–235 (2002)

    Article  Google Scholar 

  88. Fisch, R., Gravner, J., Griffeath, D.: Metastability in the Greenberg-Hastings model. Ann. Appl. Probab. 3(4), 935–967 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  89. Föllmer, H.: Tail structure of Markov chains on infinite product spaces. Z. Wahrsch. Verw. Gebiete 285(3), 273–285 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  90. Fukś, H.: Non-deterministic density classification with diffusive probabilistic cellular automata. Phys. Rev. E 66(6), 1–4 (2002)

    Article  Google Scholar 

  91. Fukś, H., Lawniczak, A.T.: Individual-based lattice model for spatial spread of epidemics. Discret. Dyn. Nat. Soc. 6(3), 191–200 (2001)

    Article  MATH  Google Scholar 

  92. Gács, P.: Reliable computation with cellular automata. J. Comput. Syst. Sci. 32, 15–78 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  93. Gács, P.: Reliable cellular automata with self-organization. J. Stat. Phys. 103(1–2), 45–267 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  94. Galves, A., Löcherbach, E.: Modeling networks of spiking neurons as interacting processes with memory of variable length. Journal de la Société Française de Statistique 157(1), 17–32 (2016)

    MathSciNet  MATH  Google Scholar 

  95. Ganguly, N., Sikdar, B.K., Deutsch, A., Canright, G., Chaudhuri, P.P.: A survey on cellular automata. Engineering pp. 1–30 (2003)

    Google Scholar 

  96. Garijo, N., Manzano, R., Osta, R., Perez, M.A.: Stochastic cellular automata model of cell migration, proliferation and differentiation: validation with in vitro cultures of muscle satellite cells. J. Theor. Biol. 314, 1–9 (2012)

    Article  MathSciNet  Google Scholar 

  97. Garzon, M.: Models of Massive Paralellism: Analysis of Cellular Automata and Neural Networks. Springer, Berlin (1985)

    Google Scholar 

  98. Gaudillière, A., den Hollander, F., Nardi, F., Olivieri, E., Scoppola, E.: Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics. Stochastic Process. Appl. 119(3), 737–774 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  99. Gaudillière, A., Nardi, F.R.: An upper bound for front propagation velocities inside moving populations. Braz. J. Probab. Stat. 24(2), 256–278 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  100. Gaudillière, A., Olivieri, E., Scoppola, E.: Nucleation pattern at low temperature for local Kawasaki dynamics in two dimensions. Markov Process. Related Fields 11, 553–628 (2005)

    MathSciNet  MATH  Google Scholar 

  101. Gaudillière, A., Scoppola, B., Scoppola, E., Viale, M.: Phase transitions for the cavity approach to the clique problem on random graphs. J. Stat. Phys. 145(5), 1127–1155 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  102. Georges, A., Le Doussal, P.: From equilibrium spin models to probabilistic cellular automata. J. Stat. Phys. 54(3–4), 1011–1064 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  103. Giesecke, K., Weber, S.: Credit contagion and aggregate losses. J. Econom. Dyn. Control 30(5), 741–767 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  104. Goldstein, S., Kuik, R., Lebowitz, J.L., Maes, C.: From PCA to equilibrium systems and back. Commun. Math. Phys. 125(1), 71–79 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  105. Goltsev, A.V., de Abreu, F.V., Dorogovtsev, S.N., Mendes, J.F.F.: Stochastic cellular automata model of neural networks. Phys. Rev. E 81(6), 61,921 (2010)

    Google Scholar 

  106. Grassberger, P.: On a self-organized critical forest-fire model. J. Phys. A 26(9), 2081 (1993)

    Article  Google Scholar 

  107. Grassberger, P.: Critical behaviour of the Drossel-Schwabl forest fire model. New J. Phys. 4(1), 17 (2002)

    Article  Google Scholar 

  108. Gray, L.F.: A reader’s guide to Gacs’s “positive rates” paper. J. Stat. Phys 103(1–2), 1–44 (2001)

    Article  MATH  Google Scholar 

  109. Griffeath, D.: Self-organization of random cellular automata: four snapshots. In: G.R. Grimmett (ed.) Probability and Phase Transition, NATO ASI Series, vol. 420, pp. 49–67. Springer, Berlin (1994)

    Google Scholar 

  110. Grimmett, G.: Percolation. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  111. Grinstein, G.: Can complex structures be generically stable in a noisy world? IBM J. Res. Develop. 48(1), 5–12 (2004)

    Article  Google Scholar 

  112. Grinstein, G., Jayaprakash, C., He, Y.: Statistical mechanics of probabilistic cellular automata. Phys. Rev. Lett. 55(23), 2527–2530 (1985)

    Article  MathSciNet  Google Scholar 

  113. Habel, L., Schreckenberg, M.: Cellular Automata: 11th International Conference on Cellular Automata for Research and Industry, ACRI 2014, Krakow, Poland, September 22–25, 2014. Proceedings. In: J. Was, G.C. Sirakoulis, S. Bandini (eds.) Lecture Notes in Computer Science, vol. 8751, pp. 620–629 (2014)

    Google Scholar 

  114. Hatzikirou, H., Deutsch, A.: Cellular automata as microscopic models of cell migration in heterogeneous environments. Curr. Top. Dev. Biol. 81(07), 401–434 (2008)

    Article  Google Scholar 

  115. Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3(4), 320–375 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  116. Hoekstra, A., Kroc, J., Sloot, P.: Introduction to modeling of complex systems using cellular automata. In: Simulating Complex Systems by Cellular Automata, pp. 1–16 (2010)

    Google Scholar 

  117. Hogeweg, P.: Cellular automata as a paradigm for ecological modeling. Appl. Math. Comput. 27(1), 81–100 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  118. den Hollander, F., Nardi, F., Olivieri, E., Scoppola, E.: Droplet growth for three-dimensional Kawasaki dynamics. Probab. Theory Related Fields 125(2), 153–194 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  119. den Hollander, F., Olivieri, E., Scoppola, E.: Metastability and nucleation for conservative dynamics. J. Math. Phys. 41(3), 1424–1498 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  120. Holroyd, A.E., Marcovici, I., Martin, J.B.: Percolation games, probabilistic cellular automata, and the hard-core model. arXiv:1503.05614 [math] (2015)

  121. Horst, U.: Stochastic CAcade, credit contagion, and large portfolio losses. J. Econ. Behav. Organ. 63(1), 25–54 (2007)

    Article  Google Scholar 

  122. Ichise, Y., Ishida, Y.: Reverse engineering of spatial patterns in cellular automata. Artif. Life Robot. 13(1), 172–175 (2008)

    Article  Google Scholar 

  123. Ignatyuk, I.A., Malyshev, V.A.: Processes with local interaction, and communication networks. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 25(1), 65–77 (1989)

    Google Scholar 

  124. Ilachinski, A.: Cellular Automata; A Discrete Universe. World Scientific, Singapore (2001)

    Book  MATH  Google Scholar 

  125. Jahnel, B., Külske, C.: A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit. Stoch. Process. Appl. 125(6), 2427–2450 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  126. Jiang, Y.: Understanding a killer: a predictive model for tumor development. Contemp. Math. 410, 173–185 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  127. Just, W.: Toom’s model with Glauber rates: an exact solution. J. Stat. Phys. 139(6), 985–990 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  128. Karafyllidis, I., Thanailakis, A.: A model for predicting forest fire spreading using cellular automata. Ecol. Model. 99, 87–97 (1997)

    Article  Google Scholar 

  129. Kari, J.: Theory of cellular automata: a survey. Theoret. Comput. Sci. 334(1–3), 3–33 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  130. Kari, J., Taati, S.: Conservation laws and invariant measures in surjective cellular automata. In: AUTOMATA-2011, 2, pp. 113–122. DMTCS Proceedings (2012)

    Google Scholar 

  131. Kari, J.J.: Basic concepts of cellular automata. In: Rozenberg, G., Bäck, T., Kok, J.N. (eds.) Handbook of Natural Computing, pp. 3–24. Springer, Berlin (2012)

    Chapter  Google Scholar 

  132. Katori, M., Konno, N., Tanemura, H.: Limit theorems for the nonattractive Domany-Kinzel model. Ann. Probab. 30(2), 933–947 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  133. Katori, M., Tsukahara, H.: Two-neighbour stochastic cellular automata and their planar lattice duals. J. Phys. A 28(14), 3935 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  134. Kesten, H.: What is Percolation? Notices of the AMS (2006)

    Google Scholar 

  135. Kinouchi, O., Copelli, M.: Optimal dynamical range of excitable networks at criticality. Nat. Phys. 2(5), 348–351 (2006)

    Article  Google Scholar 

  136. Kinzel, W.: Phase transitions of cellular automata. Zeitschrift für Physik B Condensed Matter 58(3), 229–244 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  137. Kozlov, O., Vasilyev, N.: Reversible Markov chains with local interaction. Multicompon. Random Syst. 6, 451–469 (1980)

    MathSciNet  MATH  Google Scholar 

  138. Kozma, R., Puljic, M., Balister, P., Bollobas, B., Freeman, W.J.: Neuropercolation: a random cellular automata approach to spatio-temporal neurodynamics. In: Sloot, P.M.A., Chopard, B., Hoekstra, A.G. (eds.) Lecture Notes in Computer Science: Cellular Automata, Proceedings, vol. 3305, pp. 435–443. Springer, Berlin (2004)

    Google Scholar 

  139. Kriecherbauer, T., Krug, J.: A pedestrian’s view on interacting particle systems, KPZ universality, and random matrices. J. Phys. 43(40) (2010)

    Google Scholar 

  140. Künsch, H.: Nonreversible stationary measures for infinite interacting particle systems. Z. Wahrsch. Verw. Gebiete 66(3), 407–424 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  141. Künsch, H.: Time reversal and stationary Gibbs measures. Stoch. Process. Appl. 17(1), 159–166 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  142. Kurka, P.: Topological dynamics of one-dimensional cellular automata. Technical report

    Google Scholar 

  143. Lancia, C., Scoppola, B.: Equilibrium and Non-equilibrium ising models by means of PCA. J. Stat. Phys. 153(4), 641–653 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  144. Le Ny, A.: (Non-) Gibbs Description of Mean-field Models, vol. 60, chap. 21, pp. 463–480. Birkhäuser (2008)

    Google Scholar 

  145. Lebowitz, J.L.: Emergent Phenomena. Physik J. 6(8/9) (2007)

    Google Scholar 

  146. Lebowitz, J.L., Maes, C., Speer, E.R.: Statistical mechanics of probabilistic cellular automata. J. Stat. Phys. 59(1–2), 117–170 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  147. Lebowitz, J.L., Orlandi, E., Presutti, E.: Convergence of stochastic cellular automation to Burgers’ equation: fluctuations and stability. Physica D: Nonlinear Phenomena 33(1–3), 165–188 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  148. Levin, S.A., Durrett, R.: From individuals to epidemics. Philos. Trans. B 351(1347), 1615–1621 (1996)

    Article  Google Scholar 

  149. Liggett, T.M.: Stochastic models of interacting systems. Ann. Probab. 25(1), 1–29 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  150. Liggett, T.M.: Stochastic models for large interacting systems and related correlation inequalities. Proc. Natl. Acad. Sci. U.S.A. 107, 16413–16419 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  151. Louis, P.-Y.: Automates Cellulaires Probabilistes: mesures stationnaires, mesures de Gibbs associées et ergodicité. Ph.D. thesis, Politecnico di Milano, Italy and Université Lille 1, France (2002)

    Google Scholar 

  152. Louis, P.-Y.: Ergodicity of PCA: equivalence between spatial and temporal mixing conditions. Electron. Commun. Probab. 9, 119–131 (2004)

    Google Scholar 

  153. de Maere, A., Ponselet, L.: Exponential decay of correlations for strongly coupled toom probabilistic cellular automata. J. Stat. Phys. 147(3), 634–652 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  154. Maes, C., Shlosman, S.B.: Ergodicity of probabilistic cellular automata: a constructive criterion. Commun. Math. Phys. 135(2), 233–251 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  155. Mairesse, J., Marcovici, I.: Around probabilistic cellular automata. Theor. Comput, Sci. (2014)

    Google Scholar 

  156. Mairesse, J., Marcovici, I.: Probabilistic cellular automata and random fields with i.i.d. directions. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 50(2), 455–475 (2014)

    Google Scholar 

  157. Manchanda, K., Yadav, A.C., Ramaswamy, R.: Scaling behavior in probabilistic neuronal cellular automata. Phys. Rev. E 87(1), 12,704 (2013)

    Google Scholar 

  158. Manneville, P., Boccara, N., Vichniac, G., Bidaux, R.: Cellular Automata and the Modeling of Complex Physical Systems. Springer, Berlin (1989)

    Google Scholar 

  159. Margolus, N., Toffoli, T.: Cellular Automata Machines: A New Environment for Modeling, vol. 1. MIT Press (1987)

    Google Scholar 

  160. Martin, O.C., Monasson, R., Zecchina, R.: Statistical mechanics methods and phase transitions in optimization problems. Theor. Comput. Sci. 265(1–2), 3–67 (2001)

    Google Scholar 

  161. Martínez, G.J., Seck-Tuoh Mora, J.C., Zenil, H.: Wolfram’s classification and computation in cellular automata classes III and IV. In: Zenil, H. (ed.) Irreducibility and Computational Equivalence, vol. 2, pp. 237–259. Springer, Berlin (2013)

    Chapter  Google Scholar 

  162. Bousquet Mélou, M.: New enumerative results on two-dimensional directed animals. Discret. Math. 180(1–3), 73–106 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  163. Mendonça, J.: Sensitivity to noise and ergodicity of an assembly line of cellular automata that classifies density. Phys. Rev. E 83(3) (2011)

    Google Scholar 

  164. Mendonça, J.R.G.: A Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton. Phys. Rev. E 83(1), 18–21 (2011)

    Article  Google Scholar 

  165. Mendonça, J.R.G.: The inactive-active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton. Int. J. Mod. Phys. C 27(2) (2016)

    Google Scholar 

  166. de Menibus, B.H., Sablik, M.: Self-organization in cellular automata: a particle-based approach. In: G. Mauri, A. Leporati (eds.) Developments in Language Theory. Lecture Notes in Computer Sciences, vol. 6795, pp. 251–263. Springer, Berlin (2011)

    Google Scholar 

  167. Merks, R.M.H., Perryn, E.D., Shirinifard, A., Glazier, J.a.: Contact-inhibited chemotaxis in de novo and sprouting blood-vessel growth. PLoS Comput. Biol. 4(9), e1000, 163 (2008)

    Google Scholar 

  168. Mikler, A.R., Venkatachalam, S., Abbas, K.: Modeling infectious diseases using global stochastic cellular automata. J. Biol. Syst. 13(4), 421–439 (2005)

    Article  MATH  Google Scholar 

  169. Molofsky, J., Bever, J.D.: A New Kind of Ecology? BioScience 54(5), 440 (2004)

    Article  Google Scholar 

  170. Monetti, R.A., Albano, E.V.: On the emergence of large-scale complex behavior in the dynamics of a society of living individuals: the stochastic game of life. J. Theor. Biol. 187(2), 183–194 (1997)

    Article  Google Scholar 

  171. N., W., A., R.: The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch Inst Cardiol Mex (1946)

    Google Scholar 

  172. Nardi, F.R., Olivieri, E., Scoppola, E.: Anisotropy effects in nucleation for conservative dynamics. J. Stat. Phys. 119(3), 539–595 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  173. Nardi, F.R., Spitoni, C.: Sharp asymptotics for stochastic dynamics with parallel updating rule with self-interaction. J. Stat. Phys 4(146), 701–718 (2012)

    Article  MATH  Google Scholar 

  174. von Neumann, J.: The theory of self-reproducing automata. In: Burks, A.W. (ed.) The Theory of Self-Reproducing Automata. University of Illinois Press, Urbana (1966)

    Google Scholar 

  175. Ódor, G., Szolnoki, A.: Directed-percolation conjecture for cellular automata. Phys. Rev. E 53, 2231–2238 (1996)

    Article  Google Scholar 

  176. Olivieri, E., Vares, M.E.: Large deviations and metastability, Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  177. Penington, C., Hughes, B., Landman, K.: Building macroscale models from microscale probabilistic models: A general probabilistic approach for nonlinear diffusion and multispecies phenomena. Phys. Rev. E 84(4), 41,120 (2011)

    Google Scholar 

  178. Ponselet, L.: Phase transitions in probabilistic cellular automata. Ph.D. thesis, Université catholique de Louvain (2013)

    Google Scholar 

  179. Procacci, A., Scoppola, B., Scoppola, E.: Probabilistic Cellular Automata for low temperature Ising model. J. Stat. Phys. 165, 991–1005 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  180. Rajewsky, N., Schreckenberg, M.: A probabilistic cellular automaton for evolution. J. Phys. 5(9), 1129–1134 (1995)

    MathSciNet  Google Scholar 

  181. Regnault, D.: Directed percolation arising in stochastic cellular automata analysis. In: Ochmanski, E., Tyszkiewicz, J. (eds.) Mathematical Foundations of Computer Science 2008. Lecture Notes in Computer Science, vol. 5162, pp. 563–574. Springer, Berlin (2008)

    Chapter  Google Scholar 

  182. del Rey, A.M.: A computer virus spread model based on cellular automata on graphs. In: Omatu, S., Rocha, M.P., Bravo, J., Fernández, F., Corchado, E., Bustillo, A., Corchado, J.M. (eds.) LNCS : Distributed Computing, Artificial Intelligence, Bioinformatics, Soft Computing, and Ambient Assisted Living, vol. 5518, pp. 503–506. Springer, Berlin (2009)

    Google Scholar 

  183. Rouquier, J.-B.: Robustesse et émergence dans les systèmes complexes: le modèle des automates cellulaires. Ph.D. thesis, ÉNS Lyon (2008)

    Google Scholar 

  184. Rouquier, J.-B., Morvan, M.: Coalescing cellular automata: synchronization by common random source for asynchronous updating. J. Cell. Automata 4, 55–78 (2009)

    Google Scholar 

  185. Sarkar, P.: A brief history of cellular automata. ACM Comput. Surv. 32(1), 80–107 (2000)

    Article  Google Scholar 

  186. Scalise, D., Schulman, R.: Emulating cellular automata in chemical reaction-diffusion networks. Nat. Comput. 15(2), 197–214 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  187. Schiff, J.L.: Cellular Automata: A Discrete View of the World. Wiley (2012)

    Google Scholar 

  188. Schneckenreither, G., Popper, N., Zauner, G., Breitenecker, F.: Modelling SIR-type epidemics by ODEs, PDEs, difference equations and cellular automata - A comparative study. Simul. Model. Pract. Theory 16(8), 1014–1023 (2008)

    Article  Google Scholar 

  189. Schnell, S., Grima, R., Maini, P.: Multiscale modeling in biology. Am. Sci. 95, 134–142 (2007)

    Article  Google Scholar 

  190. Schonmann, H.R., Shlosman, B.S.: Wulff droplets and the metastable relaxation of kinetic isingmodels. Commun. Math. Phys. 194(2), 389–462 (1998)

    Article  MATH  Google Scholar 

  191. Shnirman, M.: On the problem of ergodicity of a Markov chain with infinite set of states. Probl. Kibern. 20, 115–124 (1968)

    Google Scholar 

  192. Simpson, M.J., Merrifield, A., Landman, K.A., Hughes, B.D.: Simulating invasion with cellular automata: connecting cell-scale and population-scale properties. Phys. Rev. E 76(2), 021918 (2007)

    Google Scholar 

  193. Sipper, M.: Simple + parallel + local = Cellular computing. In: M. Schwefel, A. Eiben, E. Bäck, T. Schoenauer (ed.) Fifth International Conference on Parallel Problem Solving from Nature (PPSN V), vol. 1498, Lecture Notes in Computer Science, pp. 653–662. Springer, Heidelberg (1998)

    Google Scholar 

  194. Slimi, R., El Yacoubi, S.: Spreadable Probabilistic Cellular Automata model: an application in epidemiology. In: ACRI 2006. Lecture Notes in Computer Science, vol. 4173, pp. 330–336. Springer, Berlin (2006)

    Google Scholar 

  195. Słowiński, P., MacKay, R.S.: Phase diagrams of majority voter probabilistic cellular automata. J. Stat. Phys. 159(1), 43–61 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  196. Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970)

    Google Scholar 

  197. Stavskaya, O.N., Piatetsky shapiro, G.: Homogeneous networks of spontaneously active elements. Problemy Kibernet 20, 91–106 (1968)

    Google Scholar 

  198. Sutner, K.: Computational classification of cellular automata. Int. J. Gen. Syst. 41(6), 595–607 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  199. Szabó, A., Merks, R.M.H.: Cellular potts modeling of tumor growth, tumor invasion, and tumor evolution. Front. Oncol. 3, 87 (2013)

    Article  Google Scholar 

  200. Taggi, L.: Critical probabilities and convergence time of percolation probabilistic cellular automata. J. Stat. Phys. 159(4), 853–892 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  201. Toffoli, T.: Cellular automata as an alternative to (rather than an approximation of) differential equations in modeling physics. Phys. D 10, 117–127 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  202. Tomé, T., de Felício, J.R.D.: Probabilistic cellular automaton describing a biological immune system. Phys. Rev. E 53(4), 3976–3981 (1996)

    Article  Google Scholar 

  203. Toom, A.: Nonergodic multidimensional systems of automata. Probl. Inf. Trans. 10, 239–246 (1974)

    MathSciNet  MATH  Google Scholar 

  204. Toom, A.: Multicomponent Random Systems, chap. Stable and Attractive Trajectories in Multicomponent Systems, pp. 549–575. Marcel Dekker Inc (1980)

    Google Scholar 

  205. Toom, A.: Algorithmical unsolvability of the ergodicity problem for binary cellular automata. Markov process. Related Fields 6(4), 569–577 (2000)

    MathSciNet  MATH  Google Scholar 

  206. Toom, A.L., Vasilyev, N.B., Stavskaya, O.N., Mityushin, L.G., Kurdyumov, G.L., Pirogov, S.A.: Locally interacting systems and their application in biology. In: Dobrushin, Kryukov, Toom (eds.) Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, pp. 1–182. Springer, Berlin (1978)

    Google Scholar 

  207. Tristan, J.B., Zaheer, M., Steel, G.L.J., Green, S.J.: Learning topics with stochastic cellular automata dynamics. In: Neural Information Processing Systems (2015)

    Google Scholar 

  208. Ulam, S.: Random processes and transformations. In: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, pp. 264–275. American Mathematical Society, Providence, R. I. (1952)

    Google Scholar 

  209. Vaserstein, L.N.: Markov processes over denumerable products of spaces describing large system of automata. Problemy Peredači Informacii 5(3), 64–72 (1969)

    Google Scholar 

  210. Vaserstein, L.N., Leontovich, A.M.: Invariant measures of certain Markov operators that describe a homogeneous random medium. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 6(1), 71–80 (1970)

    Google Scholar 

  211. Vichniac, G.Y.: Simulating physics with cellular automata. Phys. D 10, 96–116 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  212. Werner, W.: Percolation et modèle d’Ising. Collection SMF. Société Mathématique de France, Paris (2009)

    MATH  Google Scholar 

  213. White, S.H., Del Rey, A.M., Sanchez, G.R.: Using cellular automata to simulate epidemic diseases. Appl. Math. Sci. 3, 959–968 (2009)

    MathSciNet  MATH  Google Scholar 

  214. Wolfram, S.: Statistical mechanics of cellular automata. Rev. Mod. Phys. 55(3), 601–644 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  215. Wolfram, S.: Cellular automata as models of complexity. Nature 311(5985), 419–424 (1984)

    Article  Google Scholar 

  216. Wolfram, S.: Computation theory of cellular automata. Commun. Math. Phys. 96(1), 15–57 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  217. Worsch, T.: Cellular automata as models of parallel computation. In: Computational Complexity Theory, Techniques, and Applications, pp. 298–311 (2012)

    Google Scholar 

  218. Wu, A.C., Xu, X.J., Wang, Y.H.: Excitable greenberg-hastings cellular automaton model on scale-free networks. Phys. Rev. E 75, 032,901 (2007)

    Google Scholar 

  219. Xiao, X., Shao, S.H., Chou, K.C.: A probability cellular automaton model for hepatitis B viral infections. Biochem. Biophys. Res. Commun. 342(2), 605–610 (2006)

    Article  Google Scholar 

  220. Zamith, M., Leal-Toledo, R.C.P., Clua, E., Toledo, E.M., Magales, G.V.d.: A new stochastic cellular automata model for traffic flow simulation with drivers’ behavior prediction. J. Comput. Sci. 9, 51–56 (2015)

    Google Scholar 

  221. Zinck, R.D., Johst, K., Grimm, V.: Wildfire, landscape diversity and the Drossel-Schwabl model. Ecol. Model. 221(1), 98–105 (2010)

    Article  Google Scholar 

  222. Zorzenon dos Santos, R., Coutinho, S.: Dynamics of HIV infection: a cellular automata approach. Phys. Rev. Lett. 87(16), 168,102 (2001)

    Google Scholar 

  223. Zuse, K.: The computing universe. Int. J. Theor. Phys. 21(6–7), 589–600 (1982)

    Article  Google Scholar 

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Acknowledgements

The authors of this overview chapter thank the full editorial board for their help to set up this book. [We thank, in particular, R. Merks and N. Fatés for their comments and suggestions] about this chapter.

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Authors and Affiliations

  1. Universiteit Utrecht, Utrecht, The Netherlands

    Roberto Fernández

  2. Laboratoire de Mathématiques et Applications UMR 7348, Université de Poitiers, CNRS, 11 Boulevard Marie et Pierre Curie, Téléport 2 - BP 30179, Poitiers, France

    Pierre-Yves Louis

  3. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Francesca R. Nardi

  4. Università di Firenze, Florence, Italy

    Francesca R. Nardi

  5. Dipartmento di Matematica e Informatica, Universitá di Firenze, Firenze, Italy

    Francesca R. Nardi

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    Francesca R. Nardi

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Fernández, R., Louis, PY., Nardi, F.R. (2018). Overview: PCA Models and Issues. In: Louis, PY., Nardi, F. (eds) Probabilistic Cellular Automata. Emergence, Complexity and Computation, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-65558-1_1

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