Abstract
We present a method for computing probability of occurrence of ones in a configuration obtained by iteration of a probabilistic cellular automata (PCA), starting from a random initial configuration. If the PCA is sufficiently simple, one can construct a set of words (or blocks of symbols) which is complete, meaning that probabilities of occurrence of words from this set can be expressed as linear combinations of probabilities of occurrence of these words at the previous time step. One can then set up and solve a recursion for block probabilities. We demonstrate an example of such PCA, which can be viewed as a simple model of diffusion of information or spread of rumours. Expressions for the density of ones are obtained for this rule using the proposed method.
References
Boccara, N., FukÅ›, H.: Modeling diffusion of innovations with probabilistic cellular automata. In: Delorme, M., Mazoyer, J. (eds.) Cellular Automata: A Parallel Model. Kluwer Academic Publishers, Dordrecht (1998)
Cull, P., Flahive, M., Robson, R.: Difference Equations. Springer, Berlin (2004)
Fukś, H.: Dynamics of the cellular automaton rule 142. Complex Syst. 16, 123–138 (2006)
Fukś, H.: Probabilistic initial value problem for cellular automaton rule 172. DMTCS Proc. AL, 31–44 (2010)
Fukś, H.: Construction of local structure maps for cellular automata. J. Cell. Autom. 7, 455–488 (2013)
Fukś, H., Gómez Soto, J.-M.: Exponential convergence to equilibrium in cellular automata asymptotically emulating identity. Complex Syst. 23, 1–26 (2014)
Fukś, H., Skelton, A.: Response curves for cellular automata in one and two dimensions - an example of rigorous calculations. Int. J. Nat. Comput. Res. 1, 85–99 (2010)
Fukś, H., Skelton, A.: Orbits of Bernoulli measure in asynchronous cellular automata. Discret. Math. Theor. Comput. Sci. AP, 95–112 (2011)
Kůrka, P.: On the measure attractor of a cellular automaton. Discret. Contin. Dyn. Syst. 524–535 (2005)
Kůrka, P., Maass, A.: Limit sets of cellular automata associated to probability measures. J. Stat. Phys. 100, 1031–1047 (2000)
Pivato, M.: Conservation laws in cellular automata. Nonlinearity 15(6), 1781 (2002)
Stauffer, D., Aharony, A.: Introduction to Percolation Theory. Taylor and Francis, London (1994)
Acknowledgements
The author acknowledges partial financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of Discovery Grant. Some calculations on which this work is based were made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada. The author thanks anonymous referees for suggestions leading to improvement of the article, including a simpler derivation of the cluster expansion formula.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
FukÅ›, H. (2018). An Example of Computation of the Density of Ones in Probabilistic Cellular Automata by Direct Recursion. 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_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-65558-1_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65556-7
Online ISBN: 978-3-319-65558-1
eBook Packages: EngineeringEngineering (R0)