Abstract
A construction of a finite probabilistic cellular automaton is presented and the respective evolution rule is defined for a restricted space of states. The choice of variables describing the stationary state of the automaton—expected values weighted by the limiting distribution—is justified in terms of Markov chains framework. A system of equations for stationary state in mean-field approximation is derived and then reduced, by a special choice of dynamic parameters, to special case for which the system of equations is transformed into truncated Catalan numbers recurrence. Full recurrence appear in the limit of infinite size of the automaton.
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Acknowledgements
The author has been supported by the National Science Center (Poland) under research grant No. 2017/27/B/ST10/02686 and by a subsidy from the Polish Ministry of Education and Science for the Institute of Geophysics PAS. The author thanks Toktam Zand for the comments on the text.
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Białecki, M. (2023). Catalan Numbers Recurrence as a Stationary State Equation of the Probabilistic Cellular Automaton. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_7
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