Asymptotic behaviour of the one-dimensional ``rock-paper-scissors'' cyclic cellular automaton - Archive ouverte HAL Access content directly
Journal Articles Annals of Applied Probability Year : 2020

Asymptotic behaviour of the one-dimensional ``rock-paper-scissors'' cyclic cellular automaton

Abstract

The one-dimensional three-state cyclic cellular automaton is a simple spatial model with three states in a cyclic ``rock-paper-scissors'' prey-predator relationship. Starting from a random configuration, similar states gather in increasingly large clusters; asymptotically, any finite region is filled with a uniform state that is, after some time, driven out by its predator, each state taking its turn in dominating the region (heteroclinic cycles). We consider the situation where each site in the initial configuration is chosen independently at random with a different probability for each state. We prove that the asymptotic probability that a state dominates a finite region corresponds to the initial probability of its prey. The proof methods are based on discrete probability tools, mainly particle systems and random walks.
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Origin : Files produced by the author(s)
Origin : Files produced by the author(s)

Dates and versions

hal-02084842 , version 1 (29-03-2019)

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  • HAL Id : hal-02084842 , version 1

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Benjamin Hellouin de Menibus, Yvan Le Borgne. Asymptotic behaviour of the one-dimensional ``rock-paper-scissors'' cyclic cellular automaton. Annals of Applied Probability, In press. ⟨hal-02084842⟩
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