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October 2021 Asymptotic behaviour of the one-dimensional “rock–paper–scissors” cyclic cellular automaton
Benjamin Hellouin de Menibus, Yvan Le Borgne
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Ann. Appl. Probab. 31(5): 2420-2440 (October 2021). DOI: 10.1214/20-AAP1651

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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Benjamin Hellouin de Menibus. Yvan Le Borgne. "Asymptotic behaviour of the one-dimensional “rock–paper–scissors” cyclic cellular automaton." Ann. Appl. Probab. 31 (5) 2420 - 2440, October 2021. https://doi.org/10.1214/20-AAP1651

Information

Received: 1 April 2019; Revised: 1 May 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

Digital Object Identifier: 10.1214/20-AAP1651

Subjects:
Primary: 60J10
Secondary: 37A50 , 37B15 , 60J70 , 92D25

Keywords: cellular automata , Cyclic dominance , heteroclinic cycle , Population dynamics , Random walk

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 5 • October 2021
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