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

TL;DR

This paper analyzes the long-term behavior of a one-dimensional cyclic cellular automaton with three states, showing how initial probabilities influence which state dominates regions over time.

Contribution

It provides a rigorous proof that initial state probabilities determine the asymptotic dominance in a cyclic cellular automaton, using particle systems and random walk techniques.

Findings

Initial probabilities predict long-term dominance of states.

States follow heteroclinic cycles, replacing each other over time.

Asymptotic dominance matches initial prey probabilities.

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.