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Clustering in the Three and Four Color Cyclic Particle Systems in One Dimension

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Abstract

We study the \(\kappa \)-color cyclic particle system on the one-dimensional integer lattice \(\mathbb {Z}\), first introduced by Bramson and Griffeath (Ann Prob:26–45, 1989). In that paper they show that almost surely, every site changes its color infinitely often if \(\kappa \in \{3,4\}\) and only finitely many times if \(\kappa \ge 5\). In addition, they conjecture that for \(\kappa \in \{3,4\}\) the system clusters, that is, for any pair of sites x, y, with probability tending to 1 as \(t\rightarrow \infty \), x and y have the same color at time t. Here we prove that conjecture.

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  • 15 October 2020

    There is an error in the proof of Lemma 4.1.

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Authors and Affiliations

  1. Department of Mathematics, University of Alberta, Edmonton, AB, Canada

    Eric Foxall

  2. Department of Mathematics, The Ohio State University, Columbus, OH, USA

    Hanbaek Lyu

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  1. Eric Foxall
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  2. Hanbaek Lyu
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Correspondence to Hanbaek Lyu.

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Foxall, E., Lyu, H. Clustering in the Three and Four Color Cyclic Particle Systems in One Dimension. J Stat Phys 171, 470–483 (2018). https://doi.org/10.1007/s10955-018-2004-2

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