Journal of Statistical Physics

, Volume 171, Issue 3, pp 470–483 | Cite as

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

  • Eric Foxall
  • Hanbaek Lyu


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 xy, with probability tending to 1 as \(t\rightarrow \infty \), x and y have the same color at time t. Here we prove that conjecture.


Interacting particle system Cyclic particle system Rock paper scissors model Multitype voter model Annihilating particle system Clustering 


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlbertaEdmontonCanada
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

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