Clustering in the Three and Four Color Cyclic Particle Systems in One Dimension |
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Authors: | Eric Foxall Hanbaek Lyu |
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Institution: | 1.Department of Mathematics,University of Alberta,Edmonton,Canada;2.Department of Mathematics,The Ohio State University,Columbus,USA |
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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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