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

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 (trightarrow infty ), x and y have the same color at time t. Here we prove that conjecture.