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 in \cite{bramson1989flux}. 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 $t\to\infty$, $x$ and $y$ have the same color at time $t$. Here we prove that conjecture.