Clustering in the three and four colour cyclic particle systems in one dimension
We study the $k$-color cyclic particle system on the one-dimensional integer lattice,
first introduced by Bramson and Griffeath. In their original article they show that almost surely,
every site changes its color infinitely often if $k \in \{3, 4\}$ and only finitely many times if $k \ge 5$.
In addition, they conjecture that for $k \in \{3, 4\}$ 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.
Joint work with Hanbaek Lyu.