Clustering in the three and four color cyclic particle systems in one dimension.

We study clustering in the cyclic particle systems on the one-dimensional integer lattice $\mathbb{Z}$, first introduced by Bramson and Griffeath in \cite{bramson1989flux}. To start the process, we randomly color each site of $\mathbb{Z}$ with one of the $\kappa\ge 3$ colors in $\mathbb{Z}_{\kappa}$, according to the uniform product measure, denoting the color at $y$ by $X(y)$. At exponential rate 1, each site $y\in \mathbb{Z}$ randomly chooses one of its neighbors $x\in \{y\pm 1\}$, and paints it with color $X(y)$, provided $X(y)-X(x)\equiv 1$ mod $\kappa$. It is known that every site changes its color infinitely often almost surely for $\kappa\in \{3,4\}$ and only finitely many times almost surely for $\kappa\ge 5$. In 1989, it was conjectured by Bramson and Griffeath that for $\kappa\in \{3,4\}$ the system clusters, that is, the density of dichromatic edges at time $t$ decays to 0 almost surely, as $t$ tends to infinity. We prove this conjecture.