Probability Seminar: Eric Foxall

We study the κ-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 κ ∈ {3, 4} and only finitely many times if κ ≥ 5. In addition, they conjecture that for κ ∈ {3, 4} the system clusters, that is, for any pair of sites x, y, with probability tending to 1 as t → ∞, x and y have the same color at time t. Here we prove that conjecture. Joint work with Hanbaek Lyu.