Flux and Fixation in Cyclic Particle Systems

Start by randomly coloring each site of the one-dimensional integer lattice with any of N colors, labeled 0, 1, …, N - 1. Consider the following simple continuous time Markovian evolution. At exponential rate 1, the color ξ(y) at any site y randomly chooses a neighboring site x ∈ - 1, y + 1\ and paints x with its color provided ξ(y) - ξ(x) = 1 mod N. Call this interacting process the cyclic particle system on N colors. We show that there is a qualitative change in behavior between the systems with N ≤ 4 and those with N ≥ 5. Specifically, if N ≥ 5 we show that the process fixates. That is, each site is painted a final color with probability 1. For N ≤ 4, on the other hand, we show that every site changes color at arbitrarily large times with probability 1.