Flux and Fixation in Cyclic Particle Systems

Abstract

Start by randomly coloring each site of the one-dimensional integer lattice with any of $N$ colors, labeled $0, 1, \ldots, N - 1$. Consider the following simple continuous time Markovian evolution. At exponential rate 1, the color $\xi(y)$ at any site $y$ randomly chooses a neighboring site $x \in \{y - 1, y + 1\}$ and paints $x$ with its color provided $\xi(y) - \xi(x) = 1 \operatorname{mod} N$. 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 $N \leq 4$ and those with $N \geq 5$. Specifically, if $N \geq 5$ we show that the process fixates. That is, each site is painted a final color with probability 1. For $N \leq 4$, on the other hand, we show that every site changes color at arbitrarily large times with probability 1.