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Flux and fixation in cyclic particle systems. (English) Zbl 0673.60103

Authors’ abstract: 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 \(\xi\) (y) at any site y randomly chooses a neighboring site \(x\in \{y-1,y+1\}\) and paints x with its color provided \(\xi (y)-\xi (x)=1\) 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.
Reviewer: P.A.Kučment

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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