Open Access
January, 1989 Flux and Fixation in Cyclic Particle Systems
Maury Bramson, David Griffeath
Ann. Probab. 17(1): 26-45 (January, 1989). DOI: 10.1214/aop/1176991492


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.


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Maury Bramson. David Griffeath. "Flux and Fixation in Cyclic Particle Systems." Ann. Probab. 17 (1) 26 - 45, January, 1989.


Published: January, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0673.60103
MathSciNet: MR972768
Digital Object Identifier: 10.1214/aop/1176991492

Primary: 60K35

Keywords: cellular automation , Infinite particle system

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 1 • January, 1989
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