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The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics. (English) Zbl 0701.60105
Let $$N\geq 2$$ be a fixed integer and $$X=\{0,1,2,...,N-1\}^ Z$$. The one- dimensional N-color cyclic cellular automaton $$\{\eta_ i\}$$ is a time- discrete process evolving in the following way: at the beginning, $$\eta_ 0$$ is distributed uniformly in the N-colors: $$P[\eta_ 0(x)=j]=1/N$$ for all $$x\in Z$$. Then, for each $$x\in Z$$, $\eta_{i+1}(x)=(\eta_ i(x)+1)mod N\quad if\quad \eta_ i(x+1)\quad or\quad \eta_ i(x-1)\equiv (\eta_ i(x)+1)mod N,\text{ and } \eta_{i+1}(x)=\eta_ i(x)\quad otherwise.$ This paper proves that each site changes color infinitely often if $$N\leq 4$$ but only finitely often if $$N\geq 5$$. This result is an analogue of that obtained by M. Bramson and D. Griffeath [Ann. Probab. 17, 26-45 (1989; Zbl 0673.60103) ] for the corresponding particle system with continuous time parameter. Here, for $$N=5$$, a new argument is needed.
Reviewer: Mufa Chen

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
interacting particle system
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##### References:
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