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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

60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI
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