The Annals of Probability

Clustering in the One-Dimensional Three-Color Cyclic Cellular Automaton

Robert Fisch

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Abstract

This paper investigates the dynamics of the one-dimensional three-color cyclic cellular automaton. The author has previously shown that this process fluctuates, meaning that each lattice site changes color infinitely often, so that there is no "final state" for the system. The focus of the current work is on the clustering properties of this system. This paper demonstrates that the one-dimensional three-color cyclic cellular automaton clusters, and the mean cluster size, as a function of time $t$, is asymptotic to $ct^{1/2}$, where $c$ is an explicitly calculable constant. The method of proof also allows us to compute asymptotic estimates of the mean interparticle distance for a one-dimensional system of particles which undergo deterministic motion and which annihilate upon collision. No clustering results are known about the four-color process, but evidence is presented to suggest that the mean cluster size of such systems grows at a rate different from $t^{1/2}$.

Article information

Source
Ann. Probab. Volume 20, Number 3 (1992), 1528-1548.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176989705

Digital Object Identifier
doi:10.1214/aop/1176989705

Mathematical Reviews number (MathSciNet)
MR1175276

Zentralblatt MATH identifier
0806.60096

JSTOR
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
cellular automaton particle systems cluster size interparticle distance

Citation

Fisch, Robert. Clustering in the One-Dimensional Three-Color Cyclic Cellular Automaton. Ann. Probab. 20 (1992), no. 3, 1528--1548. doi:10.1214/aop/1176989705. http://projecteuclid.org/euclid.aop/1176989705.


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