The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics
- Robert Fisch
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
Consider a cellular automaton defined on ℤ where each lattice site may take on one ofN values, referred to as colors. TheN colors are arranged in a cyclic hierarchy, meaning that colork follows colork−1 modN (k=0,...,N−1). Any two colors that are not adjacent in this hierarchy form an inert pair. In this scheme, there is symmetry in theN colors. Initialized the cellular automaton with product measure, and let time pass in discrete units. To get the configuration at timet+1 from the one at timet, each lattice site looks at the colors of its two nearest neighbors, and if it sees the color that follows its own color, then that site changes color to the color that follows; otherwise, that site does not change color. All such updates occur synchronously at timet+1. For each value ofN≥2, the fundamental question is whether each site in the cellular automaton changes color infinitely often (fluctuation) or only finitely often (fixation). We prove here that ifN≤4, then fluctuation occurs, and ifN≥5, then fixation occurs.
- Bramson, M., Griffeath, D. (1989) Flux and fixation in cyclic particle systems. Ann. Prob. 17: pp. 26-45
- Cramer, H. (1937) On a new limit theorem in the theory of probability. Hermann, Paris
- Fisch, R. (1988). One-dimensional cyclic cellular automata. Ph.D. thesis, University of Wisconsin—Madison.
- Liggett, T. M. (1985) Interacting Particle Systems. Springer-Verlag, New York
- Toffoli, T., Margolis, N. (1987) Cellular Automata Machines. MIT Press, Cambridge, Massachusetts
- Varadhan, S. R. S. (1984).Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.
- Wolfram, S. (ed.) (1986).Theory and Applications of Cellular Automata. World Scientific.
- The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics
Journal of Theoretical Probability
Volume 3, Issue 2 , pp 311-338
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- Cellular automaton
- interacting particle system
- critical value for a one-parameter family of processes
- Industry Sectors
- Robert Fisch (1)
- Author Affiliations
- 1. Department of Mathematics, University of North Carolina, 28223, Charlotte, North Carolina