The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics

Abstract

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.