One-dimensional cellular automata with random rules: longest temporal period of a periodic solution

Abstract

We study one-dimensional cellular automata whose rules are chosen at random from among r-neighbor rules with a large number n of states. Our main focus is the asymptotic behavior, as n→∞, of the longest temporal period Xσ,n of a periodic solution with a given spatial period σ. We prove, when σ≤r, that this random variable is of order nσ∕2, in that Xσ,n∕nσ∕2 converges to a nontrivial distribution. For the case σ>r, we present empirical evidence in support of the conjecture that the same result holds.