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 $n\to \mathrm{\infty }$, of the longest temporal period $X_{\mathit{\sigma },n}$ of a periodic solution with a given spatial period σ. We prove, when $\mathit{\sigma }\le r$, that this random variable is of order $n^{\mathit{\sigma }∕2}$, in that $X_{\mathit{\sigma },n}∕n^{\mathit{\sigma }∕2}$ converges to a nontrivial distribution. For the case $\mathit{\sigma }>r$, we present empirical evidence in support of the conjecture that the same result holds.