2019/09/15 by Janko Gravner, Xiaochen Liu, Gravner, Janko +1
Computer Science · Mathematics · Physics and Astronomy · #37B15 #60K35 #68Q80 #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1909.06914
openalex publication_date 2019/09/15 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28
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.