We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.
@article{10_37236_10114,
author = {Janko Gravner and Xiaochen Liu},
title = {Periodic solutions of one-dimensional cellular automata with uniformly chosen random rules},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/10114},
zbl = {1493.68234},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10114/}
}
TY - JOUR
AU - Janko Gravner
AU - Xiaochen Liu
TI - Periodic solutions of one-dimensional cellular automata with uniformly chosen random rules
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10114/
DO - 10.37236/10114
ID - 10_37236_10114
ER -
%0 Journal Article
%A Janko Gravner
%A Xiaochen Liu
%T Periodic solutions of one-dimensional cellular automata with uniformly chosen random rules
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10114/
%R 10.37236/10114
%F 10_37236_10114
Janko Gravner; Xiaochen Liu. Periodic solutions of one-dimensional cellular automata with uniformly chosen random rules. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/10114
