Comparison between simultaneous and sequential updating in 2n+1−1 cellular automata

Properties of cellular automata with Packard-Wolfram code 2n+1-1 have been studied without and with noise on square (n = 4) and triangular (n = 6) lattices. The outer totalistic rule used assign a central spin opposite to the majority spin of its neighbourhood and flips it in the case of equality. Starting from random configurations with a variable proportion r of up spins, different relaxation behaviours and equilibrium structures are obtained: a ferromagnetic (F) structure and an antiferromagnetic (AF) structure for simultaneous and sequential updatings respectively on a square lattice. When the spins are processed simultaneously in the presence of noise, a first-order transition F → AF is observed for a square lattice and a second-order transition for a triangular lattice. The results show similarities with Ising models (domains) but also strong differences (magnetization as a function of r).