On the Analysis of "Simple" 2D Stochastic Cellular Automata

We analyze the dynamics of a two-dimensional cellular automaton, 2D Minority, for the Moore neighborhood (eight neighbors per cell) under fully asynchronous dynamics (where only one random cell updates at each time step). Even if 2D Minority seems a simple rule, from the experience of Ising models and Hopfield nets, it is known that models with negative feedback are hard to study. This automaton actually presents a rich variety of behaviors, even more complex than what was observed and analyzed in a previous work on 2D Minority for the von Neumann neighborhood (four neighbors per cell) [1], including particles and a wider range of stable configurations. Nevertheless our work suggests that predicting the behavior of this automaton although difficult is possible, opening the way to analyze the class of totalistic automata.