Progresses in the analysis of stochastic 2D cellular automata: A study of asynchronous 2D minority

Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changes under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on 1D probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from 1D to 2D systems is a long lasting challenging problem. In this paper, we address the problem of analyzing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our simulations reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata.