Quick Energy Drop in Stochastic 2D Minority

Abstract

Cellular automata are usually updated synchronously and thus deterministically. The question of stochastic dynamics arises in the development of cellular automata resistant to noise [1] and in simulation of real life systems [2]. Synchronous updates may not be a valid hypothesis for such simulations and most of these studies use stochastic versions of cellular automata.

In [3,4,5,6], the authors study different classes of cellular automata under fully asynchronous dynamics (only one random cell fires at each time step) and α-asynchronous dynamics (each cell has a probability α to fire at each time step). They develop tools and methods to ease the study of other cellular automata. In [4,6], they analyze 2D Minority under fully asynchronous dynamics for Von Neumann and Moore neighborhoods. The behavior of this cellular automaton under these dynamics is surprisingly rich. The energy of a configuration is an useful information. In [4], it is proved that configurations of energy greater than $\frac{5mn}{3}$ (where m and n are the length and the width of the configuration) will not appear in the long range behavior of 2D minority for Von Neumann neighborhood. In this paper we improve this bound to $18\lceil \frac{m}{4} \rceil \lceil \frac{n}{4} \rceil$. The proof is based on an enumeration of cases made by computer. This method could be easily tuned for other cellular automata or neighborhoods.