Fully asynchronous behavior of double-quiescent elementary cellular automata

In this paper we propose a probabilistic analysis of the fully asynchronous behavior (i.e., two cells are never simultaneously updated, as in a continuous time process) of elementary finite cellular automata (i.e., {0,1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0,0,0)↦0 and (1,1,1)↦1). It has been experimentally shown in previous works that introducing asynchronism in the global function of a cellular automata was perturbing its behavior, but as far as we know, only few theoretical work exists on the subject. The cellular automata we consider live on a ring of size n and asynchronism is introduced as follows: at each time step one cell is selected uniformly at random and the transition is made on this cell while the others stay in the same state. Among the 64 cellular automata belonging to the class we consider, we show that 9 of them diverge on all non-trivial configurations while the 55 other converge almost surely to a random fixed point. We show that the exact convergence time of these 55 automata can only take the following values: either 0, Θ(nlnn), Θ(n2), Θ(n3) or Θ(n2n). Furthermore, the global behavior of each of these cellular automata is fully determined by reading its code.