Relaxation and limit cycles in a global version of the quenched Kauffman model

A combinatorial method is used to investigate all possible cellular automata, defined on a finite set of w global configurations. Relaxation time is defined as the number of steps before entering a fixed point or a cycle. For given w, we derive an analytical formula for the distribution of relaxation times t: Pw(t) = (w!/ww + 1)Σk = 0w−t−1(wk/k!). The same distribution is proved to be valid for the length of limit cycles. The results have some relevance to the quenched Kauffman model.