Critical phenomena in an one-dimensional probabilistic cellular automation

A one-dimensional probabilistic cellular automaton that models a transition from elementary rule 4 to elementary rule 22 (following Wolfram's nomenclature scheme) is studied here. The evolution of the automaton follows rule 4 with probability 1 − p and rule 22 with probability p. In course of the transition the system shows two critical points, a trivial pc1 = 0 and a nontrivial pc2 ⋍ 0.75, at which the relaxation time of the system is observed to diverge in the form of a power law τ ∼ (p − pc1)−z1 and τ ≈ (pc2 − p)−z2 with z1 ≅ 0.86 and z2 ≅ 0.92. The point pc2 is also a point of phase transition with the density of occupied sites in the equilibrium state as the order parameter; the order parameter goes to zero as n ≈ (p − pc2)ß, ß ≅ 0.32 for p → pc2+. The possible cause of the observed behaviour is discussed.