Phase transition in an elementary probabilistic cellular automaton

Cellular automata exhibit a large variety of dynamical behaviors, from fixed-point convergence and periodic motion to spatio-temporal chaos. By introducing probabilistic interactions, and regarding the asymptotic density Φ of non-quiescent cell states as an order parameter, phase transitions may be identified from a quiescent phase with Φ = 0 to a chaotic phase with non-zero Φ. We consider an elementary one-dimensional probabilistic cellular automaton (PCA) with deterministic limits given by the quiescent rule 0, the rule 72 that evolves into a non-trivial fixed point, and the chaotic rules 18 and 90. Despite the simplicity of the rules, the PCA shows a surprising number of transition phenomena. We identify ‘second-order’ phase transitions from Φ = 0 to Φ > 0 with static and dynamic exponents that differ from those of directed percolation. Moreover, we find that the non-trivial fixed-point rule 72 is a singular point in PCA space.