Supercritical probabilistic cellular automata: how effective is the synchronous updating?

Probabilistic cellular automata generalise CA by implementing in a synchronous way an updating rule defined through a probability. A probabilistic synchronous updating scheme does it mean an efficient parallel evolution mechanism? This article deals with the question of quantifying the effectiveness of the parallel updating. A good indicator of this effectiveness is the fraction of components whose value is updated between two time steps. Two classes of parameterised models are considered. Multiple stationary distributions may occur when an infinite number of interacting components is considered (ergodicity breaks/supercritical regime). As a consequence, these models both exhibit different dynamical regimes in the corresponding case when a finite number of sites are interacting. These two classes’ non trivial steady states are of different nature. One is a family of positive rates reversible PCA dynamics. The other one is the Stavskaja PCA dynamics. It exhibits an absorbing state. Thanks to numerical simulations, both these PCA dynamics are shown to behave nearly asynchronous when these phase transition phenomena occur.