Phase diagram of a stochastic cellular automaton with long-range interactions

A stochastic one-dimensional cellular automaton with long range spatial interactions is introduced. In this model the state probability of a given site at time t depends on the state of all the other sites at time t−1 through a power law of the type 1/rα, r being the distance between sites. For α→∞ this model reduces to the Domany–Kinzel cellular automaton. The dynamical phase diagram is analyzed using Monte Carlo simulations for 0⩽α⩽∞. We found the existence of two different regimes: one for 0⩽α⩽1 and the other for α>1. It is shown that in the first regime the phase diagram becomes independent of α. Regarding the frozen-active phase transition in this regime, a strong evidence is found that the mean-field prediction for this model becomes exact, a result already encountered in magnetic systems. It is also shown that, for replicas evolving under the same noise, the long-range interactions fully suppress the spreading of damage for 0⩽α⩽1.