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/r(alpha), r being the distance between sites. For alpha --> infinity this model reduces to the Domany-Kinzel cellular automaton. The dynamical phase diagram is analyzed using Monte Carlo simulations for 0 less than or equal to alpha less than or equal to infinity. We found the existence of two different regimes: one for 0 less than or equal to alpha less than or equal to 1 and the other for alpha > 1. It is shown that in the first regime the phase diagram becomes independent of alpha. 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 less than or equal to alpha less than or equal to 1. (C) 1998 Elsevier Science B.V. All rights reserved.