Locality of the interaction affects dynamics in probabilistic cellular automata

Abstract

We study a class of one-dimensional probabilistic cellular automata in which each component can be in either state zero or state one. The component interacts with two neighbors: if its neighbors are in an equal state, then the component assumes the same state as its neighbors. If its neighbors are in different states, the following can happen: a one on the right-hand side of a zero, in which case the component becomes one with probability α or zero with probability 1 − α, and conversely, a zero on the right-hand side of a one, in which case the component becomes one with probability β or zero with probability 1 − β. For a set of initial distributions when both neighbors are placed on the right-hand side (respectively, both on the left-hand side) of a component, we prove that the process always converges weakly to the measure concentrated on the configuration where all the components are zeros. When one neighbor is placed on the left-hand side and the other is on the right-hand side, the same convergence happens when β < fN(α), where N is the distance between the neighbors. However, this convergence does not happen for β > 1/2α. Thus, in this case, we get the regimes of ergodicity and non-ergodicity. Moreover, we exhibit another type of phase transition, independent of neighbors’ locations. We also present some numerical studies in which we use mean field approximation and Monte Carlo simulation.