The inactive–active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton

Abstract

We investigate the inactive–active phase transition in an array of additive (exclusive-or) cellular automata (CA) under noise. The model is closely related with the Domany-Kinzel (DK) probabilistic cellular automaton (PCA), for which there are rigorous as well as numerical estimates on the transition probabilities. Here, we characterize the critical behavior of the noisy additive cellular automaton by mean field analysis and finite-size scaling and show that its phase transition belongs to the directed percolation universality class of critical behavior. As a by-product of our analysis, we argue that the critical behavior of the noisy elementary CA 90 and 102 (in Wolfram’s enumeration scheme) must be the same. We also perform an empirical investigation of the mean field equations to assess their quality and find that away from the critical point (but not necessarily very far away) the mean field approximations provide a reasonably good description of the dynamics of the PCA.