Inefficiency of the block approximation in diploid Probabilistic Cellular Automata

Abstract

We study a probabilistic cellular automaton obtained as a mixture of the additive elementary rules 60 and 102. We prove that, for any finite periodic lattice and for mixing parameter $\lambda =1/2$, the system almost surely reaches the absorbing all-zero configuration in finitely many steps. In addition, Monte Carlo simulations indicate as well the presence of a zero-density stationary state in a finite interval around $\lambda =1/2$. Despite this absorbing behavior, both mean-field and block approximation schemes predict a stationary state with non-zero density. This failure, traced to the additive and mirror symmetries of the deterministic components, highlights a fundamental limitation of finite-block approximation in capturing the global dynamics of probabilistic cellular automata.

8 figures

References in corpus (6)

• Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models
• A Monte Carlo investigation of the critical behavior of Stavskaya's probabilistic cellular automaton
• Folding transition of the triangular lattice in a discrete three--dimensional space
• An extinction-survival-type phase transition in the probabilistic cellular automaton p182-q200
• Local structure approximation as a predictor of second order phase transitions in asynchronous cellular automata
• Phase transitions in random mixtures of elementary cellular automata