Inefficiency of the block approximation in diploid Probabilistic Cellular Automata

TL;DR

This paper investigates the limitations of block approximation methods in modeling the global behavior of a specific probabilistic cellular automaton, revealing discrepancies between predictions and actual dynamics.

Contribution

It demonstrates that finite-block approximations fail to accurately predict the stationary states of the automaton due to inherent symmetries and probabilistic effects.

Findings

Automaton almost surely reaches all-zero state at λ=1/2

Monte Carlo simulations show zero-density stationary states near λ=1/2

Block approximation predicts incorrect non-zero density stationary states

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.