Inefficiency of the block approximation in diploid probabilistic cellular automata.

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 λ=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 λ=1/2. Despite this absorbing behavior, both mean-field and block-approximation schemes predict a stationary state with nonzero density. This failure highlights a fundamental limitation of finite-block approximation in capturing the global dynamics of probabilistic cellular automata.