我们研究了一类由可加性初等规则60与102混合构成的概率元胞自动机。我们证明:对任意有限周期格点及混合参数 $λ=1/2$,该系统几乎必然在有限步内达到全零吸收态。此外,蒙特卡洛模拟还表明,在 $λ=1/2$ 附近的一个有限区间内亦存在零密度稳态。尽管系统呈现吸收行为,平均场理论与块近似方法均预测存在非零密度的稳态。这一失效源于确定性组分所具有的可加性与镜像对称性,凸显了有限块近似法在刻画概率元胞自动机全局动力学方面的一项根本局限性。
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 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.