Ergodicity and mixing rate of one-dimensional cellular automata

One- and two-dimensional cellular automata which are known to be fault-tolerant are very complex. On the other hand, only very simple cellular automata have actually been proven to lack fault-tolerance, i.e., to be mixing. The latter either have large noise probability $\varepsilon$ or belong to the small family of two-state nearest-neighbor monotonic rules which includes local majority voting. For a certain simple automaton L called the soldiers rule, this problem has intrigued researchers for the last two decades since L is clearly more robust than local voting: in the absence of noise, L eliminates any finite island of perturbation from an initial configuration of all 0's or all 1's. The same holds for a 4-state monotonic variant of L, K, called two-line voting. We will prove that the probabilistic cellular automata $K\sb\varepsilon$ and $L\sb\varepsilon$, asymptotically lose all information about their initial state when subject to small, strongly biased noise. The mixing property trivially implies that the systems are ergodic. The finite-time information-retaining quality of a mixing system can be represented by its relaxation time Relax($\cdot$), which measures the time before the onset of significant information loss. This is known to grow as (1/$\varepsilon)\sp{c}$ for noisy local voting. The impressive error-correction ability of L has prompted some researchers to conjecture that $Relax(L\sb\varepsilon) =2\sp{c/ \varepsilon}$. We prove the tight bound $2\sp{c\sb1{\rm log}\sp21/ \varepsilon} < Relax(L\sb\varepsilon) < 2\sp{c\sb2{\rm log}\sp21/ \varepsilon}$ for a biased error model. The same holds for $K\sb\varepsilon$. Moreover, the lower bound is independent of the bias assumption. The strong bias assumption makes it possible to apply sparsity/renormalization techniques, the main tools of our investigation, used earlier in the opposite context of proving fault-tolerance.