来自随机Toom轮廓的Peierls界
Peierls bounds from random Toom contours
AI总结：
针对内禀随机单调元胞自动机的稳定性公开问题，提出基于随机轮廓估算Peierls和的方法，证明了新的稳定性结果，同时构造反例说明该方法的局限性。
摘要：
对于d维整数格点上的确定性单调元胞自动机，Toom（1980）给出了全一不动点在小随机扰动下保持稳定的充要条件。我们关注将Toom的结果推广至具有内禀随机性的单调元胞自动机这一公开问题，这类系统的未受扰演化是随机的，时空点上附着独立同分布的更新规则。部分应用场景还需要考虑更通用的图结构，因此我们假设底层格点是任意可数群。Toom的稳定性证明基于Peierls论证。在之前的工作中，我们已证明该Peierls论证也可用于证明内禀随机元胞自动机的稳定性，但这种情况下估算Peierls和比确定性情形困难得多。在本文中，我们提出一种基于随机轮廓的方法来估算Peierls和，并应用该方法证明了内禀随机单调元胞自动机的新稳定性结果。我们还通过构造一个例子展示了该方法的局限性：在该例子中，尽管普遍认为系统是稳定的，但对于任意小的扰动，Peierls和都是无穷大的。
英文摘要：
For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom (1980) has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. We are interested in the open problem of extending Toom's result to monotone cellular automata with intrinsic randomness, where the unperturbed evolution is random with i.i.d. update rules attached to the space-time points. For some applications it is also desirable to consider a more general graph structure, so we assume that the underlying lattice is an arbitrary countable group. Toom's proof of stability is based on a Peierls argument. In previous work, we demonstrated that this Peierls argument can also be used to prove stability for cellular automata with intrinsic randomness, but in this case estimating the Peierls sum becomes much harder than in the deterministic case. In the present paper, we develop a method based on random contours to estimate the Peierls sum and apply it to prove new stability results for monotone cellular automata with intrisic randomness. We also demonstrate the limitations of the method by constructing an example where the Peierls sum is infinite for arbitrary small perturbations even though stability is believed to hold.
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