In this paper we prove that a uniformly distributed random circular automaton ${\mathcal{A}}_n$ of order n synchronizes with high probability (w.h.p.). More precisely, we prove that$\mathbb{P}\left[{\mathcal{A}}_n\text{ synchronizes}\right]=1-O\left(\frac{1}{n}\right).$ The main idea of the proof is to translate the synchronization problem into a problem concerning properties of a random matrix; these properties are then established with high probability by a careful analysis of the stochastic dependence structure among the random entries of the matrix. Additionally, we provide an upper bound for the probability of synchronization of circular automata in terms of chromatic polynomials of circulant graphs.

中文翻译：

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圆形自动机高概率同步

在本文中，我们证明了均匀分布的随机圆形自动机 ${\mathcal{一种}}_{ñ}$n阶的高概率（whp）同步。更确切地说，我们证明$\mathbb{P}\left[{\mathcal{一种}}_{ñ}\text{ 同步化}\right]=\mathrm{1个}-Ø（\frac{\mathrm{1个}}{ñ}）。$证明的主要思想是将同步问题转化为涉及随机矩阵性质的问题。然后，通过仔细分析矩阵随机条目之间的随机相关性结构，可以很可能地建立这些属性。此外，根据循环图的色多项式，我们提供了圆形自动机同步概率的上限。