循环自动机的高概率同步
Circular automata synchronize with high probability
摘要：
本文证明了一个均匀分布的随机循环自动机$n$阶$\mathcal{A}_n$以高概率（whp）遍历。更确切地说，我们证明，$$ \mathbb{P}\left[\mathcal{A}_n \text{ synchronizes}\right] = 1- O\left(\frac{1}{n}\right). $$-证明的主要思想是将同步问题转化为随机矩阵的属性，然后用概率方法的工具处理这些属性。此外，我们还利用循环图的色多项式给出了循环自动机同步概率的一个上界。
英文摘要：
In this paper we prove that a uniformly distributed random circular automaton $\mathcal{A}_n$ of order $n$ synchronizes with high probability (whp). 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 properties of a random matrix; these properties are then handled with tools of the probabilistic method. Additionally, we provide an upper bound for the probability of synchronization of circular automata in terms of chromatic polynomials of circulant graphs.
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