Circular automata synchronize with high probability

Research output: Contribution to journalArticle

Abstract

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.
Original language English 19 Journal of Combinatorial Theory / A Accepted/In press - 15 Oct 2020

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