# Circular automata synchronize with high probability

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### 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 arXiv.org e-Print archive Published - 6 Jun 2019

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Automata
Synchronization
Circulant Graph
Chromatic Polynomial
Probabilistic Methods
Random Matrices
Upper bound
Text

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In: arXiv.org e-Print archive, 06.06.2019.

Research output: Contribution to journalArticleResearch

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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.",
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author = "Christoph Aistleitner and Daniele D'Angeli and Abraham Gutierrez and Emanuele Rodaro and Amnon Rosenmann",
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