### Abstract

Original language | English |
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Number of pages | 19 |

Journal | arXiv.org e-Print archive |

Publication status | Published - 6 Jun 2019 |

### Fingerprint

### Keywords

- math.CO

### Cite this

**Circular automata synchronize with high probability.** / Aistleitner, Christoph; D'Angeli, Daniele; Gutierrez, Abraham; Rodaro, Emanuele; Rosenmann, Amnon.

Research output: Contribution to journal › Article › Research

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TY - JOUR

T1 - Circular automata synchronize with high probability

AU - Aistleitner, Christoph

AU - D'Angeli, Daniele

AU - Gutierrez, Abraham

AU - Rodaro, Emanuele

AU - Rosenmann, Amnon

PY - 2019/6/6

Y1 - 2019/6/6

N2 - 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.

AB - 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.

KW - math.CO

M3 - Article

JO - arXiv.org e-Print archive

JF - arXiv.org e-Print archive

ER -