Circular automata synchronize with high probability

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:178

Article

Aistleitner, Christoph¹  D'Angeli, Daniele²  Gutierrez, Abraham³  Rodaro, Emanuele⁴  Rosenmann, Amnon³ 

[1] Graz Univ Technol, Inst Anal & Number Theory, Graz, Austria

[2] Univ Niccolo Cusano, Via Don Gnocchi Roma, Rome, Italy

[3] Graz Univ Technol, Inst Discrete Math, Graz, Austria

[4] Politecn Milan, Dept Math, Milan, Italy

关键词: Automata;    Synchronization;    Random matrices;    Circulant graphs;    Chromatic polynomials;   

DOI  :  10.1016/j.jcta.2020.105356

来源: Elsevier

【 摘 要 】

In this paper we prove that a uniformly distributed random circular automaton A(n) of order n synchronizes with high probability (w.h.p.). More precisely, we prove that P[A(n) synchronizes] = 1 - O (1/n). 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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free