# Circular automata synchronize with high probability

## TL;DR

This paper proves that a random circular automaton of size n synchronizes with high probability, using probabilistic methods and properties of associated random matrices, and relates synchronization probability to chromatic polynomials of circulant graphs.

## Contribution

It establishes that random circular automata synchronize with high probability and introduces a novel approach using random matrix properties and graph chromatic polynomials.

## Key findings

- Synchronization probability approaches 1 as n increases
- Provides bounds on synchronization probability using chromatic polynomials
- Connects automaton synchronization to properties of circulant graphs

## 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.