Fast synchronization of inhomogenous random automata

TL;DR

This paper proves that automata with random mappings and two random permutation letters typically have a relatively low reset threshold, improving understanding of synchronization times in random automata.

Contribution

It provides a simple proof that such automata have a reset threshold of order  ig( \u221a{n \u2227} ig) with high probability, under partial independence assumptions.

Findings

Reset threshold is  ig(  ig( \u221a{n \u2227} ig) with high probability.

The result extends previous bounds for random automata with multiple random mappings.

The proof simplifies understanding of synchronization times in inhomogeneous random automata.

Abstract

We examine the reset threshold of randomly generated deterministic automata. We present a simple proof that an automaton with a random mapping and two random permutation letters has a reset threshold of $\mathcal{O}\big( \sqrt{n \log^3 n} \big)$ with high probability, assuming only certain partial independence of the letters. Our observation is motivated by Nicaud (2014) providing a near-linear bound in the case of two random mapping letters, among multiple other results. The upper bound for the latter case has been recently improved by the breakthrough work of Chapuy and Perarnau (2023) to $\mathcal{O}(\sqrt{n} \log n)$.