Synchronizing random automata through repeated 'a' inputs

TL;DR

This paper improves bounds on the length of synchronizing words in random automata, showing that with high probability, very short words can synchronize all states efficiently.

Contribution

It establishes tighter bounds on the length of synchronizing words in random automata, extending previous results with probabilistic guarantees.

Findings

Existence of short words reducing possible states to O(√n/ log n)

High probability that any pair of states can be synchronized by O(log n) length words

Synchronizing words of length O(√n log n) exist with probability approaching 1

Abstract

In a recent article by Chapuy and Perarnau, it was shown that a uniformly chosen automaton on $n$ states with a $2$-letter alphabet has a synchronizing word of length $O(\sqrt{n}\log n)$ with high probability. In this note, we improve this result by showing that, for any $\varepsilon>0$, there exists a synchronizing word of length $O(\varepsilon^{-1}\sqrt{n \log n})$ with probability $1-\varepsilon$. Our proof is based on two properties of random automata. First, there are words $\omega$ of length $O(\sqrt{n \log n})$ such that the expected number of possible states for the automaton, after inputting $\omega$, is $O(\sqrt{n/\log n})$. Second, with high probability, each pair of states can be synchronized by a word of length $O(\log n)$.