Synchronizing random automata through repeated 'a' inputs

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 $O(\log n)$.