Fast synchronization of inhomogenous random automata

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 $\mathcal{O}\left(\sqrt{n{\log }^{3}n}\right)$ 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 $\mathcal{O}(\sqrt{n}\log n)$.