We prove that a uniformly random automaton with $n$ states on a 2-letter alphabet has a synchronizing word of length $O(n^{1/2}\log n)$ with high probability (w.h.p.). That is to say, w.h.p. there exists a word $\omega$ of such length, and a state $v_0$, such that $\omega$ sends all states to $v_0$. Prior to this work, the best upper bound was the quasilinear bound $O(n\log^3n)$ due to Nicaud (2016). The correct scaling exponent had been subject to various estimates by other authors between $0.5$ and $0.56$ based on numerical simulations, and our result confirms that the smallest one indeed gives a valid upper bound (with a log factor). Our proof introduces the concept of $w$-trees, for a word $w$, that is, automata in which the $w$-transitions induce a (loop-rooted) tree. We prove a strong structure result that says that, w.h.p., a random automaton on $n$ states is a $w$-tree for some word $w$ of length at most $(1+\epsilon)\log_2(n)$, for any $\epsilon>0$. The existence of the (random) word $w$ is proved by the probabilistic method. This structure result is key to proving that a short synchronizing word exists.

翻译：我们证明，在2字母表上均匀随机的具有$n$个状态的自动机，以高概率（w.h.p.）存在长度为$O(n^{1/2}\log n)$的同步词。也就是说，高概率下存在一个长度为该量级的词$\omega$和一个状态$v_0$，使得$\omega$将所有状态映射到$v_0$。此前的最佳上界是Nicaud（2016）提出的拟线性界$O(n\log^3n)$。其他作者基于数值模拟对正确标度指数的估计介于0.5与0.56之间，而我们的结果证实了最小估计值（在对数因子下）确实给出了有效上界。我们的证明引入了词$w$对应的$w$树概念，即自动机中$w$迁诱导出（以环为根的）树结构。我们证明了一个强结构结果：对于任意$\epsilon>0$，高概率下，一个$n$状态的随机自动机是某个长度至多为$(1+\epsilon)\log_2(n)$的词$w$对应的$w$树。该（随机）词$w$的存在性通过概率方法得到证明。这一结构结果是证明短同步词存在性的关键。