Short Synchronizing Words for Random Automata

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 \(O(n\log^{3}n)\) due to Nicaud [ 26 ]. 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.