A synchronizing word for an automaton is a word that brings that automaton into one and the same state, regardless of the starting position. Černý conjectured in 1964 that if a n-state deterministic automaton has a synchronizing word, then it has a synchronizing word of length at most (n − 1)². Berlinkov recently made a breakthrough in the probabilistic analysis of synchronization: he proved that, for the uniform distribution on deterministic automata with n states, an automaton admits a synchronizing word with high probability. In this article, we are interested in the typical length of the smallest synchronizing word, when such a word exists: we prove that a random automaton admits a synchronizing word of length O(n log³ n) with high probability. As a consequence, this proves that most automata satisfy the Černý conjecture.