An automaton is synchronizing when there is a word that brings every state into one and the same state. Such a word is called a synchronizing word, and Černý conjectured in 1964 that if a n-state deterministic automaton is synchronizing, then it has a synchronizing word of length at most (n − 1)². The best bound known so far is cubic in n and was obtained by Szykuła in 2017. In this article, we study the synchronization properties of random deterministic automata, for the uniform distribution. Berlinkov recently proved that they are synchronizing with high probability. Our contribution is to study the typical length of the smallest synchronizing word, when such a word exists: we establish that with high probability, such an automaton with n states admits a synchronizing word of length O(n log^3 n). As a byproduct, we get that for most automata, the Černý conjecture holds.