The state complexity of random DFAs

The state complexity of a Deterministic Finite-state automaton (DFA) is the number of states in its minimal equivalent DFA. We study the state complexity of random $n$-state DFAs over a $k$-symbol alphabet, drawn uniformly from the set $[n]^{{[n]\times[k]}\times2{[n]}$} of all such automata. We show that, with high probability, the latter is $\alphak n + O(\sqrt n\log n)$ for a certain explicit constant $\alphak$.