Distribution of the number of accessible states in a random deterministic automaton

We study the distribution of the number of accessible states in deterministic and complete automata with n states over a k-letters alphabet. We show that as n tends to infinity and for a fixed alphabet size, the distribution converges in law toward a Gaussian centered around vk n and of standard deviation equivalent to sk n^(1/2), for some explicit constants vk and sk. Using this characterization, we give a simple algorithm for random uniform generation of accessible deterministic and complete automata of size n of expected complexity O(n^(3/2)), which matches the best methods known so far. Moreover, if we allow a variation around n in the size of the output automaton, our algorithm is the first solution of linear expected complexity. Finally we show how this work can be used to study accessible automata (which are difficult to apprehend from a combinatorial point of view) through the prism of the simpler deterministic and complete automata. As an example, we show how the average complexity in O(n log log n) for Moore\u27s minimization algorithm obtained by David for deterministic and complete automata can be extended to accessible automata