The graph structure of a deterministic automaton chosen at random. Issue 3 (30th December 2016) - Cai, Xing Shi; Devroye, Luc
Abstract: An n ‐state deterministic finite automaton over a k ‐letter alphabet can be seen as a digraph with n vertices which all have k labeled out‐arcs. Grusho (Publ Math Inst Hungarian Acad Sci 5 (1960), 17–61). proved that whp in a random k ‐out digraph there is a strongly connected component of linear size, i.e., a giant, and derived a central limit theorem. We show that whp the part outside the giant contains at most a few short cycles and mostly consists of tree‐like structures, and present a new proof of Grusho's theorem. Among other things, we pinpoint the phase transition for strong connectivity. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 428–458, 2017