Random Regular Expression Over Huge Alphabets

In this article, we study some properties of random regular expressions of size n, when the cardinality of the alphabet also depends on n. For this, we revisit and improve the classical Transfer Theorem from the field of analytic combinatorics. This provides precise estimations for the number of regular expressions, the probability of recognizing the empty word and the expected number of Kleene stars in a random expression. For all these statistics, we show that there is a threshold when the size of the alphabet approaches n(2), at which point the leading term in the asymptotics starts oscillating.