On Two Distributions of Subgroups of Free Groups

We study and compare two natural distributions of finitely generated subgroups of free groups. One is based on the random generation of tuples of reduced words; that is the one classically used by group theorists. The other relies on Stallings' graphical representation of subgroups and in spite of its naturality, it was only recently considered. The combinatorial structures underlying both distributions are studied in this paper with methods of analytic combinatorics. We use these methods to point out the differences between these distributions. It is particularly interesting that certain important properties of subgroups that are generic in one distribution, turn out to be negligible in the other.