  Every word in a free group induces a word measure -- a probability measure
defined via the word map -- on every compact group. This paper presents a
conjectural picture about the role of a plethora of stable invariants of words
in word measures on groups. These invariants generalize the stable commutator
length and include, among others, two invariants recently defined by Wilton:
the stable primitivity rank and a non-oriented analog of stable commutator
length we call stable square length. The conjectures say, roughly, that these
stable invariants control the asymptotics of the expected values of stable
characters, under word measures. We reinforce these conjectures by proving a
version for word measures on wreath products, and by introducing a related
formula for stable irreducible characters of the symmetric group.
