Word length statistics along random Teichmuller geodesics

Given a basepoint in Teichmuller space, a measure on the
Thurston boundary can be used
to sample at random a geodesic ray from the basepoint. In particular,
we consider two families of
measures: the ones which belong to the Lebesgue or visual measure
class, and harmonic
measures for random walks on the mapping class group generated by a
finitely supported distribution.
Along a geodesic ray, we consider the ratio between the word metric
and the relative metric of
approximating mapping class group element. We prove that this ratio
tends to infinity along a
Lebesgue-typical geodesic ray, and is finite along geodesics along a
harmonic-typical geodesic ray.
As a corollary, we give a different proof of singularity of harmonic
measure. The same proof works for
Fuchsian groups with cusps. This is joint work with Joseph Maher
(CUNY) and Giulio Tiozzo (Harvard).