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Given a measure on the Thurston boundary of Teichmueller space, one can picka geodesic ray joining some basepoint to a randomly chosen point on theboundary. Different choices of measures may yield typical geodesics withdifferent geometric properties. In particular, we consider two families ofmeasures: the ones which belong to the Lebesgue or visual measure class, andharmonic measures for random walks on the mapping class group generated by adistribution with finite first moment in the word metric. We consider the ratiobetween the word metric and the relative metric of approximating mapping classgroup elements along a geodesic ray, and prove that this ratio tends toinfinity along almost all geodesics with respect to Lebesgue measure, while thelimit is finite along almost all geodesics with respect to harmonic measure. Asa corollary, we establish singularity of harmonic measure. We show the sameresult for cofinite volume Fuchsian groups with cusps. As an application, weanswer a question of Deroin-Kleptsyn-Navas about the vanishing of the Lyapunovexpansion exponent.

Word length statistics for Teichm\"uller geodesics and singularity of harmonic measure