Given a Fuchsian group with at least one cusp, Deroin, Kleptsyn and Navas define a Lyapunov expansion exponent for a point on the boundary, and ask if it vanishes for almost all points with respect to Lebesgue measure. We give an affirmative answer to this question, by considering the behavior of the word metric along typical geodesic rays and their excursions into cusps. We also consider the behavior of the word metric along rays chosen according to harmonic measure on the boundary, arising from random walks with finite first moment. We show that the excursions have different behavior in the Lebesgue measure and harmonic measure cases, which implies that these two measures are mutually singular.

The first author was supported by a Global Research Fellowship with the Institute of Advanced Study at the University of Warwick. The second author would like to thank Kathi Crow for her generous hospitality, and was supported by PSC-CUNY award 44-178 and Simons Foundation grant CGM 234477.