Cusp excursion in hyperbolic manifolds and singularity of harmonic measure

We generalize the notion of cusp excursion geodesic rays by introducing for any \begin{document}$ k\geq 1 $\end{document} \begin{document}$ k^\text{th} excursion in cusps a hyperbolic id="M3">\begin{document}$ N $\end{document}-manifold finite volume. We show that on one hand, this is at most linear geodesics are generic with respect to hitting measure random walk. On other id="M4">\begin{document}$ k = N-1 $\end{document}, id="M5">\begin{document}$ superlinear Lebesgue measure. use and boundary space id="M6">\begin{document}$ \mathbb{H}^N id="M7">\begin{document}$ \geq 2 mutually singular.