We analyse cusp excursions of random geodesics for Weil–Petersson type incomplete metrics on orientable surfaces of finite type: in particular, we give bounds for maximal excursions.

We also give similar bounds for cusp excursions of random Weil–Petersson geodesics on non-exceptional moduli spaces of Riemann surfaces conditional on the assumption that the Weil–Petersson flow is polynomially mixing.

Moreover, we explain how our methods can be adapted to understand almost greasing collisions of typical trajectories in certain slowly mixing billiards.