  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.
