  In this article we obtain large deviation estimates for zeros of random
holomorphic sections on punctured Riemann surfaces. These estimates are then
employed to yield estimates for the respective hole probabilities. A particular
case of relevance that is covered by our setting is that of cusp forms on
arithmetic surfaces. Most of the results we obtain also allow for reasonably
general probability distributions on holomorphic sections, which shows the
universal character of these estimates. Finally, we also extend our results to
the case of certain higher dimensional complete Hermitian manifolds, which are
not necessarily assumed to be compact.
