Large deviations for zeros of holomorphic sections on punctured Riemann surfaces

TL;DR

This paper establishes large deviation estimates for zeros of random holomorphic sections on punctured Riemann surfaces, with applications to hole probabilities and extensions to higher-dimensional Hermitian manifolds.

Contribution

It provides universal large deviation estimates for zeros of random holomorphic sections, including non-compact cases and various probability distributions.

Findings

Large deviation estimates for zeros on punctured Riemann surfaces

Application to hole probability estimates

Extension to higher-dimensional Hermitian manifolds

Abstract

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.