Asymptotics of hole probability regarding open balls for random sections on compact Riemann surfaces

TL;DR

This paper investigates the asymptotic behavior of the probability that random holomorphic sections on a compact Riemann surface have no zeros within a specified open ball, focusing on how this probability changes with the hole size.

Contribution

It provides the first detailed asymptotic analysis of hole probabilities for random sections on compact Riemann surfaces, extending understanding of zero distributions in complex geometry.

Findings

Derived explicit asymptotic formulas for hole probabilities

Identified the dependence of hole probability decay on hole size

Enhanced understanding of zero distribution behavior in random holomorphic sections

Abstract

We obtain the asymptotic behavior of hole probability for random holomorphic sections on a compact Riemann surface with respect to the hole size.