Hole Probabilities for Random Zeros on Compact Riemann Surfaces (Song-Yan Xie - Academy of Mathematics and Systems Science (AMSS), CAS)

This talk explores the phenomenon of hole probabilities—the probability that a
specified region on a compact Riemann surface contains no zeros of a random
holomorphic section. A particularly compelling feature is the emergence of a
"forbidden region phenomenon", which reveals unexpected gaps in the spatial
distribution of zeros. This phenomenon was initially established by Nishry-
Ghosh in the context of Gaussian entire functions, and its analogue on Riemann
surfaces was later proved by Dinh–Ghosh–Wu. We report recent progress in
understanding this behavior using techniques from complex analysis and
algebraic geometry, including Zelditch’s density formula, Abel–Jacobi theory,
and the theory of Fekete points. Furthermore, we introduce a new perturbation
approach that greatly enhances our ability to analyze such hole probabilities.
This is joint work with Hao Wu (Nanjing University, arXiv:2406.19114).