Conditional expectations of random holomorphic fields on Riemann surfaces

TL;DR

This paper investigates the universal behavior of conditional expectations of critical points and zeros of Gaussian holomorphic sections on Riemann surfaces, revealing short-distance neutrality and long-distance similarity in their distributions.

Contribution

It establishes the universal limits of rescaling conditional expectations for critical points and zeros of Gaussian holomorphic sections as the line bundle's power increases.

Findings

Short distance between critical points and conditioned zeros is neutral.

Zeros and conditioned critical points repel each other.

Long distance behaviors of the conditional expectations are similar.

Abstract

We study two conditional expectations: the expected density of critical points of Gaussian random holomorphic sections of powers of a positive holomorphic line bundle over Riemann surfaces given that the random sections vanish at a point and the expected density of zeros given that the random sections has a fixed critical point. The main result is that these two rescaling conditional expectations have universal limits as the power of the line bundle tends to infinity. We will see that the short distance behavior between critical points and the conditioning zero is neutral while there is a repulsion between zeros and the conditioning critical point. But the long distance behaviors are the same.