Random zeros on complex manifolds: conditional expectations

TL;DR

This paper investigates the behavior of zeros of Gaussian random polynomials and holomorphic sections conditioned on vanishing at specific points, revealing unique local distribution properties and scaling asymptotics.

Contribution

It introduces universal scaling asymptotics for conditional zero distributions on complex manifolds, highlighting differences from classical pair correlation functions.

Findings

Conditional zero distribution lacks zero repulsion in dimension one

Universal scaling asymptotics are established around the conditioning point

Conditional Szego kernel asymptotics are key to the analysis

Abstract

We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros, but we show that it has quite a different small distance behavior. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension one. To prove this, we give universal scaling asymptotics for the conditional zero distribution around p. The key tool is the conditional Szego kernel and its scaling asymptotics.