Poincare-Lelong approach to universality and scaling of correlations between zeros

TL;DR

This paper proves that the scaling limits of zero correlation functions for random holomorphic sections are universal, regardless of the underlying manifold, line bundle, or point, as the degree N approaches infinity.

Contribution

It establishes the universality of the scaling limits of zero correlations for random holomorphic sections on complex manifolds, extending previous results to a broader geometric setting.

Findings

Scaling limits of correlation functions are universal

Universality holds across different line bundles and manifolds

Results apply to zeros of holomorphic sections of powers of line bundles

Abstract

This note is concerned with the scaling limit as N approaches infinity of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L^N of any positive holomorphic line bundle L over a compact Kahler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the "correlation forms" are universal, i.e. independent of the bundle L, manifold M or point on M.