Universality and scaling of correlations between zeros on complex manifolds

TL;DR

This paper investigates the universal behavior of zero correlations of random sections of line bundles over complex manifolds as the bundle power grows, revealing independence from specific line bundles and explicit correlation formulas.

Contribution

It demonstrates the universality of zero correlations in the large N limit, independent of line bundle details, and derives explicit formulas including Hannay's correlation for Riemann surfaces.

Findings

Correlation functions become universal in the large N limit.

Explicit formulas for pair correlations are provided.

Hannay's limit pair correlation applies to all compact Riemann surfaces.

Abstract

We study the limit as $N\to\infty$ of the correlations between simultaneous zeros of random sections of the powers $L^N$ of a positive holomorphic line bundle $L$ over a compact complex manifold $M$, when distances are rescaled so that the average density of zeros is independent of $N$. We show that the limit correlation is independent of the line bundle and depends only on the dimension of $M$ and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we provide an alternate derivation of Hannay's limit pair correlation function for SU(2) polynomials, and we show that this correlation function holds for all compact Riemann surfaces.