Random Riesz energies on compact K\"{a}hler manifolds

TL;DR

This paper analyzes the asymptotic behavior of the expected Riesz s-energy of zero sets of Gaussian random polynomials on compact Kähler manifolds, revealing higher energies in higher dimensions due to zero clustering.

Contribution

It extends the asymptotic analysis of Riesz energies of zero sets to all dimensions and codimensions on compact Kähler manifolds, generalizing previous one-dimensional results.

Findings

Zero sets have higher energies than random points in dimensions > 2.

Asymptotics hold for sections of any positive line bundle.

Clumping of zeros causes increased energy levels.

Abstract

This article determines the asymptotics of the expected Riesz s-energy of the zero set of a Gaussian random systems of polynomials of degree N as the degree N tends to infinity in all dimensions and codimensions. The asymptotics are proved more generally for sections of any positive line bundle over any compact Kaehler manifold. In comparison with the results on energies of zero sets in one complex dimension due to Qi Zhong (arXiv:0705.2000) (see also [arXiv:0705.2000]), the zero sets have higher energies than randomly chosen points in dimensions > 2 due to clumping of zeros.