The forbidden region for random zeros: appearance of quadrature domains

TL;DR

This paper reveals a surprising connection between quadrature domains and the zero distribution of the Gaussian Entire Function, showing that forbidden regions emerge as quadrature domains during rare hole events.

Contribution

It uncovers the emergence of quadrature domains as forbidden regions in the zero process of GEF conditioned on large hole events, linking random zeros to potential theory.

Findings

Quadrature domains appear as forbidden regions in the zero process.

The shape of holes influences the formation of quadrature domains.

A new obstacle problem approach describes the zero distribution in holes.

Abstract

Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zero process of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero density vanishes. Moreover, we give a description of those holes for which the forbidden region is a disk. The connecting link between random zeros and potential theory is supplied by a constrained extremal problem for the Zeitouni-Zelditch functional. To solve this problem, we recast it in terms of a seemingly novel obstacle problem, where the solution is forced to be harmonic inside the hole.