Gaussian complex zeros on the hole event: the emergence of a forbidden region

TL;DR

This paper studies the zero distribution of a Gaussian Entire Function conditioned on having no zeros in a large disk, revealing a forbidden region between the boundary and the equilibrium measure.

Contribution

It characterizes the limiting zero distribution conditioned on a hole, uncovering a forbidden region and providing explicit asymptotic behavior.

Findings

Convergence of zero distribution conditioned on a hole to a limiting measure.

Existence of a forbidden region between the boundary and equilibrium measure.

Explicit description of the limiting Radon measure.

Abstract

We consider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the kth coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane. We show that the law of the zero set, conditioned on the GEF having no zeros in a disk of radius r, and properly normalized, converges to an explicit limiting Radon measure in the plane, as r goes to infinity. A remarkable feature of this limiting measure is the existence of a large 'forbidden region' between a singular part supported on the boundary of the (scaled) hole and the equilibrium measure far from the hole.