Aron Wennman: Zeros of the Gaussian Entire Function: the hole event and emergent quadrature domains

Abstract

The Gaussian Entire Function (GEF) is a random Taylor series with Gaussian coefficients, whose zero distribution is invariant with respect to rigid motions of the plane. Like the somewhat similar Ginibre ensemble from random matrix theory, the GEF zeros has become a popular model in probability, analysis and mathematical physics. Curiously, both the random zeros and the eigenvalue process may be interpreted as systems of charged particles with a Coulomb interaction.

In this talk, I will discuss phenomena related to rare events for these random processes, such as the 'hole event' that a given large region is completely void of points. Natural questions include determining the probability of having a large hole of a given shape, and understanding the typical configurations conditioned on this event. 

For random zeros, the conditional zero distribution on the hole event is particularly intriguing: As the size of the hole increases, the density of zeros vanishes not just inside the hole, but also on a macroscopic region beyond the boundary - a 'forbidden region' emerges. I will describe some remarkable properties of the forbidden region, including a connection to quadrature domains from potential theory.

Based on joint work with Alon Nishry, Tel Aviv ( arxiv.org/abs/2009.08774 ).