Fluctuations of the increment of the argument for the Gaussian entire function

TL;DR

This paper investigates the fluctuations in the argument of the Gaussian entire function along planar curves, introducing a signed length concept and demonstrating asymptotic normality of these fluctuations.

Contribution

It introduces the signed length inner product and proves the asymptotic normality of argument fluctuations for the Gaussian entire function.

Findings

The signed length describes the limiting covariance of argument increments.

Fluctuations of the argument are asymptotically normal.

Provides a new framework for understanding argument fluctuations in Gaussian entire functions.

Abstract

The Gaussian entire function is a random entire function, characterised by a certain invariance with respect to isometries of the plane. We study the fluctuations of the increment of the argument of the Gaussian entire function along planar curves. We introduce an inner product on finite formal linear combinations of curves (with real coefficients), that we call the signed length, which describes the limiting covariance of the increment. We also establish asymptotic normality of fluctuations.