Zeros of a random analytic function approach perfect spacing under repeated differentiation

TL;DR

This paper studies how repeated differentiation of a random analytic function with zeros forming a Poisson process causes its zeros to become evenly spaced, converging to a shifted integer lattice.

Contribution

It demonstrates that differentiation induces a universal zero-spacing pattern, transforming a Poisson zero set into a regular lattice in distribution.

Findings

Zeros converge to a shifted integer lattice after repeated differentiation

Differentiation induces a universal spacing pattern for zeros

Zero set distribution approaches a translated integer grid

Abstract

We consider an analytic function $f$ whose zero set forms a unit intensity Poisson process on the real line. We show that repeated differentiation causes the zero set to converge in distribution to a random translate of the integers.