Nodal set of monochromatic waves satisfying the Random Wave Model

TL;DR

This paper constructs deterministic solutions to the Helmholtz equation that mimic random wave behavior and analyzes their nodal sets, including the number of nodal domains, volume, and topology, using de-randomisation and stability techniques.

Contribution

It introduces a method to produce deterministic Helmholtz solutions exhibiting pseudo-random properties and studies their nodal structures in detail.

Findings

Solutions behave according to the Random Wave Model

Quantitative analysis of nodal domains and topology

Use of de-randomisation and stability methods

Abstract

We construct deterministic solutions to the Helmholtz equation in $\mathbb{R}^m$ which behave accordingly to the Random Wave Model. We then find the number of their nodal domains, their nodal volume and the topologies and nesting trees of their nodal set in growing balls around the origin. The proof of the pseudo-random behaviour of the functions under consideration hinges on a de-randomisation technique pioneered by Bourgain and proceeds via computing their $L^p$-norms. The study of their nodal set relies on its stability properties and on the evaluation of their doubling index, in an average sense.