Topology and nesting of the zero set components of monochromatic random waves

TL;DR

This paper investigates the topological structures and nesting patterns of zero sets in monochromatic random waves, demonstrating that all configurations occur with positive probability at high frequencies.

Contribution

It introduces methods to construct Laplace eigenfunctions with zero sets of prescribed topology and nesting, advancing understanding of their geometric complexity.

Findings

Any diffeomorphism type of zero set component occurs with positive probability.

All nesting configurations of zero set components are possible at high frequencies.

Constructive techniques for eigenfunctions with specific zero set topologies are developed.

Abstract

This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of observing any diffeomorphism type, and any nesting arrangement, among the zero set components is strictly positive for waves of large enough frequencies. Our results are a consequence of building Laplace eigenfunctions in Euclidean space whose zero sets have a component with prescribed topological type, or an arrangement of components with prescribed nesting configuration.