Concentration for nodal component count of Gaussian Laplace eigenfunctions

TL;DR

This paper establishes exponential concentration results for the number of nodal components of various Gaussian Laplace eigenfunctions, including random waves and spherical harmonics, on different geometric domains.

Contribution

It extends concentration results to new settings such as monochromatic random waves on growing Euclidean balls and geodesic balls slightly larger than the wavelength scale.

Findings

Exponential concentration for MRW on Euclidean balls

Exponential concentration for RSH on geodesic balls

Exponential concentration for ARW on geodesic balls

Abstract

We study nodal component count of the following Gaussian Laplace eigenfunctions: monochromatic random waves (MRW) on $\mathbb{R}^2$, arithmetic random waves (ARW) on $\mathbb{T}^2$ and random spherical harmonics (RSH) on $\mathbb{S}^2$. Exponential concentration for nodal component count of RSH on $\mathbb{S}^2$ and ARW on $\mathbb{T}^2$ were established by Nazarov-Sodin and Rozenshein respectively. We prove exponential concentration for nodal component count in the following three cases: MRW on growing Euclidean balls in $\mathbb{R}^2$; RSH and ARW on geodesic balls, in $\mathbb{S}^2$ and $\mathbb{T}^2$ respectively, whose radius is slightly larger than the wavelength scale.