Concentration for nodal component count of Gaussian Laplace eigenfunctions

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.

References in corpus (1)

• Small-scale equidistribution for random spherical harmonics