Concentration for nodal component count of Gaussian Laplace eigenfunctions

2020/12/18 by Lakshmi Priya, Priya, Lakshmi · 1 citation
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2012.10302

Abstract

We study nodal component count of the following Gaussian Laplace eigenfunctions: monochromatic random waves (MRW) on ℝ², arithmetic random waves (ARW) on \mathbbT² and random spherical harmonics (RSH) on \mathbbS². Exponential concentration for nodal component count of RSH on \mathbbS² and ARW on \mathbbT² 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 ℝ²; RSH and ARW on geodesic balls, in \mathbbS² and \mathbbT² respectively, whose radius is slightly larger than the wavelength scale.

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