Open Access
2020 Restriction of 3D arithmetic Laplace eigenfunctions to a plane
Riccardo W. Maffucci
Electron. J. Probab. 25: 1-17 (2020). DOI: 10.1214/20-EJP457


We consider a random Gaussian ensemble of Laplace eigenfunctions on the 3D torus, and investigate the 1-dimensional Hausdorff measure (‘length’) of nodal intersections against a smooth 2-dimensional toral sub-manifold (‘surface’). A prior result of ours prescribed the expected length, universally proportional to the area of the reference surface, times the wavenumber, independent of the geometry.

In this paper, for surfaces contained in a plane, we give an upper bound for the nodal intersection length variance, depending on the arithmetic properties of the plane. The bound is established via estimates on the number of lattice points in specific regions of the sphere.


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Riccardo W. Maffucci. "Restriction of 3D arithmetic Laplace eigenfunctions to a plane." Electron. J. Probab. 25 1 - 17, 2020.


Received: 16 September 2019; Accepted: 12 April 2020; Published: 2020
First available in Project Euclid: 8 May 2020

zbMATH: 07225514
MathSciNet: MR4095056
Digital Object Identifier: 10.1214/20-EJP457

Primary: 11P21 , 60G15

Keywords: arithmetic random waves , Gaussian random fields , Kac-Rice formulas , lattice points on spheres , nodal intersections

Vol.25 • 2020
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