Nodal intersections for arithmetic random waves against a surface

TL;DR

This paper studies the length of intersections between random Laplace eigenfunctions on a 3D torus and a surface, revealing that the variance depends only on surface geometry and eigenvalue, with results linked to lattice point distribution.

Contribution

It provides the leading asymptotic of the variance of nodal intersection length for 3D arithmetic random waves against surfaces with nonzero curvature, connecting geometry and lattice point theory.

Findings

Expected intersection length proportional to eigenvalue^{1/2} and surface area.

Variance asymptotic depends only on surface geometry due to lattice point equidistribution.

Results apply to surfaces with nonvanishing Gauss-Kronecker curvature.

Abstract

Given the ensemble of random Gaussian Laplace eigenfunctions on the three-dimensional torus (`3d arithmetic random waves'), we investigate the $1$-dimensional Hausdorff measure of the nodal intersection curve against a compact regular toral surface (the `nodal intersection length'). The expected length is proportional to the square root of the eigenvalue, times the surface area, independent of the geometry. Our main finding is the leading asymptotic of the nodal intersection length variance, against a surface of nonvanishing Gauss-Kronecker curvature. The problem is closely related to the theory of lattice points on spheres: by the equidistribution of the lattice points, the variance asymptotic depends only on the geometry of the surface.