Random waves on $\mathbb{T}^3$: nodal area variance and lattice point correlations

TL;DR

This paper analyzes the variance of the nodal surface area of 3D arithmetic random waves on a torus, revealing its asymptotic behavior and connection to lattice point correlations.

Contribution

It provides the first asymptotic law for the nodal area variance of 3D random eigenfunctions, linking geometric properties to lattice point distributions.

Findings

Nodal area variance follows a specific asymptotic law.

Variance is connected to lattice point correlations on spheres.

Expected nodal surface area scales with the square root of eigenvalue.

Abstract

We consider the ensemble of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or `energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.