On the volume of nodal sets for eigenfunctions of the Laplacian on the torus

TL;DR

This paper analyzes the expected volume and fluctuations of nodal sets for Laplacian eigenfunctions on a torus, revealing that normalized volume fluctuations diminish as eigenspace dimension grows.

Contribution

It provides the first quantitative analysis of the variance of nodal set volume for high-multiplicity eigenfunctions on the torus, showing vanishing fluctuations with increasing eigenspace dimension.

Findings

Expected nodal set volume scales as constant times square root of eigenvalue

Variance of normalized volume is bounded by a term decreasing with eigenspace dimension

Normalized volume fluctuations tend to zero as eigenspace dimension increases

Abstract

We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues $4\pi^2\eigenvalue$ with growing multiplicity $\Ndim\to\infty$, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is $const \sqrt{\eigenvalue}$. Our main result is that the variance of the volume normalized by $\sqrt{\eigenvalue}$ is bounded by $O(1/\sqrt{\Ndim})$, so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.