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

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.