  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.
