The Leray measure of nodal sets for random eigenfunctions on the torus

We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in d ≥ 2 dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity N → ∞. The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to l/√2π. Our main result is that the variance of Leray measure is asymptotically 1/4pi;N, as N → ∞, at least in dimensions d = 2 and d ≥ 5.