  We study nodal sets for typical eigenfunctions of the Laplacian on the
standard torus in 2 or more 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
multiplicity N tending to infinity.
  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. Our main result is that the variance of Leray
measure is asymptotically 1/(4 pi N), as N tends to infinity, at least in
dimensions 2 and at least 5.
