  We study the asymptotic properties of the conormal cycle of nodal sets
associated to a random superposition of eigenfunctions of the Laplacian on a
smooth compact Riemannian manifold without boundary. In the case where the
dimension is odd, we show that the expectation of the corresponding current of
integration equidistributes on the fibers of the cotangent bundle. In the case
where the dimension is even, we obtain an upper bound of lower order on the
expectation. Using recent results of Alesker, we also deduce some properties on
the asymptotic expectation of any smooth valuation including the Euler
characteristic of random nodal sets.
