  We consider Riemannian random waves, i.e. Gaussian random linear combination
of eigenfunctions of the Laplacian on a compact Riemannian manifold with
frequencies from a short interval (`asymptotically fixed frequency'). We first
show that the expected limit distribution of the real zero set of a is uniform
with respect to the volume form of a compact Riemannian manifold $(M, g)$. We
then show that the complex zero set of the analytic continuations of such
Riemannian random waves to a Grauert tube in the complexification of $M$ tends
to a limit current.
