A Salem Zygmund approach to almost sure asymptotics concerning the nodal measure of Riemannian random waves.
Let $(mathcal{M},g)$ be a compact, boundaryless Riemannian manifold, and $(lambda_n,varphi_n)$ the sequences of (ordered) Laplace eigenvalues and eigenfunctions, satisfying $Delta varphi_n = -lambda_n^2 varphi_n$. We consider the model of Riemannian random waves defined by $f_lambda(x) = sum_{lambda_nleq lambda} a_n varphi_n(x)$, where $(a_n)_n$ is a iid sequence of Gaussian random variables. With probability one with respect to the Gaussian coefficients, we establish that the process $f_lambda$, properly rescaled and evaluated at an independently and uniformly chosen point X on the manifold, converges in distribution towards an universal Gaussian field as $lambda$ grows to infinity. Using the continuity of the nodal measure with respect to the $mathcal{C}^1$ topology, we deduce that almost surely with respect to the Gaussian coefficient, the nodal measure of $f_n$ weakly converges towards the Riemannian volume.