We investigate the asymptotic behavior of eigenfunctions of Riemannian manifolds, using local weak sampling. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory.

As a result, we present the first mathematically precise formulation of Berry’s conjecture for a compact negatively curved manifold, using Gaussian random eigenwaves and formulate a Berry-type conjecture for sequences of locally symmetric spaces. For the standard tori the question has been investigated by Bourgain. Already there, the picture gets nontrivial, but for a different reason, as typically one has high multiplicities in that case. We prove some weak versions of these conjectures. Using ergodic theory, we also show that Berry's conjecture implies Quantum Unique Ergodicity. Joint work with Nicolas Bergeron and Etienne le Masson.