In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in
the semiclassical limit, the local weak convergence. This allows us to give a rigorous
statement of Berry’s random wave conjecture. Using recent results of Bourgain, Buckley
and Wigman, we will prove that some deterministic families of eigenfunctions on
satisfy the conclusions of the random wave conjecture. We also show that on an
arbitrary domain, a sequence of Laplace eigenfunctions always admits local weak
limits. We explain why these local weak limits can be a powerful tool to study the
asymptotic number of nodal domains.
