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.
"Local weak limits of Laplace eigenfunctions." Tunisian J. Math. 3 (3) 481 - 515, 2021. https://doi.org/10.2140/tunis.2021.3.481