Counting Open Nodal Lines of Random Waves on Planar Domains

We compute the asymptotic expectation of the number of open nodal lines for random waves on smooth planar domains. We find that for both the long energy window [0, λ], and the short one [λ, λ +1], the expected number of open nodal lines is proportional to λ, asymptotically as λ → ∞. Our results are consistent with the predictions of Blum, Gnutzmann, and Smilansky [4] in the physics literature