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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.

Counting Open Nodal Lines of Random Waves on Planar Domains