Nodal Statistics of Planar Random Waves

We consider Berry's random planar wave model(1977) for a positiveLaplace eigenvalue E> 0 , both in thereal and complex case, and prove limit theorems for the nodalstatistics associated with a smooth compact domain, in thehigh-energy limit (E→ ∞). Our main result is that boththe nodal length (real case) and the number of nodal intersections(complex case) verify a Central Limit Theorem, which is in sharpcontrast with the non-Gaussian behaviour observed for real andcomplex arithmetic random waves on the flat 2-torus, see Marinucciet al. (2016) andDalmao et al. (2016).Our findings can be naturally reformulated in terms of the nodalstatistics of a single random wave restricted to a compact domaindiverging to the whole plane. As such, they can be fruitfullycombined with the recent results by Canzani and Hanin (2016), in order to show that, at anypoint of isotropic scaling and for energy levels divergingsufficently fast, the nodal length of any Gaussian pullbackmonochromatic wave verifies a central limit theorem with the samescaling as Berry's model. As a remarkable byproduct of ouranalysis, we rigorously confirm the asymptotic behaviour for thevariances of the nodal length and of the number of nodalintersections of isotropic random waves, as derived in Berry(2002).