A Note on the Reduction Principle for the Nodal Length of Planar Random Waves

Abstract

Inspired by the recent work [MRW20], we prove that the nodal length of a planar random wave B E , i.e. the length of its zero set B −1 E (0), is asymptotically equivalent, in the L 2 -sense and in the high-frequency limit E → ∞, to the integral of H 4 (B E (x)), H 4 being the fourth Hermite polynomial. As a straightforward consequence, we obtain a central limit theorem in Wasserstein distance. This complements recent findings in [NPR19] and [PV20].