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

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 $B_{E}^{-1}(0)$, is asymptotically equivalent, in the $L^{2}$-sense and in the high-frequency limit $E\rightarrow \infty$, 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].