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

9 pages