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

TL;DR

This paper demonstrates that the nodal length of planar random waves behaves asymptotically like a specific Hermite polynomial integral, leading to a central limit theorem in the high-frequency limit.

Contribution

It establishes a reduction principle linking the nodal length to Hermite polynomial integrals, extending previous results and providing a new CLT in Wasserstein distance.

Findings

Asymptotic equivalence of nodal length and Hermite polynomial integral

Proof of a central limit theorem in Wasserstein distance for high frequencies

Extension of recent theoretical results on random wave nodal sets

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 $E\rightarrow \infty$, 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].