# A note on the reduction principle for the nodal length of planar random waves

*Anna Vidotto*

*Statistics & Probability Letters*, 2021, vol. 174, issue C

**Abstract:**
Inspired by Marinucci et al. (2020), we prove that the nodal length of a planar random wave BE, i.e. the length of its zero set BE−1(0), is asymptotically equivalent, in the L2-sense and in the high-frequency limit E→∞, to the integral of H4(BE(x)), H4 being the fourth Hermite polynomial. As straightforward consequences, we obtain Moderate Deviation estimates and a central limit theorem in Wasserstein distance. This complements recent findings by Nourdin et al. (2019) and Peccati and Vidotto (2020).

**Keywords:** Nodal length; Random plane waves; Sample trispectrum; Berry’s cancellation; Quantitative central limit theorem (search for similar items in EconPapers)

**Date:** 2021

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**DOI:** 10.1016/j.spl.2021.109090

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