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

2020/07/08 by Anna Vidotto, Vidotto, Anna · 2 citations
Mathematics · #33C10 #34L20 #60F05 #60G60 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2007.04228

openalex publication_date 2020/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Inspired by the recent work [MRW20], we prove that the nodal length of a\nplanar random wave B_E, i.e. the length of its zero set B_E^-1(0), is\nasymptotically equivalent, in the L²-sense and in the high-frequency limit\nE\→ \∞, to the integral of H₄(B_E(x)), H₄ being the\nfourth Hermite polynomial. As a straightforward consequence, we obtain a\ncentral limit theorem in Wasserstein distance. This complements recent findings\nin [NPR19] and [PV20].\n

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• Malliavin differentiability for the excursion measure of a Gaussian field
• New chaos decomposition of Gaussian nodal volumes

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