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The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics

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Domenico Marinucci, Maurizia Rossi, Igor Wigman

Original languageEnglish
Pages (from-to)374-390
Number of pages17
JournalAnnales de l’Institut Henri Poincaré, Probabilités et Statistiques
Volume56
Issue number1
Early online date3 Feb 2020
DOIs
Accepted/In press30 Jan 2019
E-pub ahead of print3 Feb 2020
Published2020

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Abstract

We study the asymptotic behaviour of the nodal length of random 2d-spherical harmonics f of high degree ℓ → ∞, i.e. the length of their zero set f -1 (0). It is found that the nodal lengths are asymptotically equivalent, in the L 2-sense, to the "gsample trispectrum", i.e., the integral of H 4(f (x)), the fourth-order Hermite polynomial of the values of f . A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.

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