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February 2020 The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics
Domenico Marinucci, Maurizia Rossi, Igor Wigman
Ann. Inst. H. Poincaré Probab. Statist. 56(1): 374-390 (February 2020). DOI: 10.1214/19-AIHP964

Abstract

We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell}$ of high degree $\ell\rightarrow\infty$, i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It is found that the nodal lengths are asymptotically equivalent, in the $L^{2}$-sense, to the “sample trispectrum”, i.e., the integral of $H_{4}(f_{\ell}(x))$, the fourth-order Hermite polynomial of the values of $f_{\ell}$. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.

Nous étudions le comportement asymptotique de la longueur nodale de fonctions propres aléatoires $f_{\ell}$ du Laplacien sphérique pour valeurs propres très élevés $\ell\rightarrow+\infty$, c’est-à-dire la longueur de leur ensemble de niveau zéro $f_{\ell}^{-1}(0)$. Nous démontrons que la longueur nodale est asymptotiquement équivalente, au sens de $L^{2}$, au « sample trispectrum », c’est-à-dire l’intégral de $H_{4}(f_{\ell}(x))$, le polynôme de Hermite d’ordre quatre évalué en $f_{\ell}$. Une conséquence de ce résultat est un Théorème Central Limite quantitatif (dans le sens de la distance de Wasserstein) pour la longueur nodale, quand l’énergie tend vers l’infini.

Citation

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Domenico Marinucci. Maurizia Rossi. Igor Wigman. "The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics." Ann. Inst. H. Poincaré Probab. Statist. 56 (1) 374 - 390, February 2020. https://doi.org/10.1214/19-AIHP964

Information

Received: 13 February 2018; Revised: 16 January 2019; Accepted: 30 January 2019; Published: February 2020
First available in Project Euclid: 3 February 2020

zbMATH: 07199308
MathSciNet: MR4058991
Digital Object Identifier: 10.1214/19-AIHP964

Subjects:
Primary: 33C55 , 42C10 , 53C65 , 60G60 , 62M15

Keywords: Berry’s cancellation , Nodal length , Quantitative Central Limit Theorem , Sample trispectrum , Spherical harmonics

Rights: Copyright © 2020 Institut Henri Poincaré

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Vol.56 • No. 1 • February 2020
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