Translator Disclaimer
November 2020 Nodal lengths in shrinking domains for random eigenfunctions on $S^{2}$
Anna Paola Todino
Bernoulli 26(4): 3081-3110 (November 2020). DOI: 10.3150/20-BEJ1216

Abstract

We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: for example, the length of the zero set $\mathcal{Z}_{\ell,r_{\ell}}:=\mathcal{Z}^{B_{r_{\ell}}}(T_{\ell})=\operatorname{len}(\{x\in S^{2}\cap B_{r_{\ell}}:T_{\ell}(x)=0\})$, where $B_{r_{\ell}}$ is the spherical cap of radius $r_{\ell}$. We show that the variance of the nodal length is logarithmic in the high energy limit; moreover, it is asymptotically fully equivalent, in the $L^{2}$-sense, to the “local sample trispectrum”, namely, the integral on the ball of the fourth-order Hermite polynomial. This result extends and generalizes some recent findings for the full spherical case. As a consequence a Central Limit Theorem is established.

Citation

Download Citation

Anna Paola Todino. "Nodal lengths in shrinking domains for random eigenfunctions on $S^{2}$." Bernoulli 26 (4) 3081 - 3110, November 2020. https://doi.org/10.3150/20-BEJ1216

Information

Received: 1 August 2018; Revised: 1 November 2019; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256169
MathSciNet: MR4140538
Digital Object Identifier: 10.3150/20-BEJ1216

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

JOURNAL ARTICLE
30 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.26 • No. 4 • November 2020
Back to Top