Open AccessA numerical investigation on the high‐frequency geometry of spherical random eigenfunctions
Author(s) -
Fantaye Yabebal,
Cammarota Valentina,
Marinucci Domenico,
Todino Anna Paola
Publication year - 2019
Publication title -
high frequency
Language(s) - English
Resource type - Journals
ISSN - 2470-6981
DOI - 10.1002/hf2.10044
Subject(s) - eigenfunction , mathematics , excursion , spherical harmonics , eigenvalues and eigenvectors , lipschitz continuity , mathematical analysis , spin weighted spherical harmonics , geometry , connection (principal bundle) , harmonics , physics , quantum mechanics , voltage , political science , law
A lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high‐frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz‐Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler‐Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so‐called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data.