A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

Abstract

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.

On 3-dimensional Berry’s model

“…Canzani and Hanin (2020) studied the universality phenomenon in general Riemannian manifolds. The reader can find results on arithmetic random waves defined on the flat torus (Cammarota, 2019;Dalmao et al, 2019) and on random spherical harmonics in Cammarota and Marinucci (2019); Fantaye et al (2019); Marinucci and Rossi (2021) and references therein, see also Rossi (2019) for a survey on both subjects. The nodal sets of Berry's planar random waves, i.e.…”

On the correlation between nodal and boundary lengths for random spherical harmonics

“…Thus indeed the nodal lengths are asymptotically sufficient (in the high-energy limit) to characterize the measure of the boundary at any threshold level, provided that the effect of random fluctuations in the norm are properly taken into account. We refer to [9] for some numerical evidence on these and related issues.…”