The Defect of Random Hyperspherical Harmonics

Abstract

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-sphere ($d\ge 2$). We investigate the distribution of their defect i.e., the difference between the measure of positive and negative regions. Marinucci and Wigman studied the two-dimensional case giving the asymptotic variance (Marinucci and Wigman 2011) and a Central Limit Theorem (Marinucci and Wigman 2014), both in the high-energy limit. Our main results concern asymptotics for the defect variance and quantitative CLTs in Wasserstein distance, in any dimension. The proofs are based on Wiener-Itô chaos expansions for the defect, a careful use of asymptotic results for all order moments of Gegenbauer polynomials and Stein-Malliavin approximation techniques by Nourdin and Peccati. Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics.

Accepted for publication in Journal of Theoretical Probability

References in corpus (1)

• On the volume of nodal sets for eigenfunctions of the Laplacian on the torus

Cited by in corpus (1)

• A Quantitative Central Limit Theorem for the Excursion Area of Random Spherical Harmonics over Subdomains of ${\mathbb{S}}^2$