The Defect of Random Hyperspherical Harmonics

TL;DR

This paper studies the distribution of the defect of random hyperspherical harmonics on the unit sphere across all dimensions, providing new asymptotic variance formulas and quantitative CLTs using advanced probabilistic techniques.

Contribution

It extends previous two-dimensional results to higher dimensions, deriving asymptotics for defect variance and establishing quantitative CLTs with novel technical tools.

Findings

Asymptotic formulas for defect variance in any dimension

Quantitative CLTs in Wasserstein distance for the defect

Development of new integral estimates for 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-It\^o 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.