  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.
