The Defect of Random Hyperspherical Harmonics

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d -sphere ( d ≥ 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 in J Phys A Math Theor 44:355206, 2011 ) and a central limit theorem (Marinucci and Wigman in Commun Math Phys 327(3):849–872, 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 (in Prob Theory Relat Fields 145(1–2):75–118, 2009 ; Normal approximations with Malliavin calculus. Cambridge Tracts in Mathematics, vol 192, Cambridge University Press, Cambridge, 2012 ). Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics.