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-Ito 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)