The defect of random hyperspherical harmonics. (English) Zbl 1480.60139
Summary: Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit \(d\)-sphere \((d\geq 2)\). 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 [D. Marinucci and I. Wigman, J. Phys. A, Math. Theor. 44, No. 35, Article ID 355206, 16 p. (2011; Zbl 1232.60039)] and a central limit theorem [D. Marinucci and I. Wigman, Commun. Math. Phys. 327, No. 3, 849–872 (2014; Zbl 1322.60030)], 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 I. Nourdin and G. Peccati [Probab. Theory Relat. Fields 145, No. 1–2, 75–118 (2009; Zbl 1175.60053); Normal approximations with Malliavin calculus. From Stein’s method to universality. Cambridge: Cambridge University Press (2012; Zbl 1266.60001)]. Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics.
MSC:
| 60G60 | Random fields |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 60D05 | Geometric probability and stochastic geometry |
| 60B10 | Convergence of probability measures |
| 43A75 | Harmonic analysis on specific compact groups |
Keywords:
defect; Gaussian eigenfunctions; high-energy asymptotics; quantitative central limit theorem; integrals of hyperspherical harmonicsReferences:
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