×

The defect variance of random spherical harmonics. (English) Zbl 1232.60039

The framework here is the family of random functions defined as linear combinations of spherical Laplacian harmonics, in which the coefficients are independent standard Gaussian parameters with zero-mean and the unit variance. The main result is related to the so-called defect of these functions, which, in other words, is the difference between the areas of the positive and the negative inverse image of the considered function. Then, one derives the asymptotic high-frequency limit of this defect variance. The proof is based on some special properties of Legendre polynomials.

MSC:

60G60 Random fields
60G15 Gaussian processes
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
60D05 Geometric probability and stochastic geometry
33C55 Spherical harmonics