The defect variance of random spherical harmonics

Abstract

The defect of a function $f:M\to \mathbb{R}$ is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is obtaining the asymptotic shape of the defect variance, in the high frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern Cosmological data analysis.

19 pages

References in corpus (6)

• The non-Gaussian Cold Spot in the 3-year WMAP data
• Limits on Primordial Non-Gaussianity from Minkowski Functionals of the WMAP Temperature Anisotropies
• Analytic Minkowski Functionals of the Cosmic Microwave Background: Second-order Non-Gaussianity with Bispectrum and Trispectrum
• Fluctuations of the nodal length of random spherical harmonics, erratum
• On the Excursion Sets of Spherical Gaussian Eigenfunctions
• Ergodicity and Gaussianity for Spherical Random Fields

Cited by in corpus (8)

• On Nonlinear Functionals of Random Spherical Eigenfunctions
• On the Excursion Sets of Spherical Gaussian Eigenfunctions
• On the Limiting Behaviour of Needlets Polyspectra
• Fluctuations of the number of excursion sets of planar Gaussian fields
• The Defect of Random Hyperspherical Harmonics
• A Quantitative Central Limit Theorem for the Excursion Area of Random Spherical Harmonics over Subdomains of ${\mathbb{S}}^2$
• The defect of toral Laplace eigenfunctions and Arithmetic Random Waves
• Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on ${\mathbb{S}}^d$