The defect variance of random spherical harmonics

TL;DR

This paper analyzes the distribution of the defect, the difference between positive and negative regions, of random Gaussian spherical harmonics, focusing on its variance in the high frequency limit to aid understanding of spherical random fields.

Contribution

It provides the first asymptotic analysis of the defect variance for high-frequency random spherical harmonics, revealing its behavior and potential applications in cosmology.

Findings

Defect variance asymptotically characterized in high frequency limit

Expected defect value is zero for all degrees

Defect can serve as a statistical tool for spherical random fields

Abstract

The defect of a function $f:M\rightarrow \mathbb{R}$ 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.