A Quantitative Central Limit Theorem for the Excursion Area of Random Spherical Harmonics over Subdomains of $\mathbb{S}^2$

TL;DR

This paper establishes a quantitative central limit theorem for the excursion area of high-energy spherical harmonics over spherical caps, extending previous results from the full sphere and employing advanced approximation techniques.

Contribution

It generalizes the CLT for excursion areas from the full sphere to spherical caps, highlighting the dominant second-order chaos component in the asymptotic behavior.

Findings

Asymptotic behavior dominated by second-order chaos component

Established a quantitative CLT for excursion area over caps

Developed new approximation techniques for indicator functions

Abstract

In recent years, considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In this paper, we extend those results to proper subsets of the sphere $\mathbb{S}^2$, i.e., spherical caps, focussing in particular on the excursion area. Precisely, we show that the asymptotic behavior of the excursion area is dominated by the so-called second-order chaos component, and we exploit this result to establish a Quantitative Central Limit Theorem, in the high energy limit. These results generalize analogous findings for the full sphere; their proofs, however, requires more sophisticated techniques, in particular a careful analysis (of some independent interest) for smooth approximations of the indicator function for spherical caps subsets.