Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation

TL;DR

This paper investigates the geometric properties of randomly distributed points on the sphere, deriving asymptotics for various radii and separation metrics, and proposing conjectures supported by numerical evidence.

Contribution

It provides new asymptotic results for spherical cap radii, conjectures on their distributions, and precise estimates for point separation on the sphere.

Findings

Asymptotics for expected moments of spherical cap radii

Conjectures on the distribution of scaled radii and covering radius

Precise asymptotics for expected separation of points

Abstract

Geometric properties of $N$ random points distributed independently and uniformly on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as $N \to \infty$) for the expected moments of the radii of spherical caps associated with the facets of the convex hull of $N$ random points on $\mathbb{S}^{d}$. We provide conjectures for the asymptotic distribution of the scaled radii of these spherical caps and the expected value of the largest of these radii (the covering radius). Numerical evidence is included to support these conjectures. Furthermore, utilizing the extreme law for pairwise angles of Cai et al., we derive precise asymptotics for the expected separation of random points on $\mathbb{S}^{d}$.