Distributions of Angles in Random Packing on Spheres

TL;DR

This paper analyzes the asymptotic distribution of angles among randomly distributed unit vectors on spheres, revealing differences based on fixed or growing dimensions, with implications for statistics, machine learning, physics, and mathematics.

Contribution

It derives the limiting distributions of pairwise and extreme angles in high-dimensional random vector packings, clarifying their asymptotic behaviors.

Findings

Limiting empirical distribution of angles derived for fixed and growing p

Extreme angles follow specific limiting distributions

High-dimensional vectors are nearly orthogonal with high probability

Abstract

This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in R^p as the number of points n -> infinity, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that "all high-dimensional random vectors are almost always nearly orthogonal to each other". Applications to statistics and machine learning and connections with some open problems in physics and mathematics are also discussed.