## Distributions of Angles in Random Packing on Spheres

*Tony Cai, Jianqing Fan, Tiefeng Jiang*; 14(Jul):1837−1864, 2013.

### Abstract

This paper studies the asymptotic behaviors of the pairwise
angles among $n$ randomly and uniformly distributed unit vectors
in $\mathbb{R}^p$ as the number of points $n\rightarrow \infty$,
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.

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