Consider $m$ points $v\_1, \ldots, v\_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|\_2 = 1\}$. Let the minimum separation be
$$
\rho = \min\_{i,j\in{\{1,\ldots,m\}}} \|v\_i - v\_j\|\_2. 
$$
Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m,n\to\infty$?


Here are some closely related questions: 


[Mean minimum distance for N random points on a one-dimensional line](https://mathoverflow.net/questions/1294/mean-minimum-distance-for-n-random-points-on-a-one-dimensional-line)


[Mean minimum distance for N random points on a unit square (plane)](https://mathoverflow.net/questions/124579/mean-minimum-distance-for-n-random-points-on-a-unit-square-plane)


[Mean minimum distance for K random points on a N-dimensional (hyper-)cube](https://mathoverflow.net/questions/22592/mean-minimum-distance-for-k-random-points-on-a-n-dimensional-hyper-cube)


However, the case for n-dimensional sphere seems less clear. 



Minimum separation among $m$ random points on an $n$-dimensional unit sphere
The preprint ["Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation"](http://arxiv.org/abs/1512.07470) by Johann S. Brauchart, Edward B. Saff, Ian H. Sloan, Yu Guang Wang, and Robert S. Womersley gives the following result in Corollary 3.4:


$\mathbb{E}[N^{2/d}\Theta\_\text{min}]\to C\_d = (\kappa\_d/2)^{-1/d}\Gamma(1+\tfrac{1}{d})$ as $N\to\infty$


It also gives a bound on the variance.


