TITLE: Minimum separation among $m$ random points on an $n$-dimensional unit sphere
QUESTION [10 upvotes]: 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
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
However, the case for n-dimensional sphere seems less clear.

REPLY [5 votes]: The preprint "Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation" 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.