Consider 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?

Here are some closely related questions: 

http://mathoverflow.net/questions/1294/mean-minimum-distance-for-n-random-points-on-a-one-dimensional-line

http://mathoverflow.net/questions/124579/mean-minimum-distance-for-n-random-points-on-a-unit-square-plane

http://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.