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:

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.