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" 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.