Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their Grassmann geodesic distance (this is equal to $\|\mathbf{\theta}\|,$ where $\mathbf{\theta}$ is the vector of *principal angles* between the subspaces. I would like a formula (or an asymptotic expression for large $n$) for the mean of this distance.

Experiments show that this quantity seems to be normal-ish, and that the variance goes to zero - the latter is not a surprise, and is due to Gromov-Milman 1983.