# mean distance between subspaces

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.

• Probably a silly remark, but for a fixed $k$ and large $n$, doesn't it tend to the distance between $\mathrm{span}\{e_1,\dots,e_k\}$ and $\mathrm{span}\{e_{k+1},\dots,e_{2k}\}$, where $e_1,\dots,e_{2k}$ is an orthonormal basis in $\mathbb{R}^{2k}$? You can sample two subspaces as a span of $k$ independent vectors each and these will be nearly orthogonal for large $n$. – Kostya_I Jan 4 at 11:07