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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.

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    $\begingroup$ 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 $n$. $\endgroup$ – Kostya_I Jan 4 at 11:07

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