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Somehow I got wondering about the following question today:

Suppose $Q_1,\ldots,Q_n$ are random (uniformly sampled) $d \times d$ orthogonal matrices.

What is the expected value of the quantity $\|\sum_i Q_i\|$?

Additionally, suppose I actually generate random skew-symmetric matrices $S_1,\ldots,S_n$, and then obtain corresponding orthogonal matrices via the matrix exponential, $e^{S_i}$.

What is the expected value of the quantity $\|\sum_i e^{S_i}\|$

EDIT: From the comments (and from Mikael's answer) it seems like this is a tough question. But already the case with large $d$ is quite useful.

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Are you interested in an asymptotic regime (e.g. n fixed, d large; or d fixed, n large)? I doubt there will be an exact formula otherwise; the operator norm is not algebraic enough for an algebraic miracle (as opposed to, say, the Frobenius norm). In the regime where n is fixed and d is large, free probability tools should in principle answer your question (or at least predict the answer). – Terry Tao Nov 5 '11 at 19:47
Conversely, in the regime where d is fixed and n is large, the classical central limit theorem should give good results, since the sum is going to be asymptotically a gaussian matrix (possibly shifted by a multiple of the identity, in the second question). – Terry Tao Nov 5 '11 at 19:49
@Terry: Ideally, non asymptotic results would be the best. I am, however, still curious about the limited asymptotic regimes where only one of $n$ or $d$ is allowed to be large (feels like the case of large $n$ should be easier). The Frobenius norm definitely seems much friendlier. – Suvrit Nov 5 '11 at 19:52

EDIT: My answer only deals with the $d \to \infty$ regime.

This question is not too naive (or at least the answer is hard). I am almost sure that for fixed $d$ there is no exact formula. For the limit as $d \to \infty$ I think that one expects that the norm of $\sum_1^n Q_i$ almost surely converges to $2 \sqrt{n-1}$, but I don't know if a proof exists yet (my guess is that everything works the same way as for unitaries, see below).

If one replaces orthogonal by unitaries, the result is known to hold from the very recent work of Collins and Male, see part 3.2 here. Their result is more general and they compute the liming norm of any sum of products of independant random unitaries in term of free probability. In fact the proof uses a simple but clever coupling argument together with the deep work of Haagerup and Thorbjornsen A new application of Random Matrices: Ext(C*_{red}(F_2)) is not a group, where Haagerup and Thorbjornsen prove the corresponding result for gaussian hermitian matrices instead of unitaries.

The work of H-T has been generalized to random real symmetric matrices by Hanne Schulz, and this implies that the answer to the second question can be computed in finite time (assuming that you take a suitable Gaussian probability measure on the set of skew-symmetric matrices). I can try to do the exact computation if you want.

Second edit: I deleted a remark where I said that the norm of $\sum_1^N e^{S_i}$ is of order $\sqrt n$. This would be true if $E(e^{S_i})$ was $0$, which is not the case as Terry Tao points out in his comments to your question. In fact it is the norm of $\sum (e^{S_i} - E(e^{S_i}))$ which is of order $\sqrt n$.

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+1: You always have very nice answers Mikael! – Suvrit Nov 6 '11 at 9:54

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