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Let $O(N)$ be the orthogonal group, and $a,b,c\in\mathbb N$. The question is: $$\int_{O(N)}U_{11}^aU_{22}^bU_{33}^cdU=?$$

This is quite a tricky question:

(1) The first thought would go to probability, because with $N\to\infty$ the variables $U_{11},U_{22},U_{33}$ become Gaussian and independent; however, there doesn't seem to be any good analytic method for computing the correlations at $N$ fixed.

(2) The second thought would go to combinatorics, and to the Weingarten function. But that doesn't work either: we spent some time with Collins and Schlenker on this question, and just got a kind of very long (and especially unusable!) formula.

So, a new point of view on all this would be probably needed.

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Is the paper relevant? – Richard Stanley Nov 23 '12 at 17:50
What are $U_{11}, U_{22}$, and $U_{33}$? – Qiaochu Yuan Nov 24 '12 at 2:00
Seems not what you want, but may be something can be extracted... A short note about Morozov's formula Bertrand Eynard – Alexander Chervov Nov 24 '12 at 18:52
Do you want a precise answer, approximation, or anything else? I mean, once the exact formula obtained from the recursion is "ugly and unusable", it will stay this way no matter what you try. However, if you want to settle for less than an exact formula, something may be possible to do. Can you elaborate a bit on what is the absolute minimum you would settle at? – fedja Nov 26 '12 at 0:29
up vote 5 down vote accepted

This is a response to your question (1), for an analytic method to compute the integral over $O(N)$ as a power series in $1/N$. This method was developed by Prosen, Seligman and Weidenmüller in J. Math. Phys. 43, 5135-5144 (2002) [arXiv:math-ph/0203042]. Their key result can be written, in the context of your integral, as

$$\int_{O(N)}dU\;U_{11}^{a}U_{22}^{b}U_{33}^{c}=\int d\mu\; w_{\kappa}(M)M_{11}^{a}M_{22}^{b}M_{33}^{c}+{\cal O}(1/N^{z+1})$$

The order of the approximation is $z=\mbox{Int}[(\kappa+a+b+c)/2]$. The $N\times N$ real matrix $M$ has Gaussian measure $d\mu\propto\exp(-N\;\mbox{Tr}\;MM^{T})\prod_{ij}dM_{ij}$, so the integral on the right-hand-side is simply a Gaussian integral over the matrix elements $M_{ij}$ of $M$. The orthonormality constraints of the integral on the left-hand-side are accounted for by a weight function $w_{\kappa}$ of linearly independent invariants of $M$ up to order $2\kappa$. Explicit expressions for $w_{\kappa}$ for $\kappa=1,2,3,4$ are given in the cited reference.

To lowest order, one has $\kappa=1$, when $w_{1}\equiv 1$. This amounts to treating $U_{11}$, $U_{22}$, and $U_{33}$ as independent Gaussians, which is the lowest-order approximation you mentioned in your question. Increasing $\kappa$ gives you the higher order corrections in a systematic way.

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