Suppose I want to compute a quantity of the type:


where averaging is over Haar measure on the unitary group $\mathcal{U}(n)$ (one can of course consider higher order polynomials or other matrix ensembles etc.) and $A$, $B$ are some fixed matrices. Is there any standard technique for computing such averages? I'd guess people in random matrix theory or free probability compute such traces all the time, but I've been unable to find a reference. If it makes matters easier, I'm really interested in computing something for random projections (e.g. something of the form $\mathbb{E}\mathrm{tr}(APBP)$, where $P$ is a projection onto a random subspace), which of course reduces to computation of polynomials in $U$.

  • $\begingroup$ Do you mean that $A, B$ are fixed, $U$ varies over the unitary group? $\endgroup$ – Igor Rivin May 15 '12 at 15:08
  • $\begingroup$ @IR: yes, edited for clarity. $\endgroup$ – Marcin Kotowski May 15 '12 at 15:40
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    $\begingroup$ $\mathbb E tr(A U B U^*)=tr(A) tr(B)/n$. Indeed, from the property of Haar measure, $\mathbb E(U B U^*)$ is a matrix that commutes with every unitary matrix, so that it a multiple of the identity matrix. It has the same as $B$, so $\mathbb E(U B U^*) = tr(B)/n 1$. $\endgroup$ – Mikael de la Salle May 15 '12 at 16:54
  • $\begingroup$ @Mikael: can this observation be generalized to higher order polynomials (e.g. $AUBU^{\ast}CUDU^{\ast}$)? In that case, we can't put E inside the trace for both terms UXU^{\\ast} only by linearity. $\endgroup$ – Marcin Kotowski May 15 '12 at 17:17
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    $\begingroup$ Some matrix integrals over U(n) wiht Haar measure extensively computed by physitists,e.g. arxiv.org/pdf/math-ph/0406063v1.pdf A short note about Morozov’s formula B. Eynard, arxiv.org/abs/0906.3518 Unitary Integrals and Related Matrix Models A.Morozov, arxiv.org/abs/hep-th/9404005 , arxiv.org/abs/0906.3305 $\endgroup$ – Alexander Chervov May 15 '12 at 19:09

For the unitary group, the first paper I am aware of to do these sorts of averages is:


An early paper of Collins' in 2003 expresses such averages in terms of Weingarten functions, which are usually expressed as character expansions over $U(N)$, $O(N)$ or $Sp(N)$.


Some early papers calculating these character expansions are:



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    $\begingroup$ for a Mathematica program to evaluate Weingarten functions, see arXiv:1109.4244 $\endgroup$ – Carlo Beenakker Sep 6 '12 at 12:30

Answering my own question, there is a closed formula for such traces, given in: http://arxiv.org/abs/math-ph/0402073 (the formula involves representation theory of $S_n$ and gets ugly as $n$ gets bigger, but can be written down explicitly at least for small $n$)


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