# An “Itzykson-Zuber”-like integral

I been told that there exists an integration formula, which states (or something of this sort) $$\int_{U(N)} dU \det[(\mathbb I+XUYU^{-1})^{-r}\propto \frac{\det(1+x_iy_j)^{N-r-1}]_{i,j}}{\Delta_N(x)\Delta_N(y)},$$ where $dU$ is the Haar-measure, $X=\text{diag}(x_1,\ldots,x_N)$ is a positive definite diagonal matrix with $x_i\neq x_j$ for $i\neq j$ (similar for $Y$), and $$\Delta_N(x)=\prod_{1\leq i<j\leq N}(x_i-x_j)$$ is a Vandermonde determinant.

This is somewhat similar to a Harish-Chandra-Itzykson-Zuber integral, but I haven't seen this specific kind before. Are anyone familiar with this kind of integrals, and how to solve them? I would be particular interested in references to the literature.

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Since I found a reference which answers this question (and noone has provided any other solution) I will give a short answer to my own question.

This type of integrals has been studied in the context of two-matrix models. The integral in question has been discussed in arXiv:0804.0873 (section 3), while a larger class of integrals has been discussed in arXiv:0512056 [math-ph] (appendix A).

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This statement does not seem to be correct, because if X tends to the zero matrix, the left hand side is well defined but the right hand side is not (notice that HCIZ integral does not suffer from this).

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Ups! I had misunderstood the right hand side. I read it as if the numerator had the determinant of the identity plus something... Sorry – Marcel Mar 24 '13 at 23:11

This identity, and others of similar type, are proved explicitly in: J. Harnad and A. Yu. Orlov, Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions'', J. Phys. A, 39, 8783--8809 (2006). (Appendix A]

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Thank you very much for your comment. I presume that this type of integrals dates back to Gross and Richards ( Trans. Amer. Math. Soc. 301 (1987), 781 , J. Approx. Theory 59 (1989) 224 , Bulletin of AMS 24 (1991) 349 ), which, more recently, have been extended by Orlov and yourself? – Jesper Ipsen Jul 7 '14 at 11:43