Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups

Details Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups

2006-10-20

arxiv.org/abs/math-ph/0610049v5

J.Stat.Phys.129:885-935,2009

Physics

Mathematical Physics

58 pages; Acknowledgements added; small corrections in appendix A; minor changes & Note Added

Scientific paper

10.1007/s10955-007-9350-9

The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials prod tr{X^{p_1} Omega Y^{q_1} Omega^dagger X^{p_2} ... with the weight exp tr{X Omega Y Omega^dagger} are computed for the orthogonal and symplectic groups. We proceed in two steps. First, the integral over the compact group is recast into a Gaussian integral over strictly upper triangular complex matrices (with some additional symmetries), supplemented by a summation over the Weyl group. This result follows from the study of loop equations in an associated two-matrix integral and may be viewed as the adequate version of Duistermaat-Heckman's theorem for our correlation function integrals. Secondly, the Gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions.

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