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

Abstract

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.

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

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• Random matrix ensembles with column/row constraints: part I
• Scattering of Long Folded Strings and Mixed Correlators in the Two-Matrix Model
• Derivative principles for invariant ensembles
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Harish-Chandra integrals as nilpotent integrals
• Preserving Topology while Breaking Chirality: From Chiral Orthogonal to Anti-symmetric Hermitian Ensemble
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