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Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs

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Abstract

We present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. We show that the 2N×2N Gaussian Orthogonal Ensemble (GOE) and N×N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a non-orientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincaré duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.

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Authors and Affiliations

  1. Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616–8633, USA

    Motohico Mulase & Andrew Waldron

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  1. Motohico Mulase
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  2. Andrew Waldron
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Correspondence to Andrew Waldron.

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Communicated by M. Aizenman

Research supported by NSF Grant DMS-9971371 and the University of California, Davis.

Research supported by the University of California, Davis.

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Mulase, M., Waldron, A. Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs. Commun. Math. Phys. 240, 553–586 (2003). https://doi.org/10.1007/s00220-003-0918-1

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  • DOI: https://doi.org/10.1007/s00220-003-0918-1

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