A Generalisation of Dyson's Integration Theorem for Determinants

Akemann G, Shifrin L (2007)
J.Phys.A 40(32): F785-F791.

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Dyson's integration theorem is widely used in the computation of eigenvaluecorrelation functions in Random Matrix Theory. Here we focus on the variant ofthe theorem for determinants, relevant for the unitary ensembles with Dysonindex beta = 2. We derive a formula reducing the (n-k)-fold integral of an n xn determinant of a kernel of two sets of arbitrary functions to a determinantof size k x k. Our generalisation allows for sets of functions that are notorthogonal or bi-orthogonal with respect to the integration measure. In thespecial case of orthogonal functions Dyson's theorem is recovered.
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Akemann G, Shifrin L. A Generalisation of Dyson's Integration Theorem for Determinants. J.Phys.A. 2007;40(32):F785-F791.
Akemann, G., & Shifrin, L. (2007). A Generalisation of Dyson's Integration Theorem for Determinants. J.Phys.A, 40(32), F785-F791. doi:10.1088/1751-8113/40/32/F01
Akemann, G., and Shifrin, L. (2007). A Generalisation of Dyson's Integration Theorem for Determinants. J.Phys.A 40, F785-F791.
Akemann, G., & Shifrin, L., 2007. A Generalisation of Dyson's Integration Theorem for Determinants. J.Phys.A, 40(32), p F785-F791.
G. Akemann and L. Shifrin, “A Generalisation of Dyson's Integration Theorem for Determinants”, J.Phys.A, vol. 40, 2007, pp. F785-F791.
Akemann, G., Shifrin, L.: A Generalisation of Dyson's Integration Theorem for Determinants. J.Phys.A. 40, F785-F791 (2007).
Akemann, Gernot, and Shifrin, Leonid. “A Generalisation of Dyson's Integration Theorem for Determinants”. J.Phys.A 40.32 (2007): F785-F791.
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arXiv 0705.2555