On the Relation Between Orthogonal,Symplectic and Unitary Matrix Ensembles |
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Authors: | Widom Harold |
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Affiliation: | (1) Department of Mathematics, University of California, Santa Cruz, California, 95064 |
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Abstract: | For the unitary ensembles of N×N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w/w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w/w. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations |
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Keywords: | random matrices matrix kernels unitary ensembles orthogonal ensembles symplectic ensembles Laguerre ensembles |
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