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On the relation between orthogonal, symplectic and unitary matrix ensembles. (English) Zbl 0935.60090
Summary: For the unitary ensembles of $$N\times N$$ 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 $$2\times 2$$ 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.

MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 47N55 Applications of operator theory in statistical physics (MSC2000) 82B31 Stochastic methods applied to problems in equilibrium statistical mechanics 47G10 Integral operators
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