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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Widom, H. On the Relation Between Orthogonal, Symplectic and Unitary Matrix Ensembles. Journal of Statistical Physics 94, 347–363 (1999). https://doi.org/10.1023/A:1004536018336
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DOI: https://doi.org/10.1023/A:1004536018336