On Orthogonal and Symplectic Matrix Ensembles Associated with a Class of Weight Functions

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 example the n-point correlation function and the spacing probabilities have nice representations in terms of this kernel. 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 show here that whenever w/w is a rational function these entries are equal to the scalar kernel for the corresponding unitary ensemble (but with w replaced by w2 for orthogonal ensembles and N replaced by 2N for symplectic ensembles) 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 derivation.