Correlations for the Cauchy and generalized circular ensembles with orthogonal and symplectic symmetry

The generalized circular ensemble, which specifies a spectrum singularity in random matrix theory, is equivalent to the Cauchy ensemble via a stereographic projection. The Cauchy weight function is classical, and as such the n-point distribution function in the cases of orthogonal and symplectic symmetry have expressions in terms of quaternion determinants with elements given in an explicit form suitable for asymptotic analysis. The asymptotic analysis is undertaken in the neighbourhood of the spectrum singularity in both cases, and it is shown that each quaternion determinant is specified by a single function involving Bessel functions.