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

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