Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. They are characterised by a matrix-valued kernel of skew-orthogonal polynomials, however finding an appropriate set of polynomials for a given matrix ensemble is difficult.
In this talk I will present an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation. Additionally, for the symplectic elliptic Ginibre ensemble I will show how to compute the microscopic large-N limit of the kernel at the origin and prove its universality.
This is based on joint work with Gernot Akemann and Iv?n Parra. If time permits, I will also mention some results from an ongoing project with Sung-Soo Byun.