Characteristic polynomials of non-Hermitian matrices, duality, and Painlevé transcendents

We study expectations of powers and correlations for characteristic polynomials of N x N non-Hermitian random matrices. This problem is related to the analysis of planar models (log-gases) where a Gaussian (or other) background measure is perturbed by a finite number of point charges in the plane. I will discuss the critical asymptotics, for example when a point charge collides with the boundary of the support, or when two point charges collide with each other (coalesce) in the bulk. In many of these situations, we are able to express the results in terms of Painlevé transcendents. The application to certain d-fold rotationally invariant models will be discussed. This is joint work with Alfredo Deaño (University of Kent).