Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane

The purpose of this paper is to describe the strong asymptotics of orthogonal polynomials with respect to measures of the form $$ e^{-|z|^{2s}+tz^s+\overline{tz}^s}dA(z) $$ in the complex plane where $s$ is a positive integer, $t$ is a complex parameter and $dA$ stands for the area measure in the plane. Such problem has its origin from normal matrix models. After a natural symmetry reduction it is shown that the orthogonality conditions for the resulting polynomials on the complex plane can be written equivalently in terms of non-hermitian contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann-Hilbert problem by using the Deift-Zhou nonlinear steepest descent method.