Orthogonal Polynomials in the Spherical Ensemble with Two Insertions

Abstract. We consider asymptotics of planar orthogonal polynomials [Formula: see text] (where [Formula: see text]) with respect to the weight [Formula: see text] in the whole complex plane. With [Formula: see text] and [Formula: see text] fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted [Formula: see text] norm, and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift–Zhou steepest descent analysis of the associated [Formula: see text] Riemann–Hilbert problem.