Normal matrix models, dbar-problem, and orthogonal polynomials on the complex plane

Abstract

We introduce a dbar-formulation of the orthogonal polynomials on the complex plane, and hence of the related normal matrix model, which is expected to play the same role as the Riemann-Hilbert formalism in the theory of orthogonal polynomials on the line and for the related Hermitian model. We propose an analog of Deift-Kriecherbauer-McLaughlin-Venakides-Zhou asymptotic method for the analysis of the relevant dbar-problem, and indicate how familiar steps for the Hermitian model, e.g. the g-function ``undressing'', might look like in the case of the normal model. We use the particular model considered recently by P. Elbau and G. Felder as a case study.