Orthogonal polynomials in the normal matrix model with a cubic potential

We consider the normal matrix model with a cubic potential. The model is ill-defined, and in order to regularize it, Elbau and Felder introduced a model with a cut-off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut-off region on the complex plane. In the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane, without any cut-off, which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of Elbau and Felder. The main goal of this paper is to develop the Riemann-Hilbert (RH) approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree n of the polynomial goes to infinity. As the first step in the RH approach, we introduce an auxiliary vector equilibrium problem for a pair of measures (mu(1), mu(2)) on the complex plane. We then formulate a 3 x 3 matrix valued RH problem for the orthogonal polynomials in hand. and we apply the nonlinear steepest descent method of Deift-Zhou to the asymptotic analysis of the RH problem. The central steps in our study are a sequence of transformations of the RH problem, based on the equilibrium vector measure (mu(1), mu(2)), and the construction of a global parametrix. The main result of this paper is a derivation of the large n asymptotics of the orthogonal polynomials on the whole complex plane. We prove that the distribution of zeros of the orthogonal polynomials converges to the measure mu(1), the first component of the equilibrium measure. We also obtain analytical results for the measure mu(1) relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve: (C) 2012 Elsevier Inc. All rights reserved.