  We consider the normal matrix model with a cubic potential. The model is
ill-defined, and in order to reguralize 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 $3\times 3$ 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.
