Abstract. Curves in the complex plane that satisfy the S-property were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s. Rakhmanov recently showed the existence of curves with the S-property in a harmonic external field by means of a max-min variational problem in logarithmic potential theory. This is done in a fairly general setting, which however does not include the important special case of an external field Re V where V is a polynomial of degree ≥ 2. In this paper we give a detailed proof of the existence of a curve with the S-property in the external field Re V within the collection of all curves that connect two or more pre-assigned directions at infinity in which Re V → +∞. Our method of proof is very much based on the works of Rakhmanov on the max-min variational problem and of Martínez-Finkelshtein and Rakhmanov on critical measures.

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function e niλz on [−1, 1], where λ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when λ is smaller than a certain critical value, λ c . Our main goal is to compute their asymptotics when λ > λ c .We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann-Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift-Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane.Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type.In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of λ for which one cannot solve the model Riemann-Hilbert problem, and as such the corresponding polynomials fail to exist. Contents

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We consider the type I multiple orthogonal polynomials (MOPs) (An,m, Bn,m), deg An,m ≤ n − 1, deg Bn,m ≤ m − 1, and type II MOPs Pn,m, deg Pn,m = n + m, satisfying non-hermitian orthogonality with respect to the weight e −z 3 on two unbounded contours γ 1 and γ 2 on C, with (in the case of type II MOPs) n conditions on γ 1 and m on γ 2 . Under the assumption that n, m → ∞, n n + m → α ∈ (0, 1)we find the detailed (rescaled) asymptotics of An,m, Bn,m and Pn,m on C, and describe the phase transitions of this limit behavior as a function of α. This description is given in terms of the vector critical measure µα = (µ 1 , µ 2 , µ 3 ), the saddle point of an energy functional comprising both attracting and repelling forces. This critical measure is characterized by a cubic equation (spectral curve), and its components µ j live on trajectories of a canonical quadratic differential on the Riemann surface of this equation. The structure of these trajectories and their deformations as functions of α was object of study in our previous paper [Adv. Math. 302 (2016), 1137-1232, and some of the present results strongly rely on the analysis carried out there.We conclude that the asymptotic zero distribution of the polynomials An,m and Pn,m are given by appropriate combinations of the components µ j of the vector critical measure µα. However, in the case of the zeros of Bn,m the behavior is totally different, and can be described in terms of the balayage of the signed measure µ 2 − µ 3 onto certain curves on the plane. These curves are constructed with the aid of the canonical quadratic differential , and their topology has three very distinct characters, depending on the value of α, and are obtained from the critical graph of .Once the trajectories and vector critical measures are studied, the main asymptotic technical tool is the Deift-Zhou nonlinear steepest descent analysis of a 3 × 3 matrix-valued Riemann-Hilbert problem characterizing both (An,m, Bn,m) and Pn,m, which allows us to obtain the limit behavior of both types of MOPs simultaneously.We illustrate our findings with results of several numerical experiments, explain the computational methodology, and formulate some conjectures and empirical observations based on these experiments.