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We consider the type I multiple orthogonal polynomials (MOPs) $(A_{n,m},B_{n,m})$ and type II MOPs $P_{n,m}$, satisfying non-hermitian orthogonalitywith respect to the weight $e^{-z^3}$ on two unbounded contours on $\mathbb C$.Under the assumption that $$ n,m \to \infty, \quad \frac{n}{n+m}\to \alpha \in(0, 1) $$ we find the detailed asymptotics of the MOPs, and describe the phasetransitions of this limit behavior as a function of $\alpha$. This descriptionis given in terms of vector critical measures, which are saddle points of anenergy functional comprising both attracting and repelling forces. Thesecritical measures are characterized by a cubic equation (spectral curve), andtheir components $\mu_j$ live on trajectories of a canonical quadraticdifferential $\varpi$ on the Riemann surface of this equation, which was objectof study in our previous paper [Adv. Math. 302 (2016), 1137--1232].  The asymptotic zero distribution of the polynomials $A_{n,m}$ and $P_{n,m}$are given by appropriate combinations of the components of the vector criticalmeasure. However, in the case of the zeros of $B_{n,m}$ the behavior is totallydifferent, and can be described in terms of the balayage of $\mu_2 - \mu_3$onto certain curves on the plane. These curves are constructed with the aid of$\varpi$, and their topology has three very distinct characters, depending onthe value of $\alpha$, and are obtained from the critical graph of $\varpi$.  Once the trajectories and vector critical measures are studied, the mainasymptotic technical tool is the analysis of a $3\times 3$ Riemann-Hilbertproblem characterizing the MOPs.  We illustrate our findings with results of several numerical experiments, andformulate some conjectures and empirical observations based on theseexperiments.

Critical measures for vector energy: asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight