The mother body phase transition in the normal matrix model

The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth. In this paper we consider the normal matrix model with cubic plus linear potential. To regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain $\Omega$ that we determine explicitly by finding the rational parametrization of its boundary. We also study in details the mother body problem associated to $\Omega$. It turns out that the mother body measure $\mu_*$ displays a novel phase transition that we call the mother body phase transition: although $\partial \Omega$ evolves analytically, the mother body measure undergoes a "one-cut to three-cut" phase transition. To construct the mother body measure, we define a quadratic differential $\varpi$ on the associated spectral curve, and embed $\mu_*$ into its critical graph. Using deformation techniques for quadratic differentials, we are able to get precise information on $\mu_*$. In particular, this allows us to determine the phase diagram for the mother body phase transition explicitly. Following previous works of Bleher & Kuijlaars and Kuijlaars & L\'opez, we consider multiple orthogonal polynomials associated with the model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated $g$-functions significantly more involved, and the critical graph of $\varpi$ becomes the key technical tool in this analysis as well.