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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 present paper the authors consider the normal matrix model with cubic plus linear potential. In order to regularize the model, they 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 they determine explicitly by finding the rational parametrization of its boundary. The authors also study in detail 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.

Matrices --- Random matrices. --- Phase transformations (Statistical physics) --- Functions, Meromorphic. --- Riemann-Hilbert problems. --- Integral transforms. --- Norms.