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Title:
The mother body phase transition in the normal random matrix model
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Abstract:
We consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain Ω independent of the cut-off. We study in detail the mother body problem associated to Ω. It turns out that the mother body measure displays a novel phase transition, which we call the mother body phase transition: although ∂Ω evolves analytically, the mother body measure undergoes a “one-cut to three-cut” phase transition. We consider multiple orthogonal polynomials associated to the normal matrix model. Developing the Deift—Zhou nonlinear steepest descent method to the associated Riemann--Hilbert problem, we obtain strong asymptotic formulas for these polynomials, and we prove that the distribution of their zeros converges to the mother body measure.
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