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The Mother Body Phase Transition in the Normal Matrix Model
About this Title
Pavel M. Bleher, Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, Indiana 46202 and Guilherme L. F. Silva, KU Leuven, Department of Mathematics, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 265, Number 1289
ISBNs: 978-1-4704-4184-5 (print); 978-1-4704-6146-1 (online)
DOI: https://doi.org/10.1090/memo/1289
Published electronically: March 31, 2020
Keywords: Random matrix theory,
normal matrix model,
mother body problem,
Schwarz function,
Riemann-Hilbert problems,
multiple orthogonal polynomials,
trajectories of quadratic differentials
MSC: Primary 60B20; Secondary 30C99, 30Exx, 30F30, 31A15, 44A60
Table of Contents
Chapters
- 1. Introduction
- 2. Statement of main results
- 3. Limiting boundary of eigenvalues. Proofs of Propositions 2.1 and 2.7 and Theorems 2.2, 2.5 and 2.8
- 4. Geometry of the spectral curve. Proof of Theorem 2.6
- 5. Meromorphic quadratic differential on $\mathcal R$
- 6. Proofs of Theorems 2.3, 2.4, 2.9 and 2.10
- 7. Riemann-Hilbert analysis in the three-cut case
- 8. Riemann-Hilbert analysis in the one-cut case
- 9. Construction of the global parametrix
- 10. Proofs of Theorems 2.14 and 2.15
- A. Analysis of the width parameters
Abstract
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 we consider the normal matrix model with cubic plus linear potential. In order 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 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.
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 & López, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong 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.
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