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Softcover ISBN:  9781470441845 
Product Code:  MEMO/265/1289 
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Product Code:  MEMO/265/1289.E 
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Softcover ISBN:  9781470441845 
eBook ISBN:  9781470461461 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 265; 2020; 144 ppMSC: Primary 60; Secondary 30; 31; 44
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 cutoff. 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 “onecut to threecut” phase transition.

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. RiemannHilbert analysis in the threecut case

8. RiemannHilbert analysis in the onecut case

9. Construction of the global parametrix

10. Proofs of Theorems 2.14 and 2.15

A. Analysis of the width parameters


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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 cutoff. 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 “onecut to threecut” phase transition.

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. RiemannHilbert analysis in the threecut case

8. RiemannHilbert analysis in the onecut case

9. Construction of the global parametrix

10. Proofs of Theorems 2.14 and 2.15

A. Analysis of the width parameters