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The Mother Body Phase Transition in the Normal Matrix Model
 
Pavel M. Bleher Indiana University-Purdue University, Indianapolis, IN
Guilherme L. F. Silva Katholieke Universiteit Leuven, Leuven, Belgium
The Mother Body Phase Transition in the Normal Matrix Model
Softcover ISBN:  978-1-4704-4184-5
Product Code:  MEMO/265/1289
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6146-1
Product Code:  MEMO/265/1289.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-4184-5
eBook: ISBN:  978-1-4704-6146-1
Product Code:  MEMO/265/1289.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
The Mother Body Phase Transition in the Normal Matrix Model
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The Mother Body Phase Transition in the Normal Matrix Model
Pavel M. Bleher Indiana University-Purdue University, Indianapolis, IN
Guilherme L. F. Silva Katholieke Universiteit Leuven, Leuven, Belgium
Softcover ISBN:  978-1-4704-4184-5
Product Code:  MEMO/265/1289
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6146-1
Product Code:  MEMO/265/1289.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-4184-5
eBook ISBN:  978-1-4704-6146-1
Product Code:  MEMO/265/1289.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2652020; 144 pp
    MSC: 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 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.

  • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
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Volume: 2652020; 144 pp
MSC: 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 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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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