**Memoirs of the American Mathematical Society**

2020;
144 pp;
Softcover

MSC: Primary 60;
Secondary 30; 31; 44

**Print ISBN: 978-1-4704-4184-5
Product Code: MEMO/265/1289**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

**Electronic ISBN: 978-1-4704-6146-1
Product Code: MEMO/265/1289.E**

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MAA Member Price: $76.50

# The Mother Body Phase Transition in the Normal Matrix Model

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*Pavel M. Bleher; Guilherme L. F. Silva*

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

# Table of Contents

## The Mother Body Phase Transition in the Normal Matrix Model

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Statement of main results 714
- 2.1. Phase diagram of the cubic model 714
- 2.2. The limiting boundary of eigenvalues as a polynomial curve 815
- 2.3. Spectral curve 1017
- 2.4. Phase transition of the spectral curve 1118
- 2.5. The parameters (π,πβ) as a change of variables 1219
- 2.6. The mother body problem 1320
- 2.7. Associated multiple orthogonality 1623
- 2.8. Behavior at the boundary of the phase diagram 2229
- 2.9. The S-property 2330
- 2.10. Statement of Results - π‘β<0 2431
- 2.11. Phase transition along the mother body critical curve 2835
- 2.12. Setup for the remainder of the paper 3138

- Chapter 3. Limiting boundary of eigenvalues. Proofs of Propositions 2.1 and 2.7 and Theorems 2.2, 2.5 and 2.8 3340
- Chapter 4. Geometry of the spectral curve. Proof of Theorem 2.6 4350
- Chapter 5. Meromorphic quadratic differential on β 5562
- Chapter 6. Proofs of Theorems 2.3, 2.4, 2.9 and 2.10 7986
- Chapter 7. Riemann-Hilbert analysis in the three-cut case 8996
- 7.1. Multiple orthogonality in terms of Airy functions 8996
- 7.2. The Riemann-Hilbert problem π 9198
- 7.3. First transformation: π\mapstoπ 9198
- 7.4. Second transformation: π\mapstoπ 94101
- 7.5. Opening of lenses: π\mapstoπ 99106
- 7.6. The global parametrix 103110
- 7.7. The local parametrices 103110
- 7.8. Final transformation: π\mapstoπ 104111

- Chapter 8. Riemann-Hilbert analysis in the one-cut case 107114
- Chapter 9. Construction of the global parametrix 117124
- Chapter 10. Proofs of Theorems 2.14 and 2.15 125132
- Appendix A. Analysis of the width parameters 127134
- Bibliography 141148
- Back Cover Back Cover1156