The mother body phase transition in the normal matrix model /

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 c...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autores principales: Bleher, Pavel, 1947- (Autor), Silva, Guilherme L. F. (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Providence, RI : American Mathematical Society, [2020]
Colección:Memoirs of the American Mathematical Society ; no. 1289.
Temas:
Matrices > Norms > Models.
Procesos estocásticos
Funciones de variables complejas
Teoría de probabilidades
Probability theory and stochastic processes {For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX} > Probability theory on algebraic and topological structures > Random m.
Functions of a complex variable {For analysis on manifolds, see 58-XX} > Geometric function theory > None of the above, but in this section.
Functions of a complex variable {For analysis on manifolds, see 58-XX} > Miscellaneous topics of analysis in the complex domain.
Functions of a complex variable {For analysis on manifolds, see 58-XX} > Riemann surfaces > Differentials on Riemann surfaces.
Potential theory {For probabilistic potential theory, see 60J45} > Two-dimensional theory > Potentials and capacity, harmonic measure, extremal length [See also 30C85].
Integral transforms, operational calculus {For fractional derivatives and integrals, see 26A33. For Fourier transforms, see 42A38, 42B10. For integral transforms in distribution spaces, see 46F12. For.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Title page
  • Chapter 1. Introduction
  • Chapter 2. Statement of main results
  • 2.1. Phase diagram of the cubic model
  • 2.2. The limiting boundary of eigenvalues as a polynomial curve
  • 2.3. Spectral curve
  • 2.4. Phase transition of the spectral curve
  • 2.5. The parameters (, ₀) as a change of variables
  • 2.6. The mother body problem
  • 2.7. Associated multiple orthogonality
  • 2.8. Behavior at the boundary of the phase diagram
  • 2.9. The S-property
  • 2.10. Statement of Results
  • ₁<0
  • 2.11. Phase transition along the mother body critical curve
  • 2.12. Setup for the remainder of the paper
  • Chapter 3. Limiting boundary of eigenvalues. Proofs of Propositions 2.1 and 2.7 and Theorems 2.2, 2.5 and 2.8
  • 3.1. Proof of Proposition 2.1
  • 3.2. Proofs of Theorems 2.2, 2.5 and 2.8 and Proposition 2.7
  • Chapter 4. Geometry of the spectral curve. Proof of Theorem 2.6
  • 4.1. The spectral curve for ₁=0
  • 4.2. The spectral curve for ₁>0. Proof of Theorem 4.1
  • 4.3. Sheet structure for ℛ
  • Chapter 5. Meromorphic quadratic differential on ℛ
  • 5.1. Technical computations for the three-cut case
  • 5.2. Technical computations for the one-cut case
  • 5.3. Quadratic differential on the spectral curve: general principles
  • 5.4. Critical graph in the three-cut case
  • 5.5. Critical graph in the one-cut case
  • Chapter 6. Proofs of Theorems 2.3, 2.4, 2.9 and 2.10
  • Chapter 7. Riemann-Hilbert analysis in the three-cut case
  • 7.1. Multiple orthogonality in terms of Airy functions
  • 7.2. The Riemann-Hilbert problem
  • 7.3. First transformation: \mapsto
  • 7.4. Second transformation: \mapsto
  • 7.5. Opening of lenses: \mapsto
  • 7.6. The global parametrix
  • 7.7. The local parametrices
  • 7.8. Final transformation: \mapsto
  • Chapter 8. Riemann-Hilbert analysis in the one-cut case
  • Chapter 9. Construction of the global parametrix
  • 9.1. The inverse of the rational parametrization
  • 9.2. Construction of the global parametrix in the three-cut case
  • 9.3. Construction of the global parametrix in the one-cut case
  • 9.4. Explicit construction of the first row
  • Chapter 10. Proofs of Theorems 2.14 and 2.15
  • Appendix A. Analysis of the width parameters
  • A.1. Width parameters in the three-cut case
  • A.2. Width parameters in the one-cut case
  • Acknowledgements

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