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Author Bleher, Pavel, 1947- author.

Title The mother body phase transition in the normal matrix model / Pavel M. Bleher, Guilherme L.F. Silva
Published Providence, RI : American Mathematical Society, [2020]
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Description 1 online resource (v, 144 pages)
Series Memoirs of the American Mathematical Society ; number 1289
Memoirs of the American Mathematical Society ; no. 1289
Contents 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
Summary 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 bo
Notes "May 2020, volume 265, number 1289 (sixth of 7 numbers)."
Bibliography Includes bibliographical references
Subject Matrices -- Norms -- Models
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.
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.
Potential theory {For probabilistic potential theory, see 60J45} -- Two-dimensional theory -- Potentials and capacity, harmonic measure, extremal length [See also 30C85].
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.
Form Electronic book
Author Silva, Guilherme L. F., author.
ISBN 1470461463