Quadratic vector equations on complex upper half-plane
The authors consider the nonlinear equation -\frac 1m=z+Sm with a parameter z in the complex upper half plane \mathbb H , where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \mathbb H is unique and its z-dependence is conveniently des...
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| Format: | UnknownFormat |
| Sprache: | eng |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
2019
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| Schriftenreihe: | Memoirs of the American Mathematical Society
volume 261, number 1261 (September 2019) |
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| Online Zugang: | Inhaltsverzeichnis |
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Tag hinzufügen | Zusammenfassung: | The authors consider the nonlinear equation -\frac 1m=z+Sm with a parameter z in the complex upper half plane \mathbb H , where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \mathbb H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on \mathbb R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z\in \mathbb H, including the vicinity of the singularities. |
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| Beschreibung: | "September 2019, volume 261, number 1261 (fifth of 7 numbers)" Literaturverzeichnis: Seite 131-133 |
| Beschreibung: | v, 133 Seiten Diagramme, Illustrationen |
| ISBN: | 9781470436834 978-1-4704-3683-4 |