Quadratic Vector Equations on Complex Upper Half-Plane - Brossura

Ajanki, Oskari Heikki; Erdos, Laszlo; Kruger, Torben

Brossura

ISBN 10: 1470436833 ISBN 13: 9781470436834

Casa editrice: Amer Mathematical Society, 2019

Riassunto

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.

Dettagli dell'edizione

• Editore: Amer Mathematical Society
• Data di pubblicazione: 2019
• Lingua: Inglese
• ISBN 10: 1470436833
• ISBN 13: 9781470436834
• Rilegatura: Copertina flessibile
• Numero di pagine: 133
• Contatto del produttore: non disponibile
• Persona responsabile: Buchpark GmbH; gpsr@buchpark.de; ; Krügerweg 1; Trebbin; Brandenburg; 14959; Germania