Quadratic Vector Equations on Complex Upper Half-Plane

Released on by Oskari Ajanki
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  • Quadratic Vector Equations on Complex Upper Half-Plane Book Detail

  • Author : Oskari Ajanki
  • Release Date : 2019-12-02
  • Publisher : American Mathematical Soc.
  • Genre : Education
  • Pages : 133
  • ISBN 13 : 1470436833
  • File Size : 58,58 MB
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Quadratic Vector Equations on Complex Upper Half-Plane by Oskari Ajanki PDF Summary

Book Description: The authors consider the nonlinear equation −1m=z+Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on 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∈H, including the vicinity of the singularities.

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