Quadratic Vector Equations On Complex Upper Half-Plane
Oskari Ajanki, László Erdős, Torben Krüger
Source abstract
We consider the nonlinear equation − 1 m = z + S m -\frac {1}{m}=z+Sm with a parameter z z in the complex upper half plane H \mathbb {H} , where S S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H \mathbb {H} is unique and its z z -dependence is conveniently described as the Stieltjes transforms of a family of measures v v on R \mathbb {R} . In Ajanki, Erdős, and Krüger (2016b), we qualitatively identified the possible singular behaviors of v v : under suitable conditions on S S we showed that in the density of v v only algebraic singularities of degree two or three may occur. In this paper we give a comprehensive analysis of these singularities with uniform quantitative controls. We 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 companion paper, Ajanki, Erdős, and Krüger (2016c), we present a complete stability analysis of the equation for any z ∈ H z\in \mathbb {H} , including the vicinity of the singularities.
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