Quadratic vector equations on complex upper half-plane

We consider the nonlinear equation \(-\frac{1}{m}=z+Sm\) 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 [AEK17a] we 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 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 [AEK16b], we present a complete stability analysis of the equation for any \(z\in \mathbb{H}\), including the vicinity of the singularities.