Singularities of solutions to quadratic vector equations on complex upper half-plane

TL;DR

This paper analyzes the singularities of solutions to a quadratic vector equation related to random matrix eigenvalue distributions, showing they have only specific algebraic singularities under certain conditions.

Contribution

It characterizes the nature of singularities in solutions to a quadratic vector equation linked to random matrix spectra, revealing only square root and cubic root singularities.

Findings

Eigenvalue densities have only square root singularities or cubic root cusps.

Solutions are represented as Stieltjes transforms of probability measures.

Singularities are finitely many and algebraic of degree at most three.

Abstract

Let $ S $ be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation $ -\frac{1}{m}=z+Sm $ with a parameter $ z $ in the complex upper half-plane $ \mathbb{H} $ has a unique solution $ m $ with values in $ \mathbb{H} $. We show that the $ z $-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures $ v $ on $ \mathbb{R} $. Under suitable conditions on $ S $, we show that $ v $ has a real analytic density apart from finitely many algebraic singularities of degree at most three. Our motivation comes from large random matrices. The solution $ m $ determines the density of eigenvalues of two prominent matrix ensembles; (i) matrices with centered independent entries whose variances are given by $ S $ and (ii) matrices with correlated entries with a translation invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or a cubic root cusps; no other singularities occur.