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

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.