@article{721,
  abstract     = {Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in 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 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 3. 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 cubic root cusps; no other singularities occur.},
  author       = {Ajanki, Oskari H and Krüger, Torben H and Erdös, László},
  issn         = {0010-3640},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {9},
  pages        = {1672 -- 1705},
  publisher    = {Wiley},
  title        = {{Singularities of solutions to quadratic vector equations on the complex upper half plane}},
  doi          = {10.1002/cpa.21639},
  volume       = {70},
  year         = {2017},
}