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

Ajanki OH, Krüger TH, Erdös L. 2017. Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. 70(9), 1672–1705.

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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.

Publishing Year

Date Published

2017-09-01

Journal Title

Communications on Pure and Applied Mathematics

Volume

70

Issue

9

Page

1672 - 1705

ISSN

IST-REx-ID

### Cite this

Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane.

*Communications on Pure and Applied Mathematics*. 2017;70(9):1672-1705. doi:10.1002/cpa.21639Ajanki, O. H., Krüger, T. H., & Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane.

*Communications on Pure and Applied Mathematics*. Wiley-Blackwell. https://doi.org/10.1002/cpa.21639Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.”

*Communications on Pure and Applied Mathematics*. Wiley-Blackwell, 2017. https://doi.org/10.1002/cpa.21639.O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,”

*Communications on Pure and Applied Mathematics*, vol. 70, no. 9. Wiley-Blackwell, pp. 1672–1705, 2017.Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.”

*Communications on Pure and Applied Mathematics*, vol. 70, no. 9, Wiley-Blackwell, 2017, pp. 1672–705, doi:10.1002/cpa.21639.**All files available under the following license(s):**

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Open Access