# Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half-Plane

Ajanki O, Erdos L, Krueger T (2017)

**Publication Type:** Journal article

**Publication year:** 2017

### Journal

**Book Volume:** 70

**Pages Range:** 1672-1705

**Journal Issue:** 9

**DOI:** 10.1002/cpa.21639

### 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.© 2016 Wiley Periodicals, Inc.

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### How to cite

**APA:**

Ajanki, O., Erdos, L., & Krueger, T. (2017). Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half-Plane. *Communications on Pure and Applied Mathematics*, *70*(9), 1672-1705. https://doi.org/10.1002/cpa.21639

**MLA:**

Ajanki, Oskari, Laszlo Erdos, and Torben Krueger. "Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half-Plane." *Communications on Pure and Applied Mathematics* 70.9 (2017): 1672-1705.

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