---
_id: '721'
abstract:
- lang: eng
  text: '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.'
article_processing_charge: No
arxiv: 1
author:
- first_name: Oskari H
  full_name: Ajanki, Oskari H
  id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87
  last_name: Ajanki
- first_name: Torben H
  full_name: Krüger, Torben H
  id: 3020C786-F248-11E8-B48F-1D18A9856A87
  last_name: Krüger
  orcid: 0000-0002-4821-3297
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
citation:
  ama: Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector
    equations on the complex upper half plane. <i>Communications on Pure and Applied
    Mathematics</i>. 2017;70(9):1672-1705. doi:<a href="https://doi.org/10.1002/cpa.21639">10.1002/cpa.21639</a>
  apa: Ajanki, O. H., Krüger, T. H., &#38; Erdös, L. (2017). Singularities of solutions
    to quadratic vector equations on the complex upper half plane. <i>Communications
    on Pure and Applied Mathematics</i>. Wiley. <a href="https://doi.org/10.1002/cpa.21639">https://doi.org/10.1002/cpa.21639</a>
  chicago: Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of
    Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” <i>Communications
    on Pure and Applied Mathematics</i>. Wiley, 2017. <a href="https://doi.org/10.1002/cpa.21639">https://doi.org/10.1002/cpa.21639</a>.
  ieee: O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic
    vector equations on the complex upper half plane,” <i>Communications on Pure and
    Applied Mathematics</i>, vol. 70, no. 9. Wiley, pp. 1672–1705, 2017.
  ista: 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.
  mla: Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations
    on the Complex Upper Half Plane.” <i>Communications on Pure and Applied Mathematics</i>,
    vol. 70, no. 9, Wiley, 2017, pp. 1672–705, doi:<a href="https://doi.org/10.1002/cpa.21639">10.1002/cpa.21639</a>.
  short: O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics
    70 (2017) 1672–1705.
corr_author: '1'
date_created: 2018-12-11T11:48:08Z
date_published: 2017-09-01T00:00:00Z
date_updated: 2025-09-10T10:58:02Z
day: '01'
department:
- _id: LaEr
doi: 10.1002/cpa.21639
ec_funded: 1
external_id:
  arxiv:
  - '1512.03703'
  isi:
  - '000405752100002'
intvolume: '        70'
isi: 1
issue: '9'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1512.03703
month: '09'
oa: 1
oa_version: Submitted Version
page: 1672 - 1705
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication: Communications on Pure and Applied Mathematics
publication_identifier:
  issn:
  - 0010-3640
publication_status: published
publisher: Wiley
publist_id: '6959'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Singularities of solutions to quadratic vector equations on the complex upper
  half plane
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 70
year: '2017'
...