{"oa_version":"Submitted Version","day":"01","date_updated":"2025-06-04T10:12:34Z","quality_controlled":"1","publication_status":"published","arxiv":1,"month":"09","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":70,"publication_identifier":{"issn":["0010-3640"]},"article_processing_charge":"No","year":"2017","citation":{"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>.","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>","short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","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>.","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>","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."},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1512.03703"}],"type":"journal_article","date_published":"2017-09-01T00:00:00Z","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."}],"department":[{"_id":"LaEr"}],"language":[{"iso":"eng"}],"doi":"10.1002/cpa.21639","publist_id":"6959","page":"1672 - 1705","issue":"9","project":[{"call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"corr_author":"1","external_id":{"arxiv":["1512.03703"]},"publisher":"Wiley","ec_funded":1,"intvolume":"        70","status":"public","date_created":"2018-12-11T11:48:08Z","author":[{"last_name":"Ajanki","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","first_name":"Oskari H","full_name":"Ajanki, Oskari H"},{"first_name":"Torben H","orcid":"0000-0002-4821-3297","full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","last_name":"Krüger"},{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","first_name":"László","orcid":"0000-0001-5366-9603"}],"publication":"Communications on Pure and Applied Mathematics","title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","_id":"721","scopus_import":"1"}