[{"publication":"Communications on Pure and Applied Mathematics","date_published":"2017-09-01T00:00:00Z","type":"journal_article","_id":"721","oa_version":"Submitted Version","citation":{"short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","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.","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.","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>","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>"},"page":"1672 - 1705","intvolume":"        70","day":"01","abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","date_updated":"2025-09-10T10:58:02Z","volume":70,"department":[{"_id":"LaEr"}],"publication_status":"published","publisher":"Wiley","isi":1,"scopus_import":"1","arxiv":1,"year":"2017","language":[{"iso":"eng"}],"publist_id":"6959","month":"09","corr_author":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1512.03703"}],"doi":"10.1002/cpa.21639","date_created":"2018-12-11T11:48:08Z","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"issue":"9","oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_identifier":{"issn":["0010-3640"]},"external_id":{"arxiv":["1512.03703"],"isi":["000405752100002"]},"author":[{"first_name":"Oskari H","last_name":"Ajanki","full_name":"Ajanki, Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-4821-3297","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H","last_name":"Krüger","first_name":"Torben H"},{"first_name":"László","full_name":"Erdös, László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603"}],"title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","article_processing_charge":"No","ec_funded":1,"status":"public"}]