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@article{faucris.264336455,
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.},
author = {Ajanki, Oskari and Erdos, Laszlo and Krueger, Torben},
doi = {10.1002/cpa.21639},
faupublication = {no},
journal = {Communications on Pure and Applied Mathematics},
note = {CRIS-Team Scopus Importer:2021-09-24},
pages = {1672-1705},
peerreviewed = {Yes},
title = {{Singularities} of {Solutions} to {Quadratic} {Vector} {Equations} on the {Complex} {Upper} {Half}-{Plane}},
volume = {70},
year = {2017}
}