Singularities of solutions to quadratic vector equations on the complex upper half plane

article Singularities of solutions to quadratic vector equations on the complex upper half plane published yes Oskari H Ajanki author 36F2FB7E-F248-11E8-B48F-1D18A9856A87 Torben H Krüger author 3020C786-F248-11E8-B48F-1D18A9856A870000-0002-4821-3297 László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 LaEr department Random matrices, universality and disordered quantum systems project 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. Wiley2017 eng Communications on Pure and Applied Mathematics 0010-3640 1512.03703 00040575210000210.1002/cpa.21639 7091672 - 1705 Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” Communications on Pure and Applied Mathematics, vol. 70, no. 9, Wiley, 2017, pp. 1672–705, doi:<a href="https://doi.org/10.1002/cpa.21639">10.1002/cpa.21639</a>. O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705. Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” Communications on Pure and Applied Mathematics. Wiley, 2017. <a href="https://doi.org/10.1002/cpa.21639">https://doi.org/10.1002/cpa.21639</a>. Ajanki, O. H., Krüger, T. H., & Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. Wiley. <a href="https://doi.org/10.1002/cpa.21639">https://doi.org/10.1002/cpa.21639</a> 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. O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,” Communications on Pure and Applied Mathematics, vol. 70, no. 9. Wiley, pp. 1672–1705, 2017. Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. 2017;70(9):1672-1705. doi:<a href="https://doi.org/10.1002/cpa.21639">10.1002/cpa.21639</a> 7212018-12-11T11:48:08Z2025-09-10T10:58:02Z