Geodesic
Asymptotic distribution of singular values for matrices in a spherical ensemble
Matematičeskie trudy, Tome 16 (2013) no. 2, pp. 169-200.

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We consider the asymptotic behavior of the singular values of a so-called spherical ensemble of random matrices of large dimension. These are matrices of the form $\mathbf X\mathbf Y^{-1}$, where $\mathbf X$ and $\mathbf Y$ are independent matrices of dimension $n\times n$ whose symmetric entries have correlation coefficient $\rho$. We show that the limit distribution of the singular values is independent of the correlation coefficient and has the density $$ p(x)=\frac1{\pi\sqrt x(1+x)}\mathbb I\{x>0\}, $$ where $\mathbb I\{A\}$ stands for the indicator of an event $A$. 
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A. N. Tikhomirov. Asymptotic distribution of singular values for matrices in a spherical ensemble. Matematičeskie trudy, Tome 16 (2013) no. 2, pp. 169-200. http://geodesic.mathdoc.fr/item/MT_2013_16_2_a9/

[1] Marchenko V. A., Pastur L. A., “Raspredelenie sobstvennykh chisel v nekotorykh ansamblyakh sluchainykh matrits”, Matem. sb., 72(114):4 (1967), 507–536 | MR | Zbl

[2] Alastuey A., Jancovici B., “On the classical two-dimensional one-component Coulomb plasma”, J. Physique, 42:1 (1981), 1–12 | DOI | MR

[3] Bai Z. D., “Circular law”, Ann. Probab., 25:1 (1997), 494–529 | DOI | MR | Zbl

[4] Bai Z. D., Silverstein J., Spectral Analysis of Large Dimensional Random Matrices, 2nd ed., Springer, New York, 2009 | MR

[5] Bordenave Ch., “On the spectrum of sum and product of non-Hermitian random matrices”, Electron. Comm. Probab., 16 (2011), 104–113 | MR | Zbl

[6] Bordenave Ch., Chafaï D., Around the Circular Law, Preprint, 2012, arXiv: 1109.3343 | MR

[7] Edelman A., Kostlan E., Shub M., “How many eigenvalues of a random matrix are real?”, J. Amer. Math. Soc., 7:1 (1994), 247–267 | DOI | MR | Zbl

[8] Fischmann J., Forrester P., “One-component plasma on a spherical annulus and a random matrix ensemble”, J. Stat. Mech.: Theory and Experiment, 2011 (2011), P10003 | DOI

[9] Forrester P., Mays A., “Pfaffian point process for the Gaussian real generalized eigenvalue problem”, Probab. Theory Related Fields, 154:1–2 (2012), 1–47 | DOI | MR | Zbl

[10] Fyodorov Ya. V., Khoruzhenko B. A., Sommers H. Y., “Almost-Hermitian Random Matrices: Eigenvalue Density in the Complex Plane”, Phys. Lett. A, 226:1–2 (1997), 46–52 | DOI | MR | Zbl

[11] Ginibre J., “Statistical ensembles of complex, quaternion, and real matrices”, J. Mathematical Phys., 6 (1965), 440–449 | DOI | MR | Zbl

[12] Girko V. L., “The circular law”, Theory Probab. Appl., 29:4 (1985), 694–706 | DOI

[13] Girko V. L., “On the elliptic law”, Theory Probab. Appl., 30:4 (1986), 677–690 | DOI | MR | Zbl

[14] Götze F., Tikhomirov A. N., On the Circular Law, Preprint, 2007, arXiv: math/0702386v1[math.PR]

[15] Götze F., Tikhomirov A. N., “The circular law for random matrices”, Ann. Probab., 38:4 (2010), 1444–1491 | DOI | MR | Zbl

[16] Götze F., Tikhomirov A. N., On the Asymptotic Spectrum of Products of Independent Random Matrices, Preprint, arXiv: 1012.2710

[17] Hiai F., Petz D., “Asymptotic freeness almost everywhere for random matrices”, Acta. Sci. Math. (Szeged), 66:3–4 (2000), 809–834 | MR | Zbl

[18] Horn R. A., Johnson C. R., Matrix Analysis, Corrected reprint of the 1985 original, Cambridge Univ. Press, Cambridge, 1990 | MR | Zbl

[19] Krishnapur M., “From random matrices to random analytic functions”, Ann. Probab., 37:1 (2009), 314–346 | DOI | MR | Zbl

[20] Maurey B., “Some deviation inequalities”, Geom. Funct. Anal., 1:2 (1991), 188–197 | DOI | MR | Zbl

[21] Mays A., A Real Quaternion Spherical Ensemble of Random Matrices, Preprint, 2012, arXiv: 1209.0888[math-ph]

[22] Naumov A., Elliptic Law for Real Random Matrices, Preprint, 2012, arXiv: 1201.1639

[23] Patricia L. R., Biman B., Charles C., Stuart A. R., “Freezing of the classical two-dimensional, one-component plasma”, J. Chem. Phys., 81:3 (1984), 1406–1415 | DOI

[24] Tao T., Vu V., “Random matrices: Universality of ESDs and the circular law (with an appendix by M. Krishnapur)”, Ann. Probab., 38:5 (2010), 2023–2065 | DOI | MR | Zbl

[25] Voiculescu D., “Limit laws for random matrices and free products”, Invent. Math., 104:1 (1991), 201–220 | DOI | MR | Zbl

[26] Voiculescu D., “Lectures on free probability theory”, Lectures on Probability Theory and Statistics (Saint–Flour, 1998), Lectures Notes in Math., 1738, Springer, Berlin, 2000, 279–349 | MR | Zbl

[27] Wigner E., “Characteristic vectors of bordered matrices with infinite dimensions”, Ann. of Math. (2), 62 (1955), 548–564 | DOI | MR | Zbl