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References
- Adamczak, Radosław; Guédon, Olivier; Litvak, Alexander; Pajor, Alain; Tomczak-Jaegermann, Nicole. Condition number of a square matrix with i.i.d. columns drawn from a convex body. To appear in Proc. Amer. Math. Soc.
- Adamczak, Radosław; Guédon, Olivier; Litvak, Alexander; Pajor, Alain; Tomczak-Jaegermann, Nicole. Smallest singular value of random matrices with independent columns. C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 853--856. MR2441920
- Adamczak, Radosław; Litvak, Alexander E.; Pajor, Alain; Tomczak-Jaegermann, Nicole. Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Amer. Math. Soc. 23 (2010), no. 2, 535--561. MR2601042
- Anderson, Greg W.; Zeitouni, Ofer. A law of large numbers for finite-range dependent random matrices. Comm. Pure Appl. Math. 61 (2008), no. 8, 1118--1154. MR2417889
- Ané, Cécile, Blachère; Sébastien; Chafaï, Djalil; Fougères; Pierre, Gentil, Ivan; Malrieu, Florent; Roberto, Cyril and Scheffer, Grégory. Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris, Paris, 2000. MR1845806
- Aubrun, Guillaume. Random points in the unit ball of $lsp nsb p$. Positivity 10 (2006), no. 4, 755--759. MR2280648
- Bai, Zhidong; Silverstein, Jack W. Spectral analysis of large dimensional random matrices. Second edition. Springer Series in Statistics. Springer, New York, 2010. xvi+551 pp. ISBN: 978-1-4419-0660-1 MR2567175
- Bai, Z. D. Circular law. Ann. Probab. 25 (1997), no. 1, 494--529. MR1428519
- Bordenave, Charles; Caputo, Pietro; Chafaï, Djalil. Circular law theorem for random Markov matrices. Available at http://arxiv.org/abs/0808.1502 . To appear in Probab. Theory Related Fields.
- Bordenave, Charles; Caputo, Pietro; Chafaï, Djalil. Spectrum of non-hermitian heavy tailed random matrices. Available at http://arxiv.org/abs/1006.1713. To appear in Comm. Math. Phys.
- Borell, Christer. Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239--252. MR0388475
- Bryc, Włodzimierz; Dembo, Amir; Jiang, Tiefeng. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006), no. 1, 1--38. MR2206341
- Dozier, R. Brent; Silverstein, Jack W. On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices. J. Multivariate Anal. 98 (2007), no. 4, 678--694. MR2322123
- El Karoui, Noureddine. Concentration of measure and spectra of random matrices: applications to correlation matrices, elliptical distributions and beyond. Ann. Appl. Probab. 19 (2009), no. 6, 2362--2405. MR2588248
- Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer; Yin Jun. The local relaxation flow approach to universality of the local statistics for random matrices. Preprint. Available at http://arxiv.org/abs/0911.3687.
- Girko, V. L. The circle law. Teor. Veroyatnost. i Mat. Statist. No. 28, (1983), 15--21. MR0727271
- Girko, V. L. The strong circular law. Twenty years later. I. Random Oper. Stochastic Equations 12 (2004), no. 1, 49--104. MR2046403
- Götze, F.; Tikhomirov, A. Limit theorems for spectra of positive random matrices under dependence. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 311 (2004), Veroyatn. i Stat. 7, 92--123, 299; translation in J. Math. Sci. (N. Y.) 133 (2006), no. 3, 1257--1276 MR2092202
- Götze, Friedrich; Tikhomirov, Alexander. The circular law for random matrices. Ann. Probab. 38 (2010), no. 4, 1444--1491. MR2663633
- Götze, F.; Tikhomirov, A. N. Limit theorems for spectra of random matrices with martingale structure. Stein's method and applications, 181--193, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, Singapore Univ. Press, Singapore, 2005. MR2205336
- Hensley, Douglas. Slicing convex bodies—bounds for slice area in terms of the body's covariance. Proc. Amer. Math. Soc. 79 (1980), no. 4, 619--625. MR0572315
- Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman. Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp. MR0322926
- Kannan, R.; Lovász, L.; Simonovits, M. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995), no. 3-4, 541--559. MR1318794
- Klartag, B. Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 (2007), no. 1, 284--310. MR2311626
- Litvak, A. E.; Pajor, A.; Rudelson, M.; Tomczak-Jaegermann, N. Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 (2005), no. 2, 491--523. MR2146352
- Marčenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices. (Russian) Mat. Sb. (N.S.) 72 (114) 1967 507--536. MR0208649 (34 #8458)
- Pajor, A.; Pastur, L. On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution. Studia Math. 195 (2009), no. 1, 11--29. MR2539559
- Pan, Guangming; Zhou, Wang. Circular law, extreme singular values and potential theory. J. Multivariate Anal. 101 (2010), no. 3, 645--656. MR2575411
- Paouris, Grigoris. Small ball probability estimates for log-concave measures. To appear in Trans. Amer. Math. Soc.
- Péché, Sandrine. Universality results for the largest eigenvalues of some sample covariance matrix ensembles. Probab. Theory Related Fields 143 (2009), no. 3-4, 481--516. MR2475670
- Rudelson, Mark; Vershynin, Roman. The Littlewood-Offord problem and invertibility of random matrices. Adv. Math. 218 (2008), no. 2, 600--633. MR2407948
- Samson, Paul-Marie. Concentration of measure inequalities for Markov chains and $Phi$-mixing processes. Ann. Probab. 28 (2000), no. 1, 416--461. MR1756011
- Soshnikov, Alexander. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999), no. 3, 697--733. MR1727234
- Tao, Terence; Vu, Van. Random covariance matrices: Universality of local statistics of eigenvalues. Preprint. Available at http://arxiv.org/abs/0912.0966.
- Tao, Terence; Vu, Van. The condition number of a randomly perturbed matrix. STOC'07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 248--255, ACM, New York, 2007. MR2402448
- Tao, Terence; Vu, Van. Random matrices: universality of ESDs and the circular law. With an appendix by Manjunath Krishnapur. Ann. Probab. 38 (2010), no. 5, 2023--2065. MR2722794
- Yin, Y. Q.; Krishnaiah, P. R. Limit theorem for the eigenvalues of the sample covariance matrix when the underlying distribution is isotropic. Teor. Veroyatnost. i Primenen. 30 (1985), no. 4, 810--816. MR0816299

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