On finite rank deformations of Wigner matrices

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 49, Issue 1, 2013, Pages 64-94

  Pizzo, Alessandro ^a   Renfrew, David ^a   Soshnikov, Alexander ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Finite rank deformations; Ouliers in the spectrum; Random matrices

Indexed keywords

EID: 84879113729     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AIHP459     Document Type: Article

References (33)

1. G. W. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, New York, 2010. MR2760897
2. Z. D. Bai. Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 (1999) 611-677. MR1711663
3. Z. D. Bai and J. Yao. Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 447-474. MR2451053
4. Z. D. Bai and Y. Q. Yin. Necessary and sufficient conditions for the almost sure convergence of the largest eigenvalue of Wigner matrices. Ann. Probab. 16 (1988) 1729-1741. MR0958213
5. J. Baik and J. W. Silverstein. Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 (2006) 1382-1408. MR2279680
6. J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33 (2005) 1643-1697. MR2165575
7. G. Ben Arous and A. Guionnet. Wigner matrices. In Oxford Handbook on Random Matrix Theory. G. Akemann, J. Baik and P. Di Francesco (Eds). Oxford Univ. Press, New York, 2011.
8. F. Benaych-Georges and R. Rao. The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Unpublished manuscript. Available at arXiv:0910.2120v3.
9. F. Benaych-Georges, A. Guionnet and M. Maida. Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Unpublished manuscript. Available at arXiv:1009.0145.
10. F. Benaych-Georges, A. Guionnet and M. Maida. Large deviations of the extreme eigenvalues of random deformations of matrices. Unpublished manuscript. Available at arXiv:1009.0135v2.
11. M. Capitaine, C. Donati-Martin and D. Féral. The largest eigenvalue of finite rank deformation of large Wigner matrices: Convergence and non universality of the fluctuations. Ann. Probab. 37 (2009) 1-47. MR2489158
12. M. Capitaine, C. Donati-Martin and D. Féral. Central limit theorems for eigenvalues of deformations of Wigner matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 107-133. MR2919200
13. M. Capitaine, C. Donati-Martin D. Féral and M. Février. Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Unpublished manuscript. Available at arXiv:1006.3684.
14. X. Chen, H. Qi and P. Tseng. Analysis of nonsmooth symmetric-matrix- valued functions with applications to semidefinite complementary problems. SIAM J. Optim. 13 (2003) 960-985. MR2005912
15. E. B. Davies. The functional calculus. J. Lond. Math. Soc. 52 (1995) 166-176. MR1345723
16. R. Durrett. Probability. Theory and Examples, 4th edition. Cambridge Univ. Press, New York, 2010. MR2722836
17. L. Erdös, H.-T. Yau and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Unpublished manuscript. Available at arXiv:1007.4652.
18. D. Féral and S. Péché. The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys. 272 (2007) 185-228. MR2291807
19. Z. Füredi and J. Komlós. The eigenvalues of random symmetric matrices. Combinatorica 1 (1981) 233-241. MR0637828
20. A. Guionnet and B. Zegarlinski. Lectures on logarithmic Sobolev inequalities. In Seminaire de Probabilités XXXVI. Lecture Notes in Math. 1801. Springer, Paris, 2003. MR1971582
21. B. Helffer and J. Sjöstrand. Equation de Schrödinger avec champ magnetique et equation de Harper. In Schrödinger Operators 118-197.
22. H. Holden and A. Jensen (Eds). Lecture Notes in Physics 345. Springer, Berlin, 1989. MR1037319
23. K. Johansson. Universality for certain Hermitian Wigner matrices under weak moment conditions. Unpublished manuscript. Available at arXiv:0910.4467.
24. A. Khorunzhy, B. Khoruzhenko and L. Pastur. Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 (1996) 5033-5060. MR1411619
25. M. Maida. Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles. Electron. J. Probab. 12 (2007) 1131-1150. MR2336602
26. S. O'Rourke, D. Renfrew and A. Soshnikov. On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries. Unpublished manuscript. Available at arXiv:1104.1663v3.
27. D. Paul. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 (2007) 1617-1642. MR2399865
28. S. Péché. The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields 134 (2006) 127-173. MR2221787
29. A. Pizzo, D. Renfrew and A. Soshnikov. Fluctuations of matrix entries of regular functions of Wigner matrices. Unpublished manuscript. Available at arXiv:1103.1170v3.
30. M. Reed and B. Simon. Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York, 1978. MR0493421
31. M. Shcherbina. Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices. Zh. Mat. Fiz. Anal. Geom. 7 (2011) 176-192, 197, 199. MR2829615
32. M. Shcherbina. Letter from March 1, 2011.
33. T. Tao. Outliers in the spectrum of iid matrices with bounded rank perturbations. Unpublished manuscript. Available at arXiv:1012.4818v2.