no. 1
On finite rank deformations of Wigner matrices
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 64-94.

Nous étudions la distribution des valeurs propres qui sortent de l'amas du spectre de matrices de Wigner deformées par une matrice de rang fini sous l'hypothèse que les valeurs absolues des éléments non diagonaux aient un moment d'ordre cinq uniformément borné et que valeurs absolues des éléments diagonaux aient un moment d'ordre trois uniformément borné. En utilisant des travaux récents (On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries, Unpublished manuscript; Fluctuations of matrix entries of regular functions of Wigner matrices, Unpublished manuscript) et des idées de (Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices, Unpublished manuscript), nous étendons les résultats de Capitaine, Donati-Martin et Féral (Ann. Probab. 37 (2009) 1-47; Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 107-133).

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices under the assumption that the absolute values of the off-diagonal matrix entries have uniformly bounded fifth moment and the absolute values of the diagonal entries have uniformly bounded third moment. Using our recent results on the fluctuation of resolvent entries (On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries, Unpublished manuscript; Fluctuations of matrix entries of regular functions of Wigner matrices, Unpublished manuscript) and ideas from (Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices, Unpublished manuscript), we extend the results by Capitaine, Donati-Martin, and Féral (Ann. Probab. 37 (2009) 1-47; Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 107-133).

DOI : 10.1214/11-AIHP459
Classification : 60B20
Mots clés : random matrices, ouliers in the spectrum, finite rank deformations 
@article{AIHPB_2013__49_1_64_0,
     author = {Pizzo, Alessandro and Renfrew, David and Soshnikov, Alexander},
     title = {On finite rank deformations of {Wigner} matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {64--94},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {1},
     year = {2013},
     doi = {10.1214/11-AIHP459},
     mrnumber = {3060148},
     zbl = {1278.60014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/11-AIHP459/}
}
TY  - JOUR
AU  - Pizzo, Alessandro
AU  - Renfrew, David
AU  - Soshnikov, Alexander
TI  - On finite rank deformations of Wigner matrices
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2013
SP  - 64
EP  - 94
VL  - 49
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/11-AIHP459/
DO  - 10.1214/11-AIHP459
LA  - en
ID  - AIHPB_2013__49_1_64_0
ER  - 
%0 Journal Article
%A Pizzo, Alessandro
%A Renfrew, David
%A Soshnikov, Alexander
%T On finite rank deformations of Wigner matrices
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 64-94
%V 49
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/11-AIHP459/
%R 10.1214/11-AIHP459
%G en
%F AIHPB_2013__49_1_64_0
Pizzo, Alessandro; Renfrew, David; Soshnikov, Alexander. On finite rank deformations of Wigner matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 64-94. doi : 10.1214/11-AIHP459. http://www.numdam.org/articles/10.1214/11-AIHP459/

[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. | MR | Zbl

[2] Z. D. Bai. Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 (1999) 611-677. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[5] J. Baik and J. W. Silverstein. Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 (2006) 1382-1408. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[15] E. B. Davies. The functional calculus. J. Lond. Math. Soc. 52 (1995) 166-176. | MR | Zbl

[16] R. Durrett. Probability. Theory and Examples, 4th edition. Cambridge Univ. Press, New York, 2010. | MR | Zbl

[17] L. Erdös, H.-T. Yau and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Unpublished manuscript. Available at arXiv:1007.4652. | MR | Zbl

[18] D. Féral and S. Péché. The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys. 272 (2007) 185-228. | MR | Zbl

[19] Z. Füredi and J. Komlós. The eigenvalues of random symmetric matrices. Combinatorica 1 (1981) 233-241. | Zbl

[20] A. Guionnet and B. Zegarlinski. Lectures on logarithmic Sobolev inequalities. In Seminaire de Probabilités XXXVI. Lecture Notes in Math. 1801. Springer, Paris, 2003. | Numdam | MR | Zbl

[21] B. Helffer and J. Sjöstrand. Equation de Schrödinger avec champ magnetique et equation de Harper. In Schrödinger Operators 118-197. H. Holden and A. Jensen (Eds). Lecture Notes in Physics 345. Springer, Berlin, 1989. | MR | Zbl

[22] K. Johansson. Universality for certain Hermitian Wigner matrices under weak moment conditions. Unpublished manuscript. Available at arXiv:0910.4467. | Numdam | MR | Zbl

[23] A. Khorunzhy, B. Khoruzhenko and L. Pastur. Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 (1996) 5033-5060. | MR | Zbl

[24] M. Maida. Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles. Electron. J. Probab. 12 (2007) 1131-1150. | MR | Zbl

[25] 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. | MR | Zbl

[26] D. Paul. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 (2007) 1617-1642. | MR | Zbl

[27] S. Péché. The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields 134 (2006) 127-173. | MR | Zbl

[28] A. Pizzo, D. Renfrew and A. Soshnikov. Fluctuations of matrix entries of regular functions of Wigner matrices. Unpublished manuscript. Available at arXiv:1103.1170v3. | MR | Zbl

[29] M. Reed and B. Simon. Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York, 1978. | MR | Zbl

[30] 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. | MR | Zbl

[31] M. Shcherbina. Letter from March 1, 2011.

[32] T. Tao. Outliers in the spectrum of iid matrices with bounded rank perturbations. Unpublished manuscript. Available at arXiv:1012.4818v2. | MR | Zbl

Cité par Sources :