no. 1
Universality for certain hermitian Wigner matrices under weak moment conditions
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 47-79.

Nous étudions l'universalité des statistiques locales du spectre des matrices de Wigner hermitiennes divisibles par une gaussienne. Ces matrices aléatoires sont obtenues en ajoutant à une matrice de Wigner hermitienne avec des coefficients indépendants une matrice du GUE indépendante. Nous montrons que la classe d'universalité de la loi de Tracy-Widom pour les valeurs propres extrêmes est vérifiée sous la condition optimale d'une borne uniforme sur le quatrième moment des coefficients de la matrice. De plus, nous démontrons l'universalité des fluctuations dans l'intérieur du spectre dès lors que le second moment est fini.

We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy-Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.

DOI : 10.1214/11-AIHP429
Classification : 60B20, 82B44
Mots clés : Wigner matrix, gaussian divisible, optimal moment condition, universality, Tracy-Widom distribution 
@article{AIHPB_2012__48_1_47_0,
     author = {Johansson, Kurt},
     title = {Universality for certain hermitian {Wigner} matrices under weak moment conditions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {47--79},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {1},
     year = {2012},
     doi = {10.1214/11-AIHP429},
     mrnumber = {2919198},
     zbl = {1279.60014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/11-AIHP429/}
}
TY  - JOUR
AU  - Johansson, Kurt
TI  - Universality for certain hermitian Wigner matrices under weak moment conditions
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2012
SP  - 47
EP  - 79
VL  - 48
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/11-AIHP429/
DO  - 10.1214/11-AIHP429
LA  - en
ID  - AIHPB_2012__48_1_47_0
ER  - 
%0 Journal Article
%A Johansson, Kurt
%T Universality for certain hermitian Wigner matrices under weak moment conditions
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2012
%P 47-79
%V 48
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/11-AIHP429/
%R 10.1214/11-AIHP429
%G en
%F AIHPB_2012__48_1_47_0
Johansson, Kurt. Universality for certain hermitian Wigner matrices under weak moment conditions. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 47-79. doi : 10.1214/11-AIHP429. http://www.numdam.org/articles/10.1214/11-AIHP429/

[1] A. Auffinger, G. Ben Arous and S. Péché. Poisson convergence for the largest eigenvalues of heavy-taled matrices. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 589-610. | Numdam | MR | Zbl

[2] Z. D. Bai. Convergence rate of expected spectral distributions of large random matrices. Part I. Wigner matrices. Ann. Probab. 21 (1993) 625-648. | MR | Zbl

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

[4] Z. D. Bai, B. Miao and J. Tsay. Convergence rates of the spectral distribution of large Wigner matrices. Int. Math. J. 1 (2002) 65-90. | MR | Zbl

[5] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices, 2nd edition. Springer, New York, 2010. | MR

[6] G. Ben Arous and A. Guionnet. The spectrum of heavy-tailed random matrices. Comm. Math. Phys. 278 (2008) 715-751. | MR | Zbl

[7] G. Ben Arous and S. Péché. Universality of local eigenvalue statistics for some sample covariance matrices. Comm. Pure Appl. Math. 58 (2005) 1316-1357. | MR | Zbl

[8] G. Biroli, J.-P. Bouchaud and M. Potters. On the top eigenvalue of heavy-tailed random matrices. Europhys. Lett. 78 (2007) 10001. | MR | Zbl

[9] E. Brézin and S. Hikami. Spectral form factor in a random matrix theory. Phys. Rev. E 55 (1997) 4067-4083. | MR

[10] P. Cizeau and J.-P. Bouchaud. Theory of Lévy matrices. Phys. Rev. E 50 (1994) 1810-1822.

[11] L. Erdös, S. Péché, J. Ramirez, B. Schlein and H.-T. Yau. Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63 (2010) 895-925. | MR | Zbl

[12] L. Erdös, J. Ramirez, B. Schlein, T. Tao, V. Vu and H.-T. Yau. Bulk universality for Wigner Hermitian matrices with subexponential decay. Math. Res. Lett. 17 (2010) 667-674. | MR | Zbl

[13] A. Guionnet and O. Zeitouni. Concentration of the spectral measure for large matrices. Electron. Commun. Probab. 5 (2000) 119-136 (electronic). | MR | Zbl

[14] K. Johansson. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 (2001) 683-705. | MR | Zbl

[15] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003) 277-329. | MR | Zbl

[16] A. Ruzmaikina. Universality of the edge distribution of the eigenvalues of Wigner random matrices with polynomially decaying distributions of entries. Comm. Math. Phys. 261 (2006) 277-296. | MR | Zbl

[17] T. Shcherbina. On universality of bulk local regime of the deformed Gaussian Unitary Ensemble. J. Math. Phys. Anal. Geom. 5 (2009) 396-433. | MR | Zbl

[18] T. Shcherbina. On universality of local edge regime for the deformed Gaussian Uniraty Ensemble. J. Stat. Phys. 143 (2011) 455-481. | MR | Zbl

[19] B. Simon. Trace Ideals and Their Applications, 2nd edition. Math. Surveys Monogr. 120. Amer. Math. Soc., Providence, RI, 2005. | MR | Zbl

[20] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999) 697-733. | MR | Zbl

[21] A. Soshnikov. Poisson statistics for the largest eigenvaluet of Wigner matrices with heavy tails. Electron. Commun. Probab. 9 (2004) 82-91. | MR | Zbl

[22] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 (2010) 549-572. | MR | Zbl

[23] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 (2011) 127-204. | MR | Zbl

[24] T. Tao and V. Vu. Random covariance matrices: Universality of local statistics of eigenvalues. Available at arXiv:0912.0966. | MR | Zbl

Cité par Sources :