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.
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/
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