Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On finite rank deformations of Wigner matrices

Alessandro Pizzo, David Renfrew, and Alexander Soshnikov

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Abstract

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(On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries, Unpublished manuscript; Fluctuations of matrix entries of regular functions of Wigner matrices, Unpublished manuscript) and ideas from (Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices, Unpublished manuscript), we extend the results by Capitaine, Donati-Martin, and Féral (Ann.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).

Résumé

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(On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries, Unpublished manuscript; Fluctuations of matrix entries of regular functions of Wigner matrices, Unpublished manuscript) et des idées de (Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices, Unpublished manuscript), nous étendons les résultats de Capitaine, Donati-Martin et Féral (Ann.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).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 64-94.

Dates
First available in Project Euclid: 29 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1359470126

Digital Object Identifier
doi:10.1214/11-AIHP459

Mathematical Reviews number (MathSciNet)
MR3060148

Zentralblatt MATH identifier
1278.60014

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Random matrices Ouliers in the spectrum Finite rank deformations

Citation

Pizzo, Alessandro; Renfrew, David; Soshnikov, Alexander. On finite rank deformations of Wigner matrices. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 64--94. doi:10.1214/11-AIHP459. https://projecteuclid.org/euclid.aihp/1359470126


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