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Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices

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1.	Title	Title of document	Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices
2.	Creator	Author's name, affiliation, country	Mireille Capitaine; CNRS and Université de Toulouse; France
2.	Creator	Author's name, affiliation, country	Catherine Donati-Martin; Université de Versailles-St Quentin; France
2.	Creator	Author's name, affiliation, country	Delphine Féral; Université Bordeaux 1; France
2.	Creator	Author's name, affiliation, country	Maxime Février; Université de Toulouse; France
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Random matrices; Free probability; Deformed Wigner matrices; Asymptotic spectrum; Extreme eigenvalues; Stieltjes transform; Subordination property.
3.	Subject	Subject classification	15B52; 60B20,; 46L54; 15A18.
4.	Description	Abstract	We investigate the asymptotic spectrum of spiked perturbations of Wigner matrices. The entries of the Wigner matrix have a distribution which is symmetric and satisfies a Poincaré inequality. The spectral measure of the deterministic Hermitian perturbation matrix converges to some probability measure with compact support. We also assume that this perturbation matrix has a fixed number of fixed eigenvalues (spikes) outside the support of its limiting spectral measure whereas the distance between the other eigenvalues and this support uniformly goes to zero as the dimension goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of the deformed model, which will converge to some limiting points outside the support of the limiting spectral measure. This phenomenon can be fully described in terms of free probability involving the subordination function related to the free additive convolution by a semicircular distribution. Note that only finite rank perturbations had been considered up to now (even in the deformed GUE case).
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Agence Nationale de la Recherche grant ANR- 08-BLAN-0311-03
7.	Date	(YYYY-MM-DD)	2011-09-19
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/934
10.	Identifier	Digital Object Identifier	10.1214/EJP.v16-934
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 16
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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