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We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models. The limiting non-random value is shown to… (More)

- Florent Benaych-Georges, Raj Rao Nadakuditi
- J. Multivariate Analysis
- 2012

AMS 2000 subject classifications: 15A52 46L54 60F99 Keywords: Random matrices Haar measure Free probability Phase transition Random eigenvalues Random eigenvectors Random perturbation Sample covariance matrices a b s t r a c t In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random… (More)

Consider a deterministic self-adjoint matrix X n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenval-ues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigen-values of the deformed… (More)

We prove that a probability measure on the real line has a moment of order p (even integer), if and only if its R-transform admits a Taylor expansion with p terms. We also prove a weaker version of this result when p is odd. We then apply this to prove that a probability measure whose R-transform extends analytically to a ball with center zero is compactly… (More)

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra. Therefore we can define a " rectangular free convolution " , linearized… (More)

— In this paper, we provide an algorithmic method to compute the singular values of sum of rectangular matrices based on the free cumulants approach and illustrate its application to wireless communications. We first recall the algorithms working for sum/products of square random matrices, which have already been presented in some previous papers and we… (More)

In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D. Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish… (More)

In this paper, we generalize a permutation model for free random variables which was first proposed by Biane in [B95b]. We also construct its classical probability analogue, by replacing the group of permutations with the group of subsets of a finite set endowed with the symmetric difference operation. These constructions provide new discrete approximations… (More)

In a previous paper ([B-G1]), we defined the rectangular free convolution ⊞ λ. Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ⊞ λ-infinitely divisible… (More)

In this paper, we prove a result linking the square and the rectangular R-transforms, which consequence is a surprising relation between the square and rectangular free convolutions, involving the Marchenko-Pastur law. Consequences on infinite divisibility and on the arithmetics of Voiculescu's free additive and multiplicative convolutions are given.