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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

In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix. As in the prequel, where we considered the… (More)

We construct a random matrix model for the bijection Ψ between classical and free infinitely divisible distributions: for every d ≥ 1, we associate in a quite natural way to each ∗-infinitely divisible distribution μ a distribution Pμd on the space of d×d hermitian matrices such that Pμd ∗ Pνd = P μ∗ν d . The spectral distribution of a random matrix with… (More)

- Florent Benaych-Georges, SANDRINE PÉCHÉ
- 2013

We consider some random band matrices with band-width N whose entries are independent random variables with distribution tail in x−α. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when α < 2(1 + μ−1), the largest eigenvalues have order N 1+μ α , are asymptotically distributed… (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)

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all noncommutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular… (More)

- Florent Benaych-Georges, Alice Guionnet, Mylène Mäıda
- 2011

Consider a real diagonal deterministic matrix Xn of size n with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale… (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 then… (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.

We prove that independent rectangular random matrices, when embedded in a space of larger square matrices, are asymptotically free with amalgamation over a commutative finite dimensional subalgebra D (under an hypothesis of unitary invariance). Then we consider elements of a finite von Neumann algebra containing D, which have kernel and range projection in… (More)