Florent Benaych-Georges

The singular values and vectors of low rank perturbations of large rectangular random matrices

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 eigenvalues of Hermitian matrices, the non-random limiting value is shown to depend explicitly on the limiting singular value distribution of the unperturbed matrix via an integral transform that linearizes rectangular additive convolution in free probability theory. The asymptotic position of the extreme singular values of the perturbed matrix differs from that of the original matrix if and only if the singular values of the perturbing matrix are above a certain critical threshold which depends on this same aforementioned integral transform. We examine the consequence of this singular value phase transition on the associated left and right singular eigenvectors and discuss the fluctuations of the singular values around these non-random limits.

Classical and free infinitely divisible distributions and random matrices

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 P μ d on the space of d × d Hermitian matrices such that P μ d * P ν d = P μ*ν d . The spectral distribution of a random matrix with distribution P μ d converges in probability to Ψ(μ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko-Pastur distribution. In an analogous way, for every d ≥ 1, we associate to each *-infinitely divisible distribution μ, a distribution L μ d on the space of complex (non-Hermitian) d x d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |M d |, when M d is L μ d -distributed, converges in probability to Ψ(μ).

Central limit theorems for linear statistics of heavy tailed random matrices

We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of α-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.

Localization and delocalization for heavy tailed band matrices

We consider some random band matrices with band-width

On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions

N^\mu

A continuous semigroup of notions of independence between the classical and the free one

whose entries are independent random variables with distribution tail in

Kernel spectral clustering of large dimensional data

x^{-\alpha}

The breakdown point of signal subspace estimation

. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when

Local single ring theorem

\alpha 2(1+\mu^{-1})

RANDOM RIGHT EIGENVALUES OF GAUSSIAN QUATERNIONIC MATRICES

, the largest eigenvalues have order