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

Consider a $N\times n$ non-centered matrix ${𝛴}_n$ with a separable variance profile: $${𝛴}_n=\frac{D_n^{1/2}{X}_n{\tilde{D}}_n^{1/2}}{\sqrt{n}}+A_n.$$ Matrices $D_n$ and ${\tilde{D}}_n$ are non-negative deterministic diagonal, while matrix $A_n$ is deterministic, and $X_n$ is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by $Q_n\left(z\right)$ the resolvent associated to ${𝛴}_n{𝛴}_n^{*}$, i.e. $$Q_n\left(z\right)={\left({𝛴}_n{𝛴}_n^{*}-z{I}_N\right)}^{-1}.$$ Given two sequences of deterministic vectors $\left(u_n\right)$ and $\left(v_n\right)$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form: $$u_n^{*}{Q}_n\left(z\right)v_n\forall z\in ℂ-{ℝ}^{+},$$ as the dimensions...

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(On fluctuations of matrix entries of regular functions of Wigner matrices with non-identically distributed entries, Unpublished...

We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix $H{ₙ}^{\left(0\right)}$ and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of $H{ₙ}^{\left(0\right)}$ and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...

We present an approach that allows one to bound the largest and smallest singular values of an $N\times n$ random matrix with iid rows, distributed according to a measure on ${ℝ}^n$ that is supported in a relatively small ball and linear functionals are uniformly bounded in $L_p$ for some $p>8$, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of $1\pm c\sqrt{n/N}$ not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product measure...