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...

Rubinstein has produced a substantial amount of data about the even parity quadratic twists of various elliptic curves, and compared the results to predictions from random matrix theory. We use the method of Heegner points to obtain a comparable (yet smaller) amount of data for the case of odd parity. We again see that at least one of the principal predictions of random matrix theory is well-evidenced by the data.

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...

For p ≤ n, let b1(n),...,bp(n) be independent random vectors in ${ℝ}^n$ with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If ${\widehat{b}}_1^{\left(n\right)},...,{\widehat{b}}_p^{\left(n\right)}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios...

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...

A connection between representation of compact groups and some invariant ensembles of hermitian matrices is described. We focus on two types of invariant ensembles which extend the gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction multiplicities. We show that a large class of them are determinantal....