Related papers: On a surprising relation between rectangular and s…
In a previous paper (called "Rectangular random matrices. Related covolution"), we defined, for $\lambda \in [0,1]$, the rectangular free convolution with ratio $\lambda$. Here, we investigate the related notion of infinite divisiblity,…
We characterize asymptotic collective behaviour 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…
We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…
In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maida proved for Hermitian matrices…
n 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…
In this paper, we consider $m$ independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the $m$ rectangular matrices is an $n$ by…
The free convolution is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant…
We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that…
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…
We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices.…
Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators…
We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability…
We study the gaps between consecutive singular values of random rectangular matrices. Specifically, if $M$ is an $n \times p$ random matrix with independent and identically distributed entries and $\Sigma$ is a $n \times n$ deterministic…
We consider the empirical eigenvalue distribution of random real symmetric matrices with stochastically independent skew-diagonals and study its limit if the matrix size tends to infinity. We allow correlations between entries on the same…
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…
This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random…
Let $A$ be a permutation invariant random matrix and $B$ another random matrix. We give a quantitative bound on the difference between the diagonal of the resolvent of $A+B$ and the diagonal of the resolvent of the free sum with…
This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $\mu_i$ and $\nu_i, i=1,2$, are close to each other in terms of the L\'{e}vy metric and if the…
This thesis reviews recent progress on products of random matrices from the perspective of exactly solved Gaussian random matrix models. We derive exact formulae for the correlation functions for the eigen- and singular values at arbitrary…
Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in…