Related papers: Free probability and random matrices
We study the asymptotics of sums of matricially free random variables called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint…
We study the analogue of Kummer distribution in free probability. We prove characterization of free-Kummer and free Poisson distributions by freeness properties together with some assumptions about conditional moments. Our main tools are…
We show that any matrix-polynomial combination of free noncommutative random variables each having an algebraic law has again an algebraic law. Our result answers a question raised by a recent paper of Shlyakhtenko and Skoufranis. The…
We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of…
We give a method to obtain, from Voiculescu's inequality, norm estimates for sums of free variables with amalgamation in general fully symmetric spaces. We use these estimates to interpolate the Burkholder inequalities for non commutative…
We show that the limit laws of random matrices, whose entries are conditionally independent operator valued random variables having equal second moments proportional to the size of the matrices, are operator valued semicircular laws.…
We show that real second order freeness appears in the study of Haar unitary and unitarily invariant random matrices when transposes are also considered. In particular we obtain the unexpected result that a unitarily invariant random matrix…
We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on…
Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random…
In this paper we study multi-matrix models whose potentials are perturbations of the quadratic potential associated with independent GUE random matrices. More precisely, we compute the free energy and the expectation of the trace of…
Let A be a unital $C^*$-algebra, given together with a specified state $\phi:A \to C$. Consider two selfadjoint elements a,b of A, which are free with respect to $\phi$ (in the sense of the free probability theory of Voiculescu). Let us…
We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances…
Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical…
We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real…
Let $\mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $\mu^{\boxplus k}$ for any real $k \geq 1$. The purpose of this short note is…
I give a survey about my work on combinatorial and probabilistic aspects of free probability theory. In particular, I present the combinatorial description of freeness in terms of free cumulants and I give some ideas of the main results of…
We introduce a new technique to prove bounds for the spectral radius of a random matrix, based on using Jensen's formula to establish the zerofreeness of the associated characteristic polynomial in a region of the complex plane. Our…
We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many…
We review methods to calculate eigenvalue distributions of products of large random matrices. We discuss a generalization of the law of free multiplication to non-Hermitian matrices and give a couple of examples illustrating how to use…
We present a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation. This approach allows us to…