A matrix interpolation between classical and free max operations: I. The univariate case

Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings.Comment: 14 page

Freeness of Linear and Quadratic Forms in von Neumann Algebras

We characterize semicircular distribution by the freeness of linear and quadratic forms in noncommutative random variables from a tracial $W^*$-probability space with relaxed moment conditions.Comment: 15 pages; AMS-LaTeX, to appear in J Funct A

Random graph states, maximal flow and Fuss-Catalan distributions

For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.Comment: 37 pages, 24 figure

Free Energies and fluctuations for the unitary Brownian motion

We show that the Laplace transforms of traces of words in independent unitary Brownian motions converge towards an analytic function on a non trivial disc. These results allow one to study the asymptotic behavior of Wilson loops under the unitary Yang--Mills measure on the plane with a potential. The limiting objects obtained are shown to be characterized by equations analogue to Schwinger--Dyson's ones, named here after Makeenko and Migdal

Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product

Given two positive integers $n$ and $k$ and a parameter $t\in (0,1)$, we choose at random a vector subspace $V_{n}\subset \mathbb{C}^{k}\otimes\mathbb{C}^{n}$ of dimension $N\sim tnk$. We show that the set of $k$-tuples of singular values of all unit vectors in $V_n$ fills asymptotically (as $n$ tends to infinity) a deterministic convex set $K_{k,t}$ that we describe using a new norm in $\R^k$. Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high eigenvalue degeneracy.Comment: v3 changes: minor typographic improvements; accepted versio

Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Arxiv preprint arXiv:1009.0145,

Abstract. Consider a deterministic self-adjoint matrix X n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We show that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas under additional hypotheses, the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We can also generalise these results to the case when X n is random and get similar behaviour when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models

Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Arxiv preprint arXiv:1009.0145,

Abstract. Consider a deterministic self-adjoint matrix X n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when X n is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models