Lectures on the Combinatorics of Free Probability: The free commutator

The concept of freeness was introduced by Voiculescu in the context of operator algebras. Later it was observed that it is also relevant for large random matrices. We will show how the combination of various free probability results with a linearization trick allows to address successfully the problem of determining the asymptotic eigenvalue distribution of general selfadjoint polynomials in independent random matrices.

Free Probability and Random Matrices

This review covers recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT). We introduce several RMT methods and analytical techniques, such as the Replica formalism and Free Probability, with an emphasis on the Marchenko-Pastur equation that provides information on the resolvent of multiplicatively corrupted noisy matrices. Special care is devoted to the statistics of the eigenvectors of the empirical correlation matrix, which turn out to be crucial for many applications. We show in particular how these results can be used to build consistent "Rotationally Invariant" estimators (RIE) for large correlation matrices when there is no prior on the structure of the underlying process. The last part of this review is dedicated to some real-world applications within financial markets as a case in point. We establish empirically the efficacy of the RIE framework, which is found to be superior in this case to all previously proposed methods. The case of additively (rather than multiplicatively) corrupted noisy matrices is also dealt with in a special Appendix. Several open problems and interesting technical developments are discussed throughout the paper.

Cleaning large correlation matrices: tools from random matrix theory

The purpose of this review is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review and of more detailed examples—coming mainly from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.

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Random matrix techniques in quantum information theory

(2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p [282]. B−L [427]. α [216, 483]. α− z [322]. N = 2 [507]. D [222]. ẍ+ f(x)ẋ + g(x) = 0 [112, 111, 8, 5, 6]. Eτ,ηgl3 [148]. g [300]. κ [244]. L [205, 117]. L [164]. L∞ [368]. M [539]. P [27]. R [147]. Z2 [565]. Z n 2 [131]. Z2 × Z2 [25]. D(X) [166]. S(N) [110]. ∫l2 [154]. SU(2) [210]. N [196, 242]. O [386]. osp(1|2) [565]. p [113, 468]. p(x) [17]. q [437, 220, 92, 183]. R, d = 1, 2, 3 [279]. SDiff(S) [32]. σ [526]. SLq(2) [185]. SU(N) [490]. τ [440]. U(1) N [507]. Uq(sl 2) [185]. φ 2k [283]. φ [553]. φ4 [365]. ∨ [466]. VOA[M4] [33]. Z [550].

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A Complete Bibliography of Publications in the Journal of Mathematical Physics: 2005{2009

Relations between cumulants in noncommutative probability

We study random matrices acting on tensor product spaces which have been transformed by a linear block operation. Using operator-valued free probability theory, under some mild assumptions on the linear map acting on the blocks, we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution. Moreover, using recent results on operator-valued subordination, we present an algorithm that computes, numerically but in full generality, the limiting eigenvalue distribution of the modified matrices. Our analytical results cover many cases of interest in quantum information theory: we unify some known results and we obtain new distributions and various generalizations.

On the asymptotic distribution of block-modified random matrices

Nous derivons des fonctions de subordination pour la deconvolution libre additive et multiplicative sous des conditions de moment faibles. Nos resultats incluent un algorithme pour calculer ces fonctions de subordination, et donc les transformees de Cauchy associees, pour les nombres complexes ayant une partie imaginaire superieure a un parametre dependant de la mesure a deconvoler. L’existence des fonctions de subordination sur de tels domaines reduit le probleme de la deconvolution libre au probleme de la deconvolution additive classique par une distribution de Cauchy. Ainsi, nos resultats, combines a des methodes connues de deconvolution classique par une distribution de Cauchy, nous permettent de resoudre le probleme de deconvolution libre. Nous presentons egalement des extensions de ces resultats au cas des deconvolutions a valeur operateur.

Subordination methods for free deconvolution

In this paper, we give subordination functions for free additive and free multiplicative deconvolutions in some domain of the complex half-plane, under the condition that the distributions admit moments, respectively, of second order for the additive deconvolution and of fourth order for the multiplicative one. Our method of proof allows us to give an algorithm to calculate these subordinations functions, and thus the associated Cauchy transforms, for complex numbers with imaginary part bigger than a parameter depending on the measure to deconvolve. This reduces the problem of free deconvolutions to the one of the classical deconvolution with a Cauchy distribution and thus combined with known methods for the latter problem we are able to solve the deconvolution problem for the scalar case. We present also an extension of these results to the case of operator valued distributions.

In this talk, we survey on the different tools from operator-valued free probability which are used to understand the asymptotic spectral distribution of large classes of Hermitian random matrix models. As a source of examples, we consider recent matricial models for wireless communications. The (approximated) distributions of such models were usually obtained by the methods of deterministic equivalents. The applicability of free probability to understand distributions of large random matrices has extended quite a lot in the recent years. It all started with the work [Voi91] of Voiculescu, where, in particular, a multivariate Wigner’s semicircle law is established. The key concept which describes the joint non-commutative distribution (w.r.t. the normalized expected trace τn := 1 nE ◦ Tr) of several independent ensembles is the notion of freeness, introduced by Voiculescu 6 years before, in the context of operator algebras. Voiculescu developed the basic tools to compute free convolutions, which rely on transformations related to the Cauchy-Stieltjes transform Gx(z) = τ((x − z)−1), in the same way as classical convolutions are comptuted in terms of the Fourier transforms. Some years later, the broader notion of freeness with amalgamation allowed to describe the joint distribution of more general matrix models, such as band matrices and block matrices. The conceptual difference between the usual, scalar free probability and its operator-valued generalization is just replacing the functional τ : A → C by a conditional expectation E : A → B onto a larger algebra C ⊂ B ⊂ A. Definition 0.1. (1). Let A be a unital ∗-algebra and let C ⊆ B ⊆ A be a ∗-subalgebra. A Bprobability space is a pair (A,E), where E : A → B is a conditional expectation, that is, a linear map satisfying:

Carlos Vargas

Papers

Relations between cumulants in noncommutative probability

On the asymptotic distribution of block-modified random matrices

Subordination methods for free deconvolution

Subordination methods for free deconvolution

A General Solution to (Free) Deterministic Equivalents