Free Additive Convolution Research Articles

Regularization by Free Additive Convolution, Square and Rectangular Cases

The free convolution is the binary operation on the set of prob- ability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent uni- tarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free con- volution λ allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups (μt) of probability measures such that μ0 = δ0 and such that for all t> 0 and all probability measure ν, μtν (or, in the rectan- gular context, μtλν) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, for , we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semi- groups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for μ in a λ-continuous semigroup, μλν either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, and thus the expected property does not hold. However, we give suffi- cient conditions for analyticity of the density of μλν except on a negligible set of points, as well as existence and continuity of a density everywhere.

Operators related to subordination for free multiplicative convolutions

It has been shown by Voiculescu and Biane that the analytic subordina- tion property holds for free additive and multiplicative convolutions. In this paper, we present an operatorial approach to subordination for free multiplicative convolutions. This study is based on the concepts of 'freeness with subordination', or 's-free inde- pendence', and 'orthogonal independence', introduced recently in the context of free additive convolutions. In particular, we introduce and study the associated multiplica- tive convolutions and construct related operators, called 'subordination operators' and 'subordination branches'. Using orthogonal independence, we derive decompositions of subordination branches and related decompositions of s-free and free multiplicative convolutions. The operatorial methods lead to several new types of graph products, called 'loop products', associated with different notions of independence (monotone, boolean, orthogonal, s-free). We also prove that the enumeration of rooted 'alternat- ing double return walks' on the loop products of graphs and on the free product of graphs gives the moments of the corresponding multiplicative convolutions.

On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

r. Let M denote the space of Borel probability measures on R. For every t > 0 we consider the transformation B t : M → W defined by B t (μ) =(μ□ (1+t) ) U(1/1+t) , μ∈Μ, where □ and U are the operations of free additive convolution and respectively of Boolean convolution on M, and where the convolution powers with respect to □ and w are defined in the natural way. We show that B s o B t = B s+t, ∀ s, t ≥ 0 and that, quite surprisingly, every B t is a homomorphism for the operation of free multiplicative convolution □ (that is, B t (μ 14 ν) = B t (μ) □ B t (ν) for all p, ν ∈ M such that at least one of μ, ν is supported on [0, oo)). We prove that for t = 1 the transformation B 1 coincides with the canonical bijection B: M → M inf-div discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M inf-div stands for the set of probability distributions in M which are infinitely divisible with respect to the operation □. As a consequence, we have that B t (μ) is □-infinitely divisible for every μ ∈ M and every t ≥ 1. On the other hand we put into evidence a relation between the transformations B t and the free Brownian motion; indeed, Theorem 1.6 of the paper gives an interpretation of the transformations B t as a way of recasting the free Brownian motion, where the resulting process becomes multiplicative with respect to □, and always reaches □-infinite divisibility by the time t = 1.

The asymptotic behavior of free additive convolution

We provide a new proof of the limit theorem for sums of free random variables in a general infinitesimal triangular array. This result was proved by Chistyakov and Gotze using subordination functions. Our proof does not depend on subordination, and is close to the approach used in the case of arrays with identically distributed rows [5]. Mathematics subject classification (2000): 46L54, 60F05.

On freely indecomposable measures

We show that a probability measure is not a nontrivial free additive convolution if it puts no mass in an interval whose endpoints are atoms. The analogous results for free multiplicative convolutions are proved as well. The proofs use analytic subordination.

The Polynomial Method for Random Matrices

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marcenko–Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.

INFINITE DIVISIBILITY FOR THE CONDITIONALLY FREE CONVOLUTION

Infinite divisibility for the free additive convolution was studied in Ref. 20. A complete characterization of [Formula: see text]-infinitely divisible distributions was given, and it was explained in Ref. 21 that this characterization is an analogue of the classical Lévy–Khintchine characterization. In fact, the analogue of the Gaussian distribution appeared even earlier, when the central limit theorem for free additive convolution was proven in Ref. 19. In this paper we define the notion of [Formula: see text]-infinitely divisibility and give the description of infinitely divisible compactly supported probability measures relative to the conditionally free convolution. We also show that the Lévy–Khintchine measures associated with a [Formula: see text]-infinitely divisible distribution μ can be calculated, as in the classical or free case, as a weak limit of measures related with the convolution semigroup generated by (μ, φ) for [Formula: see text]-infinitely divisible.

The Lebesgue decomposition of the free additive convolution of two probability distributions

We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on $${\mathbb{R}}$$ only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of the free additive convolution of Borel probability measures on the real line have been described by Bercovici and Voiculescu in a previous paper.

DECOMPOSITIONS OF THE FREE PRODUCT OF GRAPHS

We study the free product of rooted graphs and its various decompositions using quantum probabilistic methods. We show that the free product of rooted graphs is canonically associated with free independence, which completes the proof of the conjecture that there exists a product of rooted graphs canonically associated with each notion of noncommutative independence which arises in the axiomatic theory. Using the orthogonal product of rooted graphs, we decompose the branches of the free product of rooted graphs as "alternating orthogonal products". This leads to alternating decompositions of the free product itself, with the star product or the comb product followed by orthogonal products. These decompositions correspond to the recently studied decompositions of the free additive convolution of probability measures in terms of boolean and orthogonal convolutions, or monotone and orthogonal convolutions. We also introduce a new type of quantum decomposition of the free product of graphs, where the distance partition of the set of vertices is taken with respect to a set of vertices instead of a single vertex. We show that even in the case of widely studied graphs this yields new and more complete information on their spectral properties, like spectral measures of a (usually infinite) set of cyclic vectors under the action of the adjacency matrix.

η-series and a Boolean Bercovici–Pata bijection for bounded k-tuples

Let D c ( k ) be the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a C ∗ -probability space. On D c ( k ) one has an operation ⊞ of free additive convolution, and one can consider the subspace D c inf - div ( k ) of distributions which are infinitely divisible with respect to this operation. The linearizing transform for ⊞ is the R-transform (one has R μ ⊞ ν = R μ + R ν , ∀ μ , ν ∈ D c ( k ) ). We prove that the set of R-transforms { R μ | μ ∈ D c inf - div ( k ) } can also be described as { η μ | μ ∈ D c ( k ) } , where for μ ∈ D c ( k ) we denote η μ = M μ / ( 1 + M μ ) , with M μ the moment series of μ. (The series η μ is the counterpart of R μ in the theory of Boolean convolution.) As a consequence, one can define a bijection B : D c ( k ) → D c inf - div ( k ) via the formula (I) R B ( μ ) = η μ , ∀ μ ∈ D c ( k ) . We show that B is a multi-variable analogue of a bijection studied by Bercovici and Pata for k = 1 , and we prove a theorem about convergence in moments which parallels the Bercovici–Pata result. On the other hand we prove the formula (II) B ( μ ⊠ ν ) = B ( μ ) ⊠ B ( ν ) , with μ , ν considered in a space D alg ( k ) ⊇ D c ( k ) where the operation of free multiplicative convolution ⊠ always makes sense. An equivalent reformulation of (II) is that [Display omitted] where ▪ is an operation on series previously studied by Nica and Speicher, and which describes the multiplication of free k-tuples in terms of their R-transforms. Formula (III) shows that, in a certain sense, η-series behave in the same way as R-transforms in connection to the operation of multiplication of free k-tuples of non-commutative random variables.

Berry–Esseen for Free Random Variables

An analogue of the Berry–Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n −1/2, the same as in the classical case. An example with binomial measures shows that this estimate cannot be improved without imposing further restrictions on convolved measures.

Computing moments of free additive convolution of measures

This short note explains how to use ready-to-use components of symbolic software to convert between the free cumulants and the moments of measures without sophisticated programming. This allows quick access to low order moments of free convolutions of measures, which can be used to test whether a given probability measure is a free convolution of other measures.

Decompositions of the free additive convolution

We introduce and study a new type of convolution of probability measures, denoted μ ▪ ν and called the s- free additive convolution, which is defined by the subordination functions associated with the free additive convolution. We derive an alternating decomposition of μ ▪ ν for compactly supported μ and ν, using another convolution called orthogonal additive convolution. This decomposition leads to two types of ‘complete’ alternating decompositions of the free additive convolution μ ⊞ ν . More importantly, we develop an operatorial approach to the subordination property and introduce the associated notion of s- free independence. Moreover, we establish relations between convolutions associated with the main notions of noncommutative independence (free, monotone and boolean). Finally, our result leads to natural decompositions of the free product of rooted graphs.

A note on regularity for free convolutions

Let μ ⊞ ν and μ ⊠ ν denote the free additive convolution and the free multiplicative convolution, respectively, of the Borel probability measures μ and ν. We analyze the boundary behavior of the functions G μ ⊞ ν ( z ) = ∫ 1 z − t d ( μ ⊞ ν ) ( t ) and ψ μ ⊠ ν ( z ) = ∫ z t 1 − z t d ( μ ⊠ ν ) ( t ) . We prove that, under certain conditions, these functions extend continuously to the boundary of their natural domains as functions with values in the extended complex plane C ∪ { ∞ } . As a consequence, we obtain that μ ⊞ ν (respectively μ ⊠ ν ) can never be purely singular, unless μ or ν is concentrated in one point.

Atoms and regularity for measures in a partially defined free convolution semigroup

Consider a Borel probability measure μ on the real line, and denote by {μt : t≥1} the free additive convolution semigroup defined by Nica and Speicher. We show that the singular part of μt is purely atomic and the density of μt is locally analytic, provided that t > 1. The main ingredient is a global inversion theorem for analytic functions on a half plane.

The Atoms of the Free Multiplicative Convolution of Two Probability Distributions

We determine the atoms of the free multiplicative convolution of two measures supported on the unit circle or on the positive real line. Analogous results were proved earlier for free additive convolution.

THE ORTHOGONAL POLYNOMIALS FOR A LINEAR SUM OF A FREE FAMILY OF PROJECTIONS

We calculate the probability measures for some linear combinations of a free family of projections by using Voiculescu's R-transform for free additive convolution in the free probability theory. The measures we give in this paper include many examples that have been calculated, for instance, the spectral measure for the adjacency operators of the infinite distance regular graphs, and the Plancherel measures associated with free product groups. Furthermore, we find the recursive relations for the orthogonal polynomials associated with them.

The Central Limit Theorem for Free Additive Convolution

Let {Xn}∞n=1be a sequence of free, identically distributed random variables with common distributionμ. Then there exist sequences {Bn}∞n=1and {An}∞n=1of positive and real numbers, respectively, such that sequence of random variables[formula]converges in distribution to the semicircle law if and only if the function[formula]is slowly varying in Karamata's sense. In other words, the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the Gaussian. We prove an analogous result for normal domains of attraction in the sense of Linnik.

Lévy type characterization of stable laws for free random variables

We give a description of stable probability measures relative to free additive convolution. The definition of domain of attraction is given, and a proof is provided of the noncommutative analogue of Lévy Theorem.