arXiv subjects

Romuald Lenczewski

Publications and source records attributed to Romuald Lenczewski.

At least 19 recordsLinked to original sources

arXiv2025

Infinitesimal moments in free and c-free probability and Motzkin paths

Infinitesimal moments associated with infinitesimal freeness and infinitesimal conditional freeness are studied. For free random variables, we consider continuous deformations of moment functionals associated with Motzkin paths $w$, which provide a decomposition of their moments, and we compute their derivatives at zero. We show that the first-order derivative of each functional vanishes unless the path has exactly one local maximum. Geometrically, this means that $w$ is a pyramid path, which is consistent with the characteristic formula for alternating moments of infinitesimally free centered random variables. In this framework, infinitesimal Boolean independence is also obtained and it corresponds to flat paths. A similar approach is developed for infinitesimal conditional freeness, for which we show that the only moment functionals that have a non-zero first-order derivative are associated with concatenations of a pyramid path and a flat path. This charaterization leads to a Leibniz-type definition of infinitesimal conditional freeness at the level of moments.

math.OA↗

arXiv2023

Decomposition of free cumulants

Free cumulants are multilinear functionals defined in terms of the moment functional with the use of the family of lattices of noncrossing partitions. In the univariate case, they can be identified with the coefficients of the Voiculescu transform of the moment functional which plays a role similar to that of the logarithm of the Fourier transform. The associated linearization property is connected with free independence. In turn, the family of much smaller lattices of interval partitions is used to define Boolean cumulants connected with Boolean independence. In order to bridge the gap between these two families of lattices and the associated cumulants we introduce and study the family of lattices of noncrossing partitions adapted to Motzkin paths and define the associated operator-valued `Motzkin cumulants'. We prove the corresponding M\"{o}bius inversion formula which plays the role of a lattice refinement of the formula expressing free cumulants in terms of Boolean cumulants. We apply this concept to free probability and obtain the additive decomposition of free cumulants in terms of scalar-valued counterparts of Motzkin cumulants.

math.OA↗

arXiv2022

Motzkin path decompositions of functionals in noncommutative probability

We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional $Φ$ are naturally described in terms of Motzkin paths identified with reduced Motzkin words. Using this fact, we demonstrate that the mixed moments of free random variables with respect to the free product of normalized linear functionals are sums of the mixed moments of order $n$ of the orthogonal replicas of these variables with respect to $Φ$ with summation extending over the set of reduced Motzkin words of lenght $n$. One of the applications of this formula is a decomposition formula for mixed moments of free random variables in terms of their boolean cumulants which corresponds to the decomposition of the lattice ${\rm NC}(n)$ into sublattices $\mathcal{M}(w)$ of partitions which are monotonically adapted to colors in the word $w$. The linear functionals defined by the mixed moments of orthogonal replicas and indexed by reduced Motzkin words play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product corresponds to all Motzkin paths.

math.OA↗

arXiv2020

Conditionally monotone independence and the associated products of graphs

We reduce the conditionally monotone (c-monotone) independence of Hasebe to tensor independence. For that purpose, we use the approach developed for the reduction of boolean, free and monotone independences to tensor independence. We apply the tensor product realization of c-monotone random variables to introduce the c-comb (loop) product of birooted graphs, a generalization of the comb (loop) product of rooted graphs, and we show that it is related to the c-monotone additive (multiplicative) convolution of distributions.

math.OA↗

arXiv2018

Random matrices, continuous circular systems and the triangular operator

We present a Hilbert space approach to the limit joint *-distributions of complex independent Gaussian random matrices. For that purpose, we use a suitably defined family of creation and annihilation operators living in some direct integral of Hilbert spaces. These operators are decomposed in terms of continuous circular systems of operators acting between the fibers of the considered Hilbert space direct integral. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain the triangular operators of Dykema and Haagerup. We apply this approach to give a bijective proof of a formula for *-moments of the triangular operator, using the enumeration formula of Chauve, Dulucq and Rechnizter for alternating ordered rooted trees.

math.OA↗

arXiv2017

Asymptotic distributions of Wishart type products of random matrices

We study asymptotic distributions of large dimensional random matrices of the form $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using free probability and combinatorics of colored labeled noncrossing partitions. These matrices are taken from the set of off-diagonal blocks of the family $\mathcal{Y}$ of independent Hermitian random matrices which are asymptotically free, asymptotically free against the family of deterministic diagonal matrices, and whose norms are uniformly bounded almost surely. This class includes unitarily invariant Hermitian random matrices with limit distributions given by compactly supported probability measures $ν$ on the real line. We express the limit moments in terms of colored labeled noncrossing pair partitions, to which we assign weights depending on even free cumulants of $ν$ and on asymptotic dimensions of blocks (Gaussianization). For products of $p$ independent blocks, we show that the limit moments are linear combinations of a new family of polynomials called generalized multivariate Fuss-Narayana polynomials. In turn, the product of two blocks of the same matrix leads to an example with rescaled Raney numbers.

math.PR↗

arXiv2016

Matricial circular systems and random matrices

We introduce and study `matricial circular systems' of operators which play the role of matricial counterparts of circular operators. They describe the asymptotic joint *-distributions of blocks of independent block-identically distributed Gaussian random matrices with respect to partial traces. Using these operators, we introduce `circular free Meixner distributions' as the non-Hermitian counterparts of free Meixner distributions and construct for them a random matrix model. Our approach is based on the concept of matricial freeness applied to operators on Hilbert spaces. It is closely related to freeness with amalgamation over the algebra A of r x r diagonal matrices applied to operators on Hilbert A-bimodules.

math.OA↗

arXiv2015

Limit distributions of Gaussian block ensembles

It has been shown by Voiculescu that important classes of square independent random matrices are asymptotically free, where freeness is a noncommutative analog of classical independence. Recently, we introduced the concept of matricial freeness, which is similar to freeness in free probability, but it also has some matricial features. Using this new concept of noncommutative independence, we described the asymptotics of blocks and symmetric blocks of certain classes of independent random matrices. In this paper, we present the main results obtained in this framework, concentrating on the ensembles of blocks of Gaussian random matrices.

math.OA↗

arXiv2014

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type. This framework also leads to random matrix models for boolean, monotone and s-free independences.

math.OA↗

arXiv2013

Random matrix model for free Meixner laws

Applying the concept of matricial freeness which generalizes freeness in free probability, we have recently studied asymptotic joint distributions of symmetric blocks of Gaussian random matrices (Gaussian Symmetric Block Ensemble). This approach gives a block refinement of the fundamental result of Voiculescu on asymptotic freeness of independent Gaussian random matrices. In this paper, we show that this framework is natural for constructing a random matrix model for free Meixner laws. We also demonstrate that the ensemble of independent matrices of this type is asymptotically conditionally free with respect to the pair of partial traces.

math.OA↗

arXiv2012

Multivariate Fuss-Narayana polynomials and their application to random matrices

It has been shown recently that the limit moments of $W(n)=B(n)B^{*}(n)$, where B(n) is a product of $p$ independent rectangular random matrices, are certain homogenous polynomials in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials which can be viewed as multivariate Fuss-Narayana polynomials. Using this result, we compute the moments of the n-fold free multiplicative convolution of Marchenko-Pastur distributions with arbitrary shape parameters.

math.CO↗

arXiv2011

Matricial R-transform

We study the addditon problem for strongly matricially free random variables which generalize free random variables. Using operators of Toeplitz type, we derive a linearization formula for the `matricial R-transform' related to the associated convolution. It is a linear combination of Voiculescu's R-transforms in free probability with coefficients given by internal units of the considered array of subalgebras. This allows us to view this formula as the `matricial linearization property' of the R-transform. Since strong matricial freeness unifies the main types of noncommutative independence, the matricial R-transform plays the role of a unified noncommutative analog of the logarithm of the Fourier transform for free, boolean, monotone, orthogonal, s-free and c-free independence.

math.OA↗

arXiv2010

Asymptotic properties of random matrices and pseudomatrices

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 distributions of blocks and give their Hilbert space realizations, using operators called `matricially free Gaussian operators'. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are `asymptotically matricially free' whereas the corresponding symmetric random blocks are `asymptotically symmetrically matricially free', where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, lower-block-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively.

math.OA↗

arXiv2008

Matricially free random variables

We show that the operatorial framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a generalization of both freeness and monotone independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to Gaussian random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial generalizations of semicirle laws.

math.OA↗

arXiv2007

Noncommutative Brownian motions with Kesten distributions and related Poisson processes

We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition $P$ we assign a weight depending on the numbers of disorders and orders in $P$ related to the natural partial order on the set of blocks of $P$ implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

math.QA↗

arXiv2007

Operators related to subordination for free multiplicative convolutions

It has been shown by Voiculescu and Biane that the analytic subordination 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 independence', and `orthogonal independence', introduced recently in the context of free additive convolutions. In particular, we introduce and study the associated multiplicative 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 `alternating double return walks' on the loop products of graphs and on the free product of graphs gives the moments of the corresponding multiplicative convolutions.

math.OA↗

arXiv2006

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 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.

math.CO↗

arXiv2006

Decompositions of the free additive convolution

We introduce and study a new type of convolution of probability measures called the orthogonal convolution, which is related to the monotone convolution. Using this convolution, we derive alternating decompositions of the free additive convolution of compactly supported probability measures in free probability. These decompositions are directly related to alternating decompositions of the associated subordination functions. In particular, they allow us to compute free additive convolutions of compactly supported measures without using free cumulants or R-transforms. In simple cases, representations of the corresponding Cauchy transforms as continued fractions are obtained in a natural way. Moreover, this approach establishes a clear connection between convolutions and products associated with the main notions of independence (free, monotone and boolean) in noncommutative probability. Finally, our result leads to natural decompositions of the free product of rooted graphs.

math.OA↗