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It is well known that, on a purely algebraic level, a simplified version of the central limit theorem (CLT) can be proved in the framework of a non-commutative probability space, under the hypotheses that the sequence of non-commutative random variables we consider is exchangeable and obeys a certain vanishing condition of some of its joint moments. In this approach (which covers versions for both the classical CLT and the CLT of free probability), the determination of the resulting limit law has to be addressed on a case-by-case basis. In this paper we discuss an instance of the above theorem that takes place in the framework of the group algebra $${{\mathbb {C}}}[ S_{\infty } ]$$ of the infinite symmetric group: The exchangeable sequence is provided by the star-generators of $$S_{\infty }$$ , and the expectation functional used on $${{\mathbb {C}}}[ S_{\infty } ]$$ depends in a natural way on a parameter $$d \in {{\mathbb {N}}}$$ . We identify precisely the limit distribution $$\mu _d$$ for this special instance of CLT, via a connection that $$\mu _d$$ turns out to have with the average empirical eigenvalue distribution of a random $$d \times d$$ GUE matrix. Moreover, we put into evidence a multivariate version of this result which follows from the observation that, on the level of calculations with pair-partitions, the (non-centered) star-generators are related to a (centered) exchangeable sequence of GUE matrices with independent entries. more...

We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the ∗ * -distribution of the product of two selfadjoint freely independent random variables, and in connection to the distribution of the anti-commutator of such random variables. A key instrument in our considerations is a new combinatorial object, coloured noncrossing partitions with a structural property which we call vertical no-repeat property. As a byproduct, we obtain several results concerning enumeration of some special sets of noncrossing partitions. more...

We consider two extensions of free probability that have been studied in the research literature, and are based on the notions of c-freeness and respectively of infinitesimal freeness for noncommutative random variables. In a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these two frameworks, at the level of their operations of 1-dimensional free additive convolution. Motivated by that, we propose a construction which produces a multi-variate version of the Belinschi-Shlyakhtenko result, together with a result concerning free products of multi-variate noncommutative distributions. Our arguments are based on the combinatorics of the specific types of cumulants used in c-free and in infinitesimal free probability. They work in a rather general setting, where the initial data consists of a vector space $V$ given together with a linear map $\Delta : V \to V \otimes V$. In this setting, all the needed brands of cumulants live in the guise of families of multilinear functionals on $V$, and our main result concerns a certain transformation $\Delta^{*}$ on such families of multilinear functionals., Comment: Version 2: Minor revision, added references more...

We consider closed meandric systems, and their equivalent description in terms of the Hasse diagrams of the lattices of non-crossing partitions $NC(n)$. In this equivalent description, the number of components of a random meandric system of order $n$ translates into the distance between two partitions in $NC(n)$. We focus on a class of couples $(\pi,\rho)\in NC(n)^2$ -- namely the ones where $\pi$ is conditioned to be an interval partition -- for which it turns out to be tractable to study distances in the Hasse diagram. As a consequence, we observe a non-trivial class of meanders (i.e. connected meandric systems), which we call "meanders with shallow top", and which can be explicitly enumerated. Moreover, the expected number of components for a random "meandric system with shallow top", is asymptotically $(9n+28)/27$. Our calculations concerning expected number of components are related to the idea of taking the derivative at $t=1$ in a semigroup for the operation $\boxplus$ of free probability (but the underlying considerations are presented in a self-contained way, and can be followed without assuming a free probability background). Let $c_{n}'$ denote the expected number of components of a general, unconditioned, meandric system of order $n$. A variation of the methods used in the shallow-top case allows us to prove that $\mathrm{lim\ inf}_{n\to\infty}c_{n}'/n\geq0.17$. We also note that, by a direct elementary argument, one has $\mathrm{lim\ sup}_{n\to\infty}c_{n}'/n\leq0.5$. These bounds support the conjecture that $c_{n}'$ follows a regime of "constant times $n$" (where numerical experiments suggest that the constant should be $\approx0.23$)., Comment: 36 pages, 9 Figures, added Remark 7.3 giving a shorter proof to Thm 1.1, journal version more...

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map \eta from B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever \eta is completely positive, while the free additive convolution power is defined whenever \eta - 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent \eta. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion., Comment: 33 pages, no figures more...

We show that for all q in the interval (-1,1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C*-algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (-1,1)., 20 pages more...

Let k be a positive integer and let D_k denote the space of joint distributions for k-tuples of selfadjoint elements in C*-probability space. The paper studies the concept of "subordination distribution of \mu \boxplus \nu with respect to \nu" for \mu, \nu \in D_k, where \boxplus is the operation of free additive convolution on D_k. The main tools used in this study are combinatorial properties of R-transforms for joint distributions and a related operator model, with operators acting on the full Fock space Multi-variable subordination turns out to have nice relations to a process of evolution towards \boxplus-infinite divisibility on D_k that was recently found by Belinschi and Nica (arXiv:0711.3787). Most notably, one gets better insight into a connection which this process was known to have with free Brownian motion., Comment: Minor modifications and reference added to 1st version. 34 pages, no figures more...

Let Dc(k) be the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a C∗-probability space. On Dc(k) one has an operation ⊞ of free additive convolution, and one can consider the subspace Dcinf-div(k) of distributions which are infinitely divisible with respect to this operation. The linearizing transform for ⊞ is the R-transform (one has Rμ⊞ν=Rμ+Rν, ∀μ,ν∈Dc(k)). We prove that the set of R-transforms {Rμ|μ∈Dcinf-div(k)} can also be described as {ημ|μ∈Dc(k)}, where for μ∈Dc(k) we denote ημ=Mμ/(1+Mμ), with Mμ the moment series of μ. (The series ημ is the counterpart of Rμ in the theory of Boolean convolution.) As a consequence, one can define a bijection B:Dc(k)→Dcinf-div(k) via the formula(I)RB(μ)=ημ,∀μ∈Dc(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 Dalg(k)⊇Dc(k) where the operation of free multiplicative convolution ⊠ always makes sense. An equivalent reformulation of (II) is that(III)ημ⊠ν=ημην,∀μ,ν∈Dalg(k), 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. more...

The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an eta-diagonal distribution that is the Boolean counterpart of an R-diagonal distribution. We establish a number of properties of eta-diagonal distributions, then we examine the canonical bijection relating eta-diagonal distributions to infinitely divisible R-diagonal ones. The overall result is a parametrization of an arbitrary $\boxplus$-infinitely divisible R-diagonal distribution that can arise in a C*-probability space, by a pair of compactly supported Borel probability measures on $[ 0, \infty )$. Among the applications of this parametrization, we prove that the set of $\boxplus$-infinitely divisible R-diagonal distributions is closed under the operation $\boxtimes$ of free multiplicative convolution., Comment: 33 pages more...

We study joint free cumulants of u_t and u_t^{*}, where u_t is a free unitary Brownian motion at time t. We determine explicitly some special families of such cumulants. On the other hand, for a general joint cumulant of u_t and u_t^{*}, we "calculate the derivative" for t going to infinity, when u_t approaches a Haar unitary. In connection to the latter calculation we put into evidence an "infinitesimal determining sequence" which naturally accompanies an arbitrary R-diagonal element in a tracial *-probability space., 35 pages. This version has added details in Sections 5 and 6 more...

The concept of R-diagonal element was introduced in [5], and was subsequently found to have applications to several problems in free probability. In this paper we describe a new approach to R-diagonality, which relies on freeness with amalgamation. The class of R-diagonal elements is enlarged to contain examples living in non-tracial *-probability spaces, such as the generalized circular elements of [7]. more...

LetΓ2nbe the set of paths with 2nsteps of unit length inZ2, which begin and end at (0,0). Forγ∈Γ2n, letarea(γ)∈Zdenote the oriented area enclosed byγ. We show that for every positive even integerk, there exists a rational functionRkwith integer coefficients, such that:1|Γ2n|∑γ∈Gamma;2n[area(γ)]k=Rk(n,n>2k.We calculate explicitly the degree and leading coefficient ofRk. We show how as a consequence of this (and by also using the enumeration of up-down permutations, and the exponential formula for cycles of permutations) one can derive the asymptotic distribution of the area enclosed by a random path inΓ2n. The formula for the asymptotic distribution can be stated as follows: forα more...

We follow the guiding line offered by canonical operators on the full Fock space, in order to identify what kind of cumulant functionals should be considered for the concept of bi-free independence introduced in the recent work of Voiculescu. By following this guiding line we arrive to consider, for a general noncommutative probability space (A, phi), a family of "(l,r)-cumulant functionals" which enlarges the family of free cumulant functionals of the space. In the motivating case of canonical operators on the full Fock space we find a simple formula for a relevant family of (l,r)-cumulants of a (2d)-tuple (A_1, ..., A_d, B_1, ..., B_d), with A_1, ... , A_d canonical operators on the left and B_1, ... , B_d canonical operators on the right. This extends a known one-sided formula for free cumulants of A_1, ..., A_d, which establishes a basic operator model for the R-transform of free probability., Comment: In this (final) version, the introduction was re-written to better show the motivation for the question considered in the paper more...

We consider the concept of irreducible meandric system introduced by Lando and Zvonkin. We place this concept in the lattice framework of [Formula: see text]. As a consequence, we show that the even generating function for irreducible meandric systems is the [Formula: see text]-transform of [Formula: see text], where [Formula: see text] and [Formula: see text] are classically (commuting) independent random variables, and each of [Formula: see text] has centred semicircular distribution of variance [Formula: see text]. Following this point of view, we make some observations about the symmetric linear functional on [Formula: see text] which has [Formula: see text]-transform given by the even generating function for meanders. more...

Free probabilistic considerations of type B first appeared in a paper by Biane, Goodman and Nica in 2003. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko (arXiv:0903.2721). The interplay between "type B" and "infinitesimal" is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A_1, ..., A_k in an infinitesimal noncommutative probability space (A, phi, phi'), and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, phi, phi'), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A_1, ... A_k is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from "usual" free probability. We show that the lattices of non-crossing partitions of type B appear in the combinatorial study of (A, phi, phi'), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A_1, ..., A_k in (A, phi) can be naturally upgraded to infinitesimal freeness in (A, phi, phi'), for a suitable choice of a "companion functional" phi'., Comment: 38 pages, 1 figure more...

Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect to free additive convolution and then the 1/(1+t) power with respect to Boolean convolution of the given element. We show that the family of maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism for the operation of free multiplicative convolution. We prove that for t=1 the transformation $\mathbb B_1$ coincides with the canonical bijection $\mathbb B : M \to M_{inf-div}$ 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 free additive convolution. As a consequence, we have that $\mathbb B_t(\mu)$ is infinitely divisible with respect to free additive convolution for any for every $\mu$ in M and every t greater than or equal to one. On the other hand we put into evidence a relation between the transformations $\mathbb B_t$ and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations $\mathbb B_t$ as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to free multiplicative convolution, and always reaches infinite divisibility with respect to free additive convolution by the time t=1., Comment: 30 pages, minor changes; to appear in Indiana University Mathematics Journal more...

We introduce the concept of ``R-cyclic family'' of matrices with entries in a non-commutative probability space; the definition consists in asking that only the ``cyclic'' non-crossing cumulants of the entries of the matrices are allowed to be non-zero. Let A_{1}, ..., A_{s} be an R-cyclic family of d \times d matrices over a non-commutative probability space. We prove a convolution-type formula for the explicit computation of the joint distribution of A_{1}, ..., A_{s} (considered in M_{d} (\A) with the natural state), in terms of the joint distribution (considered in the original space) of the entries of the s matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula. Moreover, let A_{1}, ..., A_{s} be a family of d \times d matrices over a non-commutative probability space, let \D \subset M_{d} (\A) denote the algebra of scalar diagonal matrices, and let {\cal C} be the subalgebra of M_{d} (\A) generated by \{A_{1}, ..., A_{s} \} \cup \D. We prove that the R-cyclicity of A_{1}, ..., A_{s} is equivalent to a property of {\cal C} -- namely that {\cal C} is free from M_{d} (\C), with amalgamation over \D. more...

We consider the concept of a q-circular system, which is a deformation of the circular system from free probability, taking place in the framework of the so-called 'q-commutation relations'. We show that certain averages of random unitaries in non-commutative tori behave asymptotically like a q-circular system., Comment: 32 pages, LaTeX 2.09 more...

A non-commutative non-selfadjoint random variable z is called R-diagonal, if its *-distribution is invariant under multiplication by free unitaries: if a unitary w is *-free from z, then the *-distribution of z is the same as that of wz. Using Voiculescu's microstates definition of free entropy, we show that the R-diagonal elements are characterized as having the largest free entropy among all variables y with a fixed distribution of y^*y. More generally, let Z be a d*d matrix whose entries are non-commutative random variables X_{ij}. Then the free entropy of the family {X_{ij}} of the entries of Z is maximal among all Z with a fixed distribution of Z^*Z, if and only if Z is R-diagonal and is *-free from the algebra of scalar d*d matrices. The results of this paper are analogous to the results of our paper "Some minimization problems for the free analogue of the Fisher information", where we considered the same problems in the framework of the non-microstates definition of free entropy., 12 pages, latex2e (using amsart) more...

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 denote $c:=i(ab-ba)$, where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper funct-an/9604012., Comment: LaTeX, 38 pages with 2 figures more...

We study the set $\sncb (p,q)$ of annular non-crossing permutations of type B, and we introduce a corresponding set $\ncb (p,q)$ of annular non-crossing partitions of type B, where $p$ and $q$ are two positive integers. We prove that the natural bijection between $\sncb (p,q)$ and $\ncb (p,q)$ is a poset isomorphism, where the partial order on $\sncb (p,q)$ is induced from the hyperoctahedral group $B_{p+q}$, while $\ncb (p,q)$ is partially ordered by reverse refinement. In the case when $q=1$, we prove that $\ncb (p,1)$ is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between $\sncd (p,q)$ and $\ncd (p,q)$. For $q=1$, the poset $\ncd (p,1)$ coincides with a poset ``$NC^{(D)} (p+1)$'' constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper., Comment: Revised version (shortened Introduction, corrected typos), 31 pages, 4 figures, to appear in Discrete Mathematics more...

We give a combinatorial proof of Harer and Zagier's formula for the disjoint cycle distribution of a long cycle multiplied by an involution with no fixed points, in the symmetric group on a set of even cardinality. The main result of this paper is a direct bijection of a set Bp,k, the enumeration of which is equivalent to the Harer-Zagier formula. The elements of Bp,k are of the form (µ, π), where µ is a pairing on {1,...,2p}, π is a partition into k blocks of the same set, and a certain relation holds between µ and π. (The set partitions π that can appear in Bp,k are called "shift-symmetric", for reasons that are explained in the paper.) The direct bijection for Bp,k identifies it with a set of objects of the form (ρ, t), where ρ is a pairing on a 2(p - k + 1)-subset of {1, ..., 2p} (a "partial pairing"), and t is an ordered tree with k vertices. If we specialize to the extreme case when p = k - 1, then ρ is empty, and our bijection reduces to a well-known tree bijection. more...

We consider the free non-commutative analogue Phi^*, introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value of Phi^*(a,a^*), if a is a non-commutative random variable subject to the constraint that the distribution of aa^* is prescribed. More generally, we obtain the minimal possible value of Phi^*({a_{ij},a_{ij}^*), if {a_{ij}} is a family of non-commutative random variables such that the distribution of AA^* is prescribed, where A is the matrix (a_{ij}). The d*d-generalization is obtained from the case d=1 via a result of independent interest, concerning the minimal value of Phi^*({a_{ij},a_{ij}^*), when the matrix A=(a_{ij}) and its adjoint have a given joint distribution. We then show how the minimization results obtained for Phi^* lead to maximization results concerning the free entropy chi^*, also defined by Voiculescu., 31 pages, Latex more...