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'.
Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulan…
We define higher order infinitesimal noncommutative probability space and infinitesimal non-crossing cumulant functionals. In this framework, we generalize to higher order the notion of infinitesimal freeness, via a vanishing of mixed cumulants condition. We also introduce and study some non-crossing partitions related to this notion. Finally, as an application, we show how to compute the successive derivatives of the free convolution of two time-indexed families of distributions from their individual derivatives.
We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (GOE). In particular we show how the infinitesimal moments are described by non-crossing partitions, but not of type B. We demonstrate the asymptotic infinitesimal freeness of independent complex Wishart matrices. With the combinatorial picture we can easily compute the infinitesimal cumulants of the GOE and demonstrate the lack of asymptotic infinitesimal freeness of independent GOE ensembles. I have added a new figure and some additional explanatory remarks. This update corrects a number…
We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the non-crossing cumulants of R. Speicher. Here we introduce the concept of {\em non-crossing cumulant of type B;} the inspiration for its definition is found by looking at an operation of ``restricted convolution of multiplicative functions'', studied in parallel for functions o…
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.
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 d…
We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple proofs for a lot of results about $R$-diagonal elements. Our investigations do not assume the trace property for the considered linear functionals.
In this paper we collect a few results about exchangeability systems in which crossing cumulants vanish, which we call noncrossing exchangeability systems. The main result is a free version of De Finetti's theorem, characterising amalgamated free products as noncrossing exchangeability systems which satisfy a so-called weak singleton condition. The main tool in the proof is an $L^p$-inequality with uniformly bounded constants for i.i.d. sequences in noncrossing exchangeability systems.
The paper establishes a connection between two recent combinatorial developments in free probability: the non-crossing linked partitions introduced by Dykema in 2007 to study the S-transform, and the partial order << on NC(n) introduced by Belinschi and Nica in 2008 in order to study relations between free and Boolean probability. More precisely, one has a canonical bijection between NCL(n) (the set of all non-crossing linked partitions of {1, ..., n}) and the set {(p,q) | p,q in NC(n), p<<q}. As a consequence of this bijection, one gets an alternative description of Dykema's formula expressin…
We introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not a multiple of k. First, we consider the combinatorial convolution \ast in the lattices NC of non-crossing partitions and NC^k of k-divisible non-crossing partitions and show that convolving k times with the zeta-function in NC is equivalent to convolving once with the zeta-function in NC^k. Furthermore, when x is k-divisible, we derive a formula for the free cumulants of x^k in terms of the fr…
The present material addresses several problems left open in the Trans. AMS paper " Non-crossing cumulants of type B" of P. Biane, F. Goodman and A. Nica. The main result is that a type B non-commutative probability space can be studied in the framework of freeness with amalgamation. This view allows easy ways of constructing a version of the S-transform as well as proving analogue results to Central Limit Theorem and Poisson Limit Theorem.
We derive a formula for the moments and the free cumulants of the multiplication of $k$ free random variables in terms of $k$-equal and $k$-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge of the support of the free multiplicative convolution $μ^{\boxtimes k}$, given by Kargin which show that the support grows at most linearly with $k$. Moreover, this combinatorial approach generalize the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theore…
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 t…
Let k be a positive integer and let G_k denote the set of non-commutative k-variable distributions μsuch that μ(X_1) = ... = μ(X_k) = 1. G_k is a group under the operation of free multiplicative convolution. We identify G_k as the group of characters of a certain Hopf algebra Y_k. Then, by using the log map from characters to infinitesimal characters of Y_k, we introduce a transform LS_μ for distributions μin G_k. The main property of the LS-transform is that it linearizes commuting products in G_k. For μin G_k, the transform LS_μ is a power series in k non-commuting indeterminates; its coeffi…
