Free stochastic measures via noncrossing partitions II

Epsilon-noncrossing partitions and cumulants in free probability

Motivated by recent work on mixtures of classical and free probabilities, we introduce and study the notion of $ε$-noncrossing partitions. It is shown that the set of such partitions forms a lattice, which interpolates as a poset between the poset of partitions and the one of noncrossing partitions. Moreover, $ε$-cumulants are introduced and shown to characterize the notion of $ε$-independence.

On the representation theory of some noncrossing partition quantum groups

We compute the representation theory of two families of noncrossing partition quantum groups connected to amalgamated free products and free wreath products. This illustrates the efficiency of the methods developed in our previous joint work with M. Weber.

Free stochastic measures via noncrossing partitions

We consider free multiple stochastic measures in the combinatorial framework of the lattice of all diagonals of an n-dimensional space. In this free case, one can restrict the analysis to only the noncrossing diagonals. We give definitions of what free multiple stochastic measures are, and calculate them for the free Poisson and free compound Poisson processes. We also derive general combinatorial Itô-type relationships between free stochastic measures of different orders. These allow us to calculate, for example, free Poisson-Charlier polynomials, which are the orthogonal polynomials with res…

自由累积量的分解

Decomposition of free cumulants

引入适应Motzkin路径的非交叉分拆格，定义算子值Motzkin累积量，通过Möbius反演公式细化自由累积量与布尔累积量的关系，并应用于自由概率得到自由累积量的加性分解。

Free nested cumulants and an analogue of a formula of Brillinger

We prove a free analogue of Brillinger's formula (sometimes called "law of total cumulance") which expresses classical cumulants in terms of conditioned cumulants. As expected, the formula is obtained by replacing the lattice of set partitions by the lattice of noncrossing set partitions and using and an appropriate notion of noncommutative nested products. As an application we reprove a characterization of freeness due to Nica, Shlyakhtenko and Speicher by Möbius inversion techniques, without recourse to the Fock space model for free random variables.

Traces On Diagram Algebras I: Free Partition Quantum Groups, Random Lattice Paths And Random Walks On Trees

We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica-Speicher (arXiv:0808.2628) and Weber (arXiv:1201.4723). For the infinite-dimensional Temperley-Lieb-algebra (corresponding to the quantum group $O^+_N$) and the Motzkin algebra ($B^+_N$), the classification of extremal traces implies a classification result for well-known types of central random lattice paths. For the $2$-Fuss-Catalan algebra ($H_N^+$) we solve the classification problem by computing the \emph{minimal or exit bounda…