## An operad of non-commutative independences defined by trees

### Volume 553 / 2020

#### Abstract

We study certain notions of $N$-ary non-commutative independence, which generalize free, Boolean, and monotone independence. For every rooted subtree $\mathcal{T}$ of an $N$-regular rooted tree, we define the $\mathcal{T}$-free product of $N$ non-commutative probability spaces and the $\mathcal{T}$-free additive convolution of $N$ non-commutative laws.

These $N$-ary additive convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities such as $\mu \mathbin{\boxplus} \nu = \mu \rhd (\nu \mathbin{\boxright } \mu)$ can be reduced to combinatorial manipulations of trees. In particular, we obtain a decomposition of the $\mathcal{T}$-free convolution into iterated Boolean and orthogonal convolutions, which generalizes work of Lenczewski.

We also develop a theory of $\mathcal{T}$-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. This includes combinatorial moment formulas, cumulants, a central limit theorem, and classification of distributions that are infinitely divisible with bounded support. In particular, we study the case where the root vertex of $\mathcal{T}$ has $n$ children and each other vertex has $d$ children, and we relate the $\mathcal{T}$-free convolution powers to free and Boolean convolution powers and the Belinschi–Nica semigroup.