Combinatorics of NC-Probability Spaces with Independent Constants

TL;DR

This paper explores the combinatorial structure of boolean and monotone independence in non-commutative probability spaces, focusing on the property of independent constants and their implications for cumulants and operator-valued extensions.

Contribution

It characterizes independent constants combinatorially, introduces variations of boolean and monotone cumulants with independent constants, and connects these to operator-valued c-free probability.

Findings

Characterization of independent constants via cumulants and set partitions.

Introduction of mild variations of boolean and monotone cumulants with independent constants.

Analysis of combinatorial aspects such as Möbius functions and their relation to known sequences.

Abstract

Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz. We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension. Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probability theories are closely related to the usual notions. Hence, an important part of the boolean/monotone probability theories can be imported directly. We describe some standard combinatorial aspects of these variations (and their cyclic versions), such as their Mobius functions, which feature well-known combinatorial integer sequences. The new notions with independent constants seem also more strongly related to the operator-valued extension of c-free probability.