Cumulants, free cumulants and half-shuffles

TL;DR

This paper presents a unified algebraic and combinatorial framework for understanding classical and free cumulants using half-shuffles and fixed point equations, revealing their underlying structures.

Contribution

It introduces a novel approach to cumulants via (half-)shuffles and (half-)unshuffles, connecting classical and free cumulants through fixed point equations.

Findings

Cumulants can be characterized by linear fixed point equations.

Exponential solutions reflect the commutative or non-commutative nature of cumulants.

Provides a unified algebraic perspective on classical and free cumulants.

Abstract

Free cumulants were introduced as the proper analog of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, the one of free cumulants is described, and often introduced in terms of non-crossing set partitions. The formal series approach to classical and free cumulants also largely differ. It is the purpose of the present article to put forward a different approach to these phenomena. Namely, we show that cumulants, whether classical or free, can be understood in terms of the algebra and combinatorics underlying commutative as well as non-commutative (half-)shuffles and (half-)unshuffles. As a corollary, cumulants and free cumulants can be characterized through linear fixed point equations. We study the exponential solutions of these linear fixed point equations, which display well the commutative, respectively non-commutative, character of classical, respectively free, cumulants.