The splitting process in free probability theory

TL;DR

This paper introduces a new combinatorial Hopf algebra approach to studying free cumulants in free probability, providing an alternative to the classical Moebius calculus method.

Contribution

It develops a Hopf algebra framework for free cumulants and moments, offering a novel combinatorial perspective and a fixed point characterization.

Findings

Hopf algebra structure on non-crossing partitions elucidates free cumulants

Characterization of free moments via a fixed point equation

Connection to arborification processes in dynamical systems

Abstract

Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley's work.