The splitting process in free probability theory

International audienceFree 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 M ̈obius 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