The splitting process in free probability theory

Kurusch Ebrahimi-Fard 1 Frédéric Patras 2
2 LJAD
JAD - Laboratoire Jean Alexandre Dieudonné
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 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.
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Kurusch Ebrahimi-Fard, Frédéric Patras. The splitting process in free probability theory. International Mathematics Research Notices, Oxford University Press (OUP), 2015, pp.rnv209. ⟨10.1093/imrn/rnv209 ⟩. ⟨hal-01141525⟩

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