Cumulants, free cumulants and half-shuffles

Kurusch Ebrahimi-Fard, Frederic Patras

Published 2014-09-19Version 1

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 noncommutative (half-)shuffles and (half-)unshuffles. As a corollary, cumulants and free cumulants can be characterized through linear fixed point equations, from which various new properties thereof can be automatically deduced. As a first step in that direction, we study the exponential solutions of these linear fixed point equations, which display well the commutative, respectively noncommutative, character of classical, respectively free, cumulants.