Cumulants, free cumulants and half-shuffles

Abstract : Free cumulants were introduced as the proper analog of class ical cumulants in the theory of free probability. There is a mix of similarities an d differences, when one considers the two families of cumulants. Whereas the combinatorics of cla ssical cumulants is well expressed in terms of set partitions, the one of free cumulants is describ ed, 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 differ ent approach to these phenomena. Namely, we show that cumulants, whether classical or free, c an be understood in terms of the algebra and combinatorics underlying commutative as well as non-co mmutative (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 thes e linear fixed point equations, which display well the commutative, respectively non-commutati ve, character of classical, respectively free, cumulants.
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Kurusch Ebrahimi-Fard, Frédéric Patras. Cumulants, free cumulants and half-shuffles. Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2015, ⟨10.1098/rspa.2014.0843⟩. ⟨hal-01287031⟩

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