Free cumulants, Schr\"oder trees, and operads
Matthieu Josuat-Verg\`es, Fr\'ed\'eric Menous, Jean-Christophe, Novelli, and Jean-Yves Thibon
TL;DR
This paper explores the combinatorial structures underlying free cumulants in free probability, extending classical formulas to operads and Schr"oder trees, and providing new algebraic and combinatorial insights.
Contribution
It introduces a novel lifting of free cumulant equations to operads over Schr"oder trees, unifying various combinatorial formulas in free probability.
Findings
Derived new combinatorial expressions for free cumulants
Connected operadic structures with free probability formulas
Extended Speicher's formula to operator-valued free probability
Abstract
The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Fa\`a di Bruno algebra, and then to the group of a free operad over Schr\"oder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.
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