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arXiv · 1604.04759

Free cumulants, Schröder trees, and operads

Matthieu Josuat-Vergès·Frédéric Menous·Jean-Christophe Novelli·Jean-Yves Thibon
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Abstract

The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder 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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Matthieu Josuat-Vergès, Frédéric Menous, Jean-Christophe Novelli, Jean-Yves Thibon. 2017-05-23. Free cumulants, Schröder trees, and operads. https://doi.org/10.1016/j.aam.2017.02.002

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