Free cumulants, Schr\"oder trees, and operads
Combinatorics
2017-05-24 v2
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.
Cite
@article{arxiv.1604.04759,
title = {Free cumulants, Schr\"oder trees, and operads},
author = {Matthieu Josuat-Vergès and Frédéric Menous and Jean-Christophe Novelli and Jean-Yves Thibon},
journal= {arXiv preprint arXiv:1604.04759},
year = {2017}
}
Comments
23 pages