Decomposition of free cumulants

TL;DR

This paper introduces a new lattice framework based on Motzkin paths to decompose free cumulants, bridging the gap between free and Boolean cumulants, and providing a refined understanding of their relationships in free probability.

Contribution

It develops a novel family of lattices of noncrossing partitions adapted to Motzkin paths and defines associated Motzkin cumulants, extending the theory of cumulant decompositions.

Findings

Proves a Möbius inversion formula for Motzkin cumulants.

Provides an additive decomposition of free cumulants using Motzkin cumulants.

Establishes a lattice refinement connecting free and Boolean cumulants.

Abstract

Free cumulants are multilinear functionals defined in terms of the moment functional with the use of the family of lattices of noncrossing partitions. In the univariate case, they can be identified with the coefficients of the Voiculescu transform of the moment functional which plays a role similar to that of the logarithm of the Fourier transform. The associated linearization property is connected with free independence. In turn, the family of much smaller lattices of interval partitions is used to define Boolean cumulants connected with Boolean independence. In order to bridge the gap between these two families of lattices and the associated cumulants we introduce and study the family of lattices of noncrossing partitions adapted to Motzkin paths and define the associated operator-valued `Motzkin cumulants'. We prove the corresponding M\"{o}bius inversion formula which plays the role of a lattice refinement of the formula expressing free cumulants in terms of Boolean cumulants. We apply this concept to free probability and obtain the additive decomposition of free cumulants in terms of scalar-valued counterparts of Motzkin cumulants.