Epsilon-noncrossing partitions and cumulants in free probability

TL;DR

This paper introduces epsilon-noncrossing partitions and epsilon-cumulants, providing a new framework that interpolates between classical and free probability structures, and characterizes epsilon-independence.

Contribution

It defines epsilon-noncrossing partitions forming a lattice and introduces epsilon-cumulants to characterize epsilon-independence, bridging classical and free probability.

Findings

Epsilon-noncrossing partitions form a lattice structure.

Epsilon-cumulants characterize epsilon-independence.

The framework interpolates between classical and free probability.

Abstract

Motivated by recent work on mixtures of classical and free probabilities, we introduce and study the notion of $\epsilon$-noncrossing partitions. It is shown that the set of such partitions forms a lattice, which interpolates as a poset between the poset of partitions and the one of noncrossing partitions. Moreover, $\epsilon$-cumulants are introduced and shown to characterize the notion of $\epsilon$-independence.