${\epsilon}$ -Noncrossing Partitions and Cumulants in Free Probability

Abstract

Motivated by recent work on mixtures of classical and free probabilities, we introduce and study the notion of ϵ-noncrossing partitions. It is shown that the set of such partitions forms a lattice. It is a subposet of the poset of partitions and it contains the one of noncrossing partitions. Moreover, ϵ-cumulants are introduced and shown to characterize the notion of ϵ-independence.