On two-faced families of non-commutative random variables

Abstract

We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

Revised for publication

References in corpus (2)

• Free Probability for Pairs of Faces I
• Double-ended queues and joint moments of left-right canonical operators on full Fock space

Cited by in corpus (9)

• Combinatorics of Bi-Freeness with Amalgamation
• Conditionally Bi-Free Independence for Pairs of Algebras
• On Operator-Valued Bi-Free Distributions
• Independences and Partial $R$-Transforms in Bi-Free Probability
• A Combinatorial Approach to Voiculescu's Bi-Free Partial Transforms
• An alternating moment condition for bi-freeness
• Conditionally Bi-Free Independence with Amalgamation
• Analogues of Entropy in Bi-Free Probability Theory: Microstates
• Bi-Free Entropy with Respect to Completely Positive Maps