On Two-faced Families of Non-commutative Random Variables

By Ian Charlesworth, Brent Nelson, P. Skoufranis. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2014

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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.

Subfields: Discrete Mathematics and Combinatorics, Statistics and Probability

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