Free Probability for Pairs of Faces II: 2-Variables Bi-free $R$-Transform and Systems with Rank $\le 1$ Commutation

Abstract

We compute the generating series for the simplest class of bi-free cumulants, beyond free cumulants, the two-bands bi-free cumulants of a pair of a left and a right variable. We also consider two-faced systems with a commutation condition implying that two-bands moments, that is expectation values of the product of a monomial of left and a monomial of right variables determine the other moments. Examples include hyponormal operators, dual systems in free entropy theory and bi-partite systems.

21 pages

References in corpus (1)

• Random Matrices in 2D, Laplacian Growth and Operator Theory

Cited by in corpus (4)

• Fock space associated to Coxeter group of type B
• On Operator-Valued Bi-Free Distributions
• Principal functions for bi-free central limit distributions
• Two-faced Families of Non-commutative Random Variables Having Bi-free Infinitely Divisible Distributions