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

2013/08/09 by Dan-Virgil Voiculescu, Voiculescu, Dan-Virgil · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Operator Algebras (math.OA) #Primary : 46L54 #Probability (math.PR) #Random Matrices and Applications #Secondary:47B20

paper · pdf · doi:10.48550/arxiv.1308.2035

openalex publication_date 2013/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute the generating series for the simplest class of bi-free cumulants,\nbeyond free cumulants, the two-bands bi-free cumulants of a pair of a left and\na right variable. We also consider two-faced systems with a commutation\ncondition implying that two-bands moments, that is expectation values of the\nproduct of a monomial of left and a monomial of right variables determine the\nother moments. Examples include hyponormal operators, dual systems in free\nentropy theory and bi-partite systems.\n

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