Bigraph independence : a mixture of the five natural independences
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912838458540032 |
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| author | Gilliers, Nicolas Jekel, David |
| author_facet | Gilliers, Nicolas Jekel, David |
| contents | We introduce a notion of non-commutative joint independence for multiple algebras in a non-commutative probability space. The pairwise relationships between these algebras are encoded by a graph with two edge sets -- a combinatorial structure we call a bigraph -- and naturally encompass the five fundamental types of independence: tensor, free, (anti)monotone, and Boolean. It subsumes the BMT independence of Arizmendi--Mendoza--Vazquez-Becerra (when all pairwise relationships are Boolean, (anti)monotone, or tensor) and the $ε$ or $Λ$-independence of Mlotkowski (when the pairwise relationships are tensor and free). We present explicit combinatorial moment formulas, a Hilbert space construction, and natural associativity relations within this setting. Furthermore, we demonstrate that bigraph independence emerges in the asymptotic behavior of tensor product random matrix models with respect to a vector state, encompassing the Charlesworth--Collins model for $\varepsilon$-independence as a special case and offering a random matrix perspective on BMT independence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15215 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bigraph independence : a mixture of the five natural independences Gilliers, Nicolas Jekel, David Probability Operator Algebras We introduce a notion of non-commutative joint independence for multiple algebras in a non-commutative probability space. The pairwise relationships between these algebras are encoded by a graph with two edge sets -- a combinatorial structure we call a bigraph -- and naturally encompass the five fundamental types of independence: tensor, free, (anti)monotone, and Boolean. It subsumes the BMT independence of Arizmendi--Mendoza--Vazquez-Becerra (when all pairwise relationships are Boolean, (anti)monotone, or tensor) and the $ε$ or $Λ$-independence of Mlotkowski (when the pairwise relationships are tensor and free). We present explicit combinatorial moment formulas, a Hilbert space construction, and natural associativity relations within this setting. Furthermore, we demonstrate that bigraph independence emerges in the asymptotic behavior of tensor product random matrix models with respect to a vector state, encompassing the Charlesworth--Collins model for $\varepsilon$-independence as a special case and offering a random matrix perspective on BMT independence. |
| title | Bigraph independence : a mixture of the five natural independences |
| topic | Probability Operator Algebras |
| url | https://arxiv.org/abs/2601.15215 |