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Bigraph independence : a mixture of the five natural independences

Nicolas Gilliers, David Jekel

TL;DR

Bigraph independence introduces a unified framework to capture mixed non-commutative independence among many algebras via a two-edge bigraph G=(V,E1,E2). It provides a joint moment formula Phi(a1,...,ak)=sum_{pi in P(c,G)} K_pi that encodes pairwise relations and yields Boolean cumulant based and cumulant-translation reformulations, with a Hilbert space model and a tensor-product random matrix model realizing G-independence in the large-N limit. The work shows how the five Muraki independences can co-exist in a single state, recovers special cases such as epsilon-independence and BMT independence, and connects to tree/digraph independence frameworks and Weingarten calculus for unitaries. Collectively, these results offer a robust toolkit for combinatorial, operator-algebraic, and random-matrix analyses of mixed non-commutative independence and open avenues for limit theorems and operadic formulations.

Abstract

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.

Bigraph independence : a mixture of the five natural independences

TL;DR

Bigraph independence introduces a unified framework to capture mixed non-commutative independence among many algebras via a two-edge bigraph G=(V,E1,E2). It provides a joint moment formula Phi(a1,...,ak)=sum_{pi in P(c,G)} K_pi that encodes pairwise relations and yields Boolean cumulant based and cumulant-translation reformulations, with a Hilbert space model and a tensor-product random matrix model realizing G-independence in the large-N limit. The work shows how the five Muraki independences can co-exist in a single state, recovers special cases such as epsilon-independence and BMT independence, and connects to tree/digraph independence frameworks and Weingarten calculus for unitaries. Collectively, these results offer a robust toolkit for combinatorial, operator-algebraic, and random-matrix analyses of mixed non-commutative independence and open avenues for limit theorems and operadic formulations.

Abstract

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 or -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 -independence as a special case and offering a random matrix perspective on BMT independence.
Paper Structure (21 sections, 28 theorems, 135 equations, 3 figures)

This paper contains 21 sections, 28 theorems, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Bigraph independence
  4. Matrix models
  5. Further questions
  6. Organization
  7. Acknowledgements
  8. Bigraph independence and relations to other independences
  9. Combinatorial setup
  10. The five natural independences, $\varepsilon$-independence, and BMT independence
  11. Bigraph independence via Boolean and classical cumulants
  12. Free--(anti)monotone--Boolean mixtures in the tree framework
  13. Operad associativity
  14. Hilbert space model
  15. Construction of the product Hilbert space
  16. ...and 6 more sections

Key Result

Proposition 1.5

Consider the situation of Definition def: bigraph independence. Let $v,w\in\mathcal{V}$ with $v\neq w$. Then $A_v$ and $A_w$ are:

Figures (3)

  • Figure 1: On the left, example of a bigraph $\mathcal{G}$ : edges of type $1$ are drawn as solid black edges and edges of type $2$ are drawn as dashed edges. On the right : partitions in the leftmost column belong to $\mathcal{P}(c,\mathcal{G})$, those in the rightmost column do not belong to $\mathcal{P}(c,\mathcal{G})$.
  • Figure 2: Relation of the $\mathcal{G}$-independences to other independences. We have assumed $\mathcal{E}_2 \subset \mathcal{E}_1\cap\bar{\mathcal{E}}_1$.
  • Figure 3: An unfinished partition. Unfinished blocks are pictured with a segment extending to the left.

Theorems & Definitions (87)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4: Bigraph independence
  • Proposition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 77 more