Bigraph independence : a mixture of the five natural independences

TL;DR

This paper introduces a new framework called bigraph independence that unifies five fundamental types of non-commutative independence using a graph-based structure, with applications to random matrix models.

Contribution

It defines bigraph independence, unifying various independence notions, and provides combinatorial formulas, a Hilbert space construction, and asymptotic analysis in random matrix models.

Findings

Bigraph independence encompasses tensor, free, monotone, Boolean, and their combinations.

Explicit combinatorial moment formulas are derived.

Connections to random matrix models and asymptotic behaviors are established.

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 $\epsilon$ or $\Lambda$-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.