Matrix models for cyclic monotone and monotone independences

TL;DR

This paper introduces a unified matrix model that captures both cyclic monotone and monotone independence in noncommutative probability, revealing their close algebraic relationship.

Contribution

It demonstrates that the same random matrix model can represent both cyclic monotone and monotone independence, unifying their conceptual framework.

Findings

The model provides a nonrandom matrix representation for both independences.

Cyclic monotone and monotone independence are closely related algebraically.

The approach simplifies understanding of noncommutative independence structures.

Abstract

Cyclic monotone independence is an algebraic notion of noncommutative independence, introduced in the study of multi-matrix random matrix models with small rank. Its algebraic form turns out to be surprisingly close to monotone independence, which is why it was named cyclic monotone independence. This paper conceptualizes this notion by showing that the same random matrix model is also a model for the monotone convergence with an appropriately chosen state. This observation provides a unified nonrandom matrix model for both types of monotone independences.