Matrix models for $\varepsilon$-free independence

TL;DR

This paper demonstrates that tensor products of random matrices with independent entries asymptotically exhibit -free independence, linking matrix models to a mixture of classical and free independence, and provides a new proof for -embeddability preservation under graph products.

Contribution

It introduces a novel connection between tensor product structures of random matrices and -free independence, offering explicit matrix models and a new proof for -embeddability preservation.

Findings

Asymptotic -free independence from tensor products of random matrices.

Explicit construction of matrix models realizing arbitrary -independence.

New proof that -embeddability is preserved under graph products.

Abstract

We investigate tensor products of random matrices, and show that independence of entries leads asymptotically to $\varepsilon$-free independence, a mixture of classical and free independence studied by M{\l}otkowski and by Speicher and Wysocza\'nski. The particular $\varepsilon$ arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary $\varepsilon$ may be realized in this way. As a result we obtain a new proof that $\mathcal{R}^\omega$-embeddability is preserved under graph products of von Neumann algebras, along with an explicit recipe for constructing matrix models.