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