Limit joint distributions of SYK models with partial interactions, mixed Q-Gaussian models and asymptotic ε-freeness

We study the joint distribution of the SYK Hamiltonians for different systems with specified overlaps. We demonstrate that, in the large-system limit, the joint distribution converges in moments to a mixed q-Gaussian system. Furthermore, we explain that the graph product of diffusive abelian von Neumann algebras is isomorphic to a W * -probability space generated by the corresponding ε-freely independent random variables with semicircular laws. Because these variables form a special case of mixed q-Gaussian systems, they can be approximated by our SYK Hamiltonian models. As a byproduct, we obtain Gaussian random matrix models for asymptotic ε-freeness.