Limit joint distributions of SYK Models with partial interactions, Mixed q-Gaussian Models and Asymptotic $arepsilon$-freeness

We study the joint distribution of SYK Hamiltonians for different systems with specified overlaps. We show that, in the large-system limit, their joint distribution converges in distribution to a mixed $q$-Gaussian system. We explain that the graph product of diffusive abelian von Neumann algebras is isomorphic to a $W^*$-probability space generated by the corresponding $\varepsilon$-freely independent random variables with semicircular laws which form a special case of mixed $q$-Gaussian systems that can be approximated by our SYK Hamiltonian models. Thus, we obtain a random model for asymptotic $\varepsilon$-freeness.

💡 Research Summary

The paper investigates the joint distribution of SYK (Sachdev‑Ye‑Kitaev) Hamiltonians belonging to several quantum systems that may share a prescribed amount of overlap. For each system (k) a family of Majorana fermions ({\psi_i}_{i=1}^n) is fixed and a random Hamiltonian is defined by
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