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

TL;DR

This paper investigates the joint distributions of SYK Hamiltonians with overlaps, showing their convergence to mixed q-Gaussian systems and providing a random model for asymptotic ε-freeness in large systems.

Contribution

It introduces a novel connection between SYK models, mixed q-Gaussian systems, and ε-freeness, expanding understanding of their joint distributions in the large-system limit.

Findings

Joint distribution of SYK Hamiltonians converges to mixed q-Gaussian systems.

Graph product of diffusive abelian von Neumann algebras is isomorphic to a W*-probability space.

Provides a random model for asymptotic ε-freeness.

Abstract

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.