Graphon-Theoretic Approach to Central Limit Theorems for $\epsilon$-Independence

We establish a central limit theorem for the sum of $\epsilon$-independent random variables, extending both the classical and free probability setting. Central to our approach is the use of graphon limits to characterize the limiting distribution, which depends on the asymptotic structure of the underlying graphs governing $\epsilon$-independence. This framework yields a range of exotic limit laws, interpolating between free and classical cases and allowing for mixtures such as free and classical convolutions of the semi-circle and Gaussian distributions. We provide a complete characterization of the limiting law, captured as the distribution of an operator on the full Fock space. We extend our main result to the multivariate setting as well as the non-identically distributed case. The proof provides insights into the combinatorial structure of $\epsilon$-independence, shedding light on a natural connection between the graph of crossings of the partitions involved and the graphon limit of the graphs of $\epsilon$-independence.