Universal Central Limit Theorem for non-exchangeable interacting diffusions

We study non-exchangeable interacting diffusions with pairwise interaction strengths encoded by a sequence of matrices.

Under suitable structural and denseness conditions on these matrices, we prove a universal Central Limit Theorem for the global fluctuation field.

As the number of particles $n$ becomes large, it converges in distribution to the unique solution of a stochastic partial differential equation (SPDE), the same Gaussian limit as in the exchangeable mean field case.

The result applies, for instance, to scaled adjacency matrices of $m_n$-regular graphs when $m_n/\sqrt{n}\to\infty$.

A spatial interaction model shows that the $n^{-1/2}$ denseness threshold is sharp.

The proof proceeds with an analysis in negative Sobolev spaces, building on sharp quantitative propagation of chaos results together with functional inequalities.