Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction

We consider a system of $N$ disordered mean-field interacting diffusions within spatial constraints: each particle $θ_i$ is attached to one site $x_i$ of a periodic lattice and the interaction between particles $θ_i$ and $θ_j$ decreases as $| x_i-x_j|^{-α}$ for $α\in[0,1)$. In a previous work, it was shown that the empirical measure of the particles converges in large population to the solution of a nonlinear partial differential equation of McKean-Vlasov type. The purpose of the present paper is to study the fluctuations associated to this convergence. We exhibit in particular a phase transition in the scaling and in the nature of the fluctuations: when $α\in[0,\frac{1}{2})$, the fluctuations are Gaussian, governed by a linear SPDE, with scaling $\sqrt{N}$ whereas the fluctuations are deterministic with scaling $N^{1-α}$ in the case $α\in(\frac{1}{2},1)$.

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