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 ${\theta }_i$ is attached to one site $x_i$ of a periodic lattice and the interaction between particles ${\theta }_i$ and ${\theta }_j$ decreases as $\mathrm{\mid }{x}_i-x_j{\mathrm{\mid }}^{-\alpha }$ for $\alpha \in [0,1)$. 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 $\alpha \in [0,\frac{1}{2})$, the fluctuations are Gaussian, governed by a linear SPDE, with scaling $\sqrt{N}$ whereas the fluctuations are deterministic with scaling $N^{1-\alpha }$ in the case $\alpha \in (\frac{1}{2},1)$.