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We consider a system of $N$ disordered mean-field interacting diffusionswithin 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 $| x_i-x_j|^{-\alpha}$ for $\alpha\in[0,1)$. In aprevious work, it was shown that the empirical measure of the particlesconverges in large population to the solution of a nonlinear partialdifferential equation of McKean-Vlasov type. The purpose of the present paperis to study the fluctuations associated to this convergence. We exhibit inparticular a phase transition in the scaling and in the nature of thefluctuations: when $\alpha\in[0,\frac{1}{2})$, the fluctuations are Gaussian,governed by a linear SPDE, with scaling $\sqrt{N}$ whereas the fluctuations aredeterministic with scaling $N^{1-\alpha}$ in the case$\alpha\in(\frac{1}{2},1)$.

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