  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 $| x_i-x_j|^{-\alpha}$ 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
$\alpha\in(\frac{1}{2},1)$.
