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

Abstract

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 $\vert x_{i}-x_{j}\vert^{-\alpha}$ for $\alpha\in[0,1)$. In a previous work [Ann. Appl. Probab. 24 (2014) 1946–1993], 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 $\alpha\in(\frac{1}{2},1)$.