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**Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction.**
*(English)*
Zbl 1358.60104

Summary: 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 [the authors, ibid. 24, No. 5, 1946–1993 (2014; Zbl 1309.60096)], 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 \(\alpha \in (\frac{1}{2},1)\).

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60J60 | Diffusion processes |

60G15 | Gaussian processes |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60F05 | Central limit and other weak theorems |

60G57 | Random measures |

82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |