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

TL;DR

This paper investigates the fluctuation behavior of mean-field diffusions with spatial interactions, revealing a phase transition from Gaussian to deterministic fluctuations depending on the decay rate of interactions.

Contribution

It characterizes the phase transition in fluctuation types and scalings for spatially constrained mean-field diffusions with power-law interactions.

Findings

Gaussian fluctuations for , governed by a linear SPDE

Deterministic fluctuations for , with scaling N^{1-\u03b1}

Identification of a phase transition in fluctuation behavior

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