Quenched Limits and Fluctuations of the Empirical Measure for Plane Rotators in Random Media.

Abstract

The Kuramoto model has been introduced to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc. The model consists of $N$ interacting oscillators on the one dimensional sphere $S^1$, driven by independent Brownian Motions with constant drift chosen at random. This quenched disorder is chosen independently for each oscillator according to the same law $\mu$. The behaviour of the system for large $N$ can be understood via its empirical measure: we prove here the convergence as $N\to\infty$ of the quenched empirical measure to the unique solution of coupled McKean-Vlasov equations, under weak assumptions on the disorder $\mu$ and general hypotheses on the interaction. The main purpose of this work is to address the issue of quenched fluctuations around this limit, motivated by the dynamical properties of the disordered system for large but fixed $N$: hence, the main result of this paper is a quenched Central Limit Theorem for the empirical measure. Whereas we observe a self-averaging for the law of large numbers, this no longer holds for the corresponding central limit theorem: the trajectories of the fluctuations process are sample-dependent.