Scaling Limit of the Kuramoto Model on Random Geometric Graphs

Abstract

We consider the Kuramoto model on a graph with nodes given by $n$ i.i.d. points uniformly distributed on the $d$ dimensional torus. Two nodes are declared neighbors if they are at distance less than $\epsilon$. We prove a scaling limit for this model in compact time intervals as $n\to\infty$ and $\epsilon \to 0$ such that $\epsilon^{d+2}n/\log n \to \infty$. The limiting object is given by the heat equation. On the one hand this shows that the nonlinearity given by the sine function disappears under this scaling and on the other hand, provides evidence that stable equilibria of the Kuramoto model on these graphs are, as $n\to\infty$, in correspondence with those of the heat equation, which are explicit and given by twisted states. In view of this, we conjecture the existence of twisted stable equilibria with high probability as $n\to \infty$.