Scaling Limit of the Kuramoto Model on Random Geometric Graphs
Francisco Cirelli, Pablo Groisman, Ruojun Huang, Hernán Vivas
Source abstract
Abstract. We consider the Kuramoto model on a graph with nodes given by [Formula: see text] i.i.d. points uniformly distributed on the [Formula: see text] dimensional torus. Two nodes are declared neighbors if they are at distance less than [Formula: see text]. We prove a scaling limit for this model in compact time intervals as [Formula: see text] and [Formula: see text] such that [Formula: see text]. 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 [Formula: see text], 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 [Formula: see text].
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