| Abstract: |
| We'll consider the energy function of the Kuramoto model in random geometric graphs on a given manifold, with particular focus on the d-dimensional torus and the 2-sphere. As the number of nodes diverges, with high probability there are no (smooth) local minima other than the phase-locked state in the case of the sphere, while in the torus there is at least one local minimum for each homotopy class of continuous functions from the manifold to the circle. This is proved for d = 1 and conjectured for every d > 1. We'll then specialize to the cycle graph, where we can go beyond existence and describe the volume and geometry of the basins of attraction of each twisted state, shedding light on the long-time behavior of the system. |
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