Statistics of coherent subsets in the Kuramoto model for uniform distributions

Experiments in synchronization and simulations of the Kuramoto model reveal that small subsets of oscillators tend to synchronize into coherent clumps at moderate coupling strengths below the critical coupling strength. Such clumps are not present in the thermodynamic limit, and so are intrinsically a finite size effect. We develop a theory that predicts the critical coupling for the formation of individual coherent subsets. Applying these new conditions to a finite population sampled from a uniform distribution, we obtain known results, such as a discontinuous transition to synchrony. We also obtain a lower estimate for the critical coupling of this transition that is consistent with simulation and previous results for uniformly sampled oscillators. This opens a new route for computing finite size effects, which we will present.