Volume bounds for the phase-locking region in the kuramoto model

The Kuramoto model is a fundamental model for the study of phase locking in systems of coupled oscillators. It is well understood that when the natural frequencies of the oscillators are close together the oscillators will phase lock; conversely when the frequencies are spread out then phase locking will not occur. Much of the work on this model has focused on understanding the set of frequencies for which phase locking occurs. In this paper we consider the problem of computing the volume of the set of frequencies that exhibit phase locking. We prove upper and lower bounds on the measure of the set of frequencies that exhibit phase locking in terms of a weighted sum of spanning trees in the oscillator network. In the case where all of the edge weights are equal this shows that the number of spanning trees is in some sense a useful proxy for the volume of the phase-locking set. For families of dense graphs, where the number of edges is proportional to the square of the number of vertices, we further show that the controlling asymptotics of the volume of the synchronization region is given by the number of spanning trees.