  We present a detailed analysis of the stability of synchronized solutions to
the Kuramoto system of oscillators. We derive an analytical expression counting
the dimension of the unstable manifold associated to a given stationary
solution. From this we are able to derive a number of consequences, including:
analytic expressions for the first and last frequency vectors to synchronize,
upper and lower bounds on the probability that a randomly chosen frequency
vector will synchronize, and very sharp results on the large $N$ limit of this
model. One of the surprises in this calculation is that for frequencies that
are Gaussian distributed the correct scaling for full synchrony is not the one
commonly studied in the literature---rather, there is a logarithmic correction
to the scaling which is related to the extremal value statistics of the random
frequency vector.
