  We apply a recently developed renormalization group (RG) method to study
synchronization in a one-dimensional chain of phase-coupled oscillators in the
regime of weak randomness. The RG predicts how oscillators with randomly
distributed frequencies and couplings form frequency-synchronized clusters.
Although the RG was originally intended for strong randomness, i.e. for
distributions with long tails, we find good agreement with numerical
simulations even in the regime of weak randomness. We use the RG flow to derive
how the correlation length scales with the width of the coupling distribution
in the limit of large coupling. This leads to the identification of a
universality class of distributions with the same critical exponent $\nu$. We
also find universal scaling for small coupling. Finally, we show that the RG
flow is characterized by a universal approach to the unsynchronized fixed
point, which provides physical insight into low-frequency clusters.
