  The space-time process whereby one-dimensional systems of self-sustained
oscillators synchronize is shown to display generic scale invariance, with
scaling properties characteristic of the Kardar-Parisi-Zhang equation with
columnar noise, and phase fluctuations that follow a Tracy-Widom probability
distribution. This is revealed by a numerical exploration of rings of
Stuart-Landau oscillators (the universal representation of an oscillating
system close to a Hopf bifurcation) and rings of van der Pol oscillators, both
of which are paradigms of self-sustained oscillators. The critical behavior is
very well-defined for limit-cycle oscillations near the bifurcation point, and
still dominates the behavior comparatively far from the bifurcation. In
particular, the Tracy-Widom fluctuation distribution seems to be an extremely
robust feature of the synchronization process. The nonequilibrium criticality
here described appears to transcend the details of the coupled dynamical
systems that synchronize, making plausible its experimental observation.
