  The synchronization of oscillator ensembles is pervasive throughout nonlinear
science, from classical or quantum mechanics to biology, to human assemblies.
Traditionally, the main focus has been the identification of threshold
parameter values for the transition to synchronization as well as the nature of
such transition. Here, we show that considering an oscillator lattice as a
discrete growing interface provides unique insights into the dynamical process
whereby the lattice reaches synchronization for long times. Working on a
generalization of the Kuramoto model that allows for odd or non-odd couplings,
we elucidate synchronization of oscillator lattices as an instance of generic
scale invariance, whereby the system displays space-time criticality, largely
irrespective of parameter values. The critical properties of the system (like
scaling exponent values and the dynamic scaling Ansatz which is satisfied) fall
into universality classes of kinetically rough interfaces with columnar
disorder, namely, those of the Edwards-Wilkinson (equivalently, the Larkin
model of an elastic interface in a random medium) or the Kardar-Parisi-Zhang
(KPZ) equations, for Kuramoto (odd) coupling and generic (non-odd) couplings,
respectively. From the point of view of kinetic roughening, the critical
properties we find turn out to be quite innovative, especially concerning the
statistics of the fluctuations as characterized by their probability
distribution function (PDF) and covariance. While the latter happens to be that
of the Larkin model irrespective of the symmetry of the coupling, in the
generic non-odd coupling case the PDF turns out to be the Tracy-Widom
distribution associated with the KPZ nonlinearity. This brings the
synchronization of oscillator lattices into a remarkably large class of
strongly-correlated, low-dimensional (classical and quantum) systems with
strong universal fluctuations.
