  We show that the synchronization transition of a large number of noisy
coupled oscillators is an example for a dynamic critical point far from
thermodynamic equilibrium. The universal behaviors of such critical
oscillators, arranged on a lattice in a $d$-dimensional space and coupled by
nearest neighbors interactions, can be studied using field theoretical methods.
The field theory associated with the critical point of a homogeneous
oscillatory instability (or Hopf bifurcation of coupled oscillators) is the
complex Ginzburg-Landau equation with additive noise. We perform a perturbative
renormalization group (RG) study in a $4-\epsilon$ dimensional space. We
develop an RG scheme that eliminates the phase and frequency of the
oscillations using a scale-dependent oscillating reference frame. Within a
Callan-Symanzik RG scheme to two-loop order in perturbation theory, we find
that the RG fixed point is formally related to the one of the model $A$
dynamics of the real Ginzburg-Landau theory with an O(2) symmetry of the order
parameter. Therefore, the dominant critical exponents for coupled oscillators
are the same as for this equilibrium field theory. This formal connection with
an equilibrium critical point imposes a relation between the correlation and
response functions of coupled oscillators in the critical regime. Since the
system operates far from thermodynamic equilibrium, a strong violation of the
fluctuation-dissipation relation occurs and is characterized by a universal
divergence of an effective temperature. The formal relation between critical
oscillators and equilibrium critical points suggests that long-range phase
order exists in critical oscillators above two dimensions.

Statistical Mechanics;Biological Physics