  We present a renormalization group study of two dimensional arrays of
oscillators, with dissipative, short range interactions. We consider the case
of non-identical oscillators, with distributed intrinsic frequencies within the
array and study the steady-state properties of the system. In two dimensions no
macroscopic mutual entrainment is found but, for identical oscillators,
critical behavior of the Berezinskii-Kosterlitz-Thouless type is shown to be
present. We then discuss the stability of (BKT) order in the physical case of
distributed quenched random frequencies. In order to do that, we show how the
steady-state dynamical properties of the two dimensional array of non-identical
oscillators are related to the equilibrium properties of the XY model with
quenched randomness, that has been already studied in the past. We propose a
novel set of recursion relations to study this system within the Migdal
Kadanoff renormalization group scheme, by mean of the discrete clock-state
formulation. We compute the phase diagram in the presence of random dissipative
coupling, at finite values of the clock state parameter. Possible experimental
applications in two dimensional arrays of microelectromechanical oscillators
are briefly suggested.
