  We investigate the appearance of spontaneous coherence in the parametric
emission from planar semiconductor microcavities in the strong coupling regime.
Calculations are performed by means of a Quantum Monte Carlo technique based on
the Wigner representation of the coupled exciton and cavity-photon fields. The
numerical results are interpreted in terms of a non-equilibrium phase
transition occurring at the parametric oscillation threshold: below the
threshold, the signal emission is incoherent, and both the first and the
second-order coherence functions have a finite correlation length which becomes
macroscopic as the threshold is approached. Above the threshold, the emission
is instead phase-coherent over the whole two-dimensional sample and intensity
fluctuations are suppressed. Similar calculations for quasi-one-dimensional
microcavities show that in this case the phase-coherence of the signal emission
has a finite extension even above the threshold, while intensity fluctuations
are suppressed.
