  We analyse the spatial and temporal coherence properties of a two-dimensional
and finite sized polariton condensate with parameters tailored to the recent
experiments which have shown spontaneous and thermal equilibrium polariton
condensation in a CdTe microcavity [J. Kasprzak, M. Richard, S. Kundermann, A.
Baas, P. Jeambrun, J.M.J. Keeling, F.M. Marchetti, M.H. Szymanska, R. Andre,
J.L. Staehli, et al., Nature 443 (7110) (2006) 409]. We obtain a theoretical
estimate of the thermal length, the lengthscale over which full coherence
effectively exists (and beyond which power-law decay of correlations in a
two-dimensional condensate occurs), of the order of 5 micrometers. In addition,
the exponential decay of temporal coherence predicted for a finite size system
is consistent with that found in the experiment. From our analysis of the
luminescence spectra of the polariton condensate, taking into account pumping
and decay, we obtain a dispersionless region at small momenta of the order of 4
degrees. In addition, we determine the polariton linewidth as a function of the
pump power. Finally, we discuss how, by increasing the exciton-photon detuning,
it is in principle possible to move the threshold for condensation from a
region of the phase diagram where polaritons can be described as a weakly
interacting Bose gas to a region where instead the composite nature of
polaritons becomes important.
