  The influence of polariton-polariton scattering on the statistics of the
polariton condensate in a non-resonantly excited semiconductor quantum well
embedded in a CdTe semiconductor microcavity is discussed. Taking advantage of
the existence of a bottleneck in the polariton dispersion curve, the polariton
states are separated into two domains: reservoir polaritons inside the
bottleneck and active polaritons with wave vector q whose energy lies below the
bottleneck. In the framework of the master equation formalism, the
non-equilibrium stationary reduced density matrix is calculated and the
statistics of polaritons in the condensate at q=0 is determined. The anomalous
correlations between the polaritons in the condensate and those with wave
vectors q, -q leads to an enhancement of the noise in the condensate. As a
consequence, the second order correlation function of the condensate does not
show the full coherence that is characteristic of laser emission.
