  We adopt a kinetic theory of polariton non-equilibrium Bose-Einstein
condensation, to describe the formation of off-diagonal long-range order. The
theory accounts properly for the dominant role of quantum fluctuations in the
condensate. In realistic situations with optical excitation at high energy, it
predicts a significant depletion of the condensate caused by long-wavelength
fluctuations. As a consequence, the one-body density matrix in space displays a
partially suppressed long-range order and a pronounced dependence on the finite
size of the system.
