disorder effects even at high densities. Numerical simulations allow to compare with experimental data confirming our theoretical predictions.

For an equilibrium BEC our observations would be unexpected, since an increasing density screens the disorder potential and leads to an ordered superfluid state [@Nattermann.2008; @Falco.2009; @Malpuech.2007]. Analogously, for a polariton BEC, interactions also can lead to superfluidity, as observed in clean samples [@Amo.2009; @Amo.2009b; @Sanvitto.2010]. However, as mentioned above, in the presence of disorder the polariton BEC is strictly speaking not a superfluid and long-range order is destroyed. [@Janot.2013] Thus, we expect and observe that disorder affects a dissipative polariton BEC much more than an equilibrium one. Several further observations found in literature seem to support this. For example, in one-dimensional CdTe MCs [@Manni.2011; @Stepnicki.2013] and ZnO MCs [@Trichet.2013] the spatial first-order correlation function of polariton BEC emission was analyzed in the presence of disorder and significant changes due to disorder were found. In the CdTe MCs the disorder effects remain present even with increasing excitation power, similarly to our findings in two-dimensional ZnO MCs. We note that the correlation length of the assumed disorder potential discussed in Ref.  is of the order of microns, which enables the trapping of the 