size ${L_{c}}\ll {\mathcal{L}_{\phi}}$ will behave completely different from one of size ${L_{c}}\sim {\mathcal{L}_{\phi}}$.

In the following, we discuss these two scenarios sketched schematically in Fig. \[fig:sketch\_scenarios\]B.

For scenario I with ${L_{c}}\ll {\mathcal{L}_{\phi}}$ (called quasi-equilibrium in the following) the phase is correlated over the entire condensate region, and disorder induces mainly density fluctuations. As discussed above, the impact of disorder will decrease with increasing density, which should be directly observable by increasing the excitation power. Such kind of percolation transition from a disordered to an ordered regime was predicted (for a polariton BEC in equilibrium) in Ref. .

In the presence of gain and loss disorder induces phase fluctuations as explained above. For scenario II we assume that the phase correlation length ${\mathcal{L}_{\phi}}$ is comparable to the condensate size ${L_{c}}$, i.e. ${\mathcal{L}_{\phi}}\sim {L_{c}}$, such that spatial correlations and superfluidity are destroyed. The ratio ${L_{c}}/{\mathcal{L}_{\phi}}\propto (V_0 \ \xi_{{\mathrm{P}}} \xi_{{\mathrm{V}}} m / \hbar^2) \: (\Gamma / U)$ does not depend on the condensate density and, thus, is independent of the excitation power. A similar conclusion holds for the ratio ${L_{c}}/ {\mathcal{L}_{s}}$. As a consequence, a condensate stabilization with increasing density, as present in an equilibrium system, is strongly suppressed.

![(color online) Expectation value 