size of the Gaussian pump spot. We note that for the case of a spatially homogeneous excitation the eGPE  was successfully used to analyze a driven dissipative condensate. [@Sieberer.2013; @Altman.2013]

Because of interactions, the condensate energy is blueshifted by $n_0U$ where $n_0$ is the mean condensate density determined by the balance of gain and loss (for a definition of $n_0$ see [Eq. ]{}). The healing length $\xi \equiv \hbar / \sqrt{2 m n_0 U}$ is obtained by comparing kinetic and interaction energy of [Eq. ]{}.

The disordered environment is described by a random potential $V(\vec{x})$. We choose Gaussian-distributed delta-correlated disorder with zero mean and variance $\xi_V^2 V_0^2$, see Appendix \[sec:theory\_detailsmodel\] for details. We introduce an effective dimensionless disorder parameter, $$\begin{aligned}
  \label{eq:kappa}
  \kappa \equiv \frac{\xi_V \: V_0}{\xi\: n_0U} \ .\end{aligned}$$ An analysis of the gain and loss terms in [Eq. ]{} allows us to define a ’non-equilibrium parameter’ $$\begin{aligned}
  \label{eq:alpha}
  \alpha \equiv \frac{\Gamma}{U} \ .\end{aligned}$$ Its magnitude parametrizes the influence of gain and loss on the polariton BEC. For example, in the limit $\alpha \to 0$ (keeping $n_0$ finite) the equilibrium mean field description of a BEC is obtained, and, on the other hand, in the limit $\alpha \to \infty$ the 