$3$                                       $5$                                  $3$                            $0.4$

  : Parameters extracted from experiment and corresponding parameters used for simulations as well as relevant length scales. For definitions see Appendix \[sec:theory\_detailsmodel\].[]{data-label="tab:parameters"}

![(color online) Snap shots of the normalized two-dimensional intensity distribution $I_P(\vec{k})$ of a polariton BEC with ${\mathcal{L}_{\phi}}\sim {L_{c}}$ in a disordered environment for increasing excitation power $P$. The used parameters are presented in Tab. \[tab:parameters\]. []{data-label="fig:snapshot"}](./{fig_Snap_Ik2d_L120_cP3_ckappa0.125_alpha7_snap8}.pdf){width="0.95\hsize"}

For a typical disorder realization, a series of numerically obtained snap shots of the two-dimensional intensity distribution $I_P(\vec{k})$ for increasing excitation power is shown in Fig. \[fig:snapshot\]. These images correspond to a polariton BEC described by scenario II. We clearly observe a disorder-induced deviation from the ideally radial distribution, which does not converge to a symmetric intensity distribution while increasing the excitation power. Such an asymmetry as well as its persistence is also observed experimentally, see Fig. \[fig:Disorder\], and thus agrees qualitatively with our simulations. We note that the experimental data represent the intensity distribution of one disordered sample, and correspond to a one-dimensional cut along a given line crossing the origin of the two-dimensional $k$-space distribution, for example the $x$-axis.

For a quantitative analysis we compare directly the experimental measurements with the numerically computed expectation value $\mu_{{\mathrm{P}}}$ 