such that the expectation values $\mu_P(k)$ and $\sigma^2_P(k)$ depend on the magnitude $k = |\vec{k}|$ of wavevector only.

In Fig. \[fig:IkMeanVar\], the results for $\mu_P$ and $\sigma_P^2$ are shown for scenario I (left panels) and II (right panels). We find that the intensity $I_P$ vanishes for all wavevectors outside of the lower-polariton dispersion ( $k \gtrsim \xi^{-1}$) and that its average value does not change qualitatively as compared to a disorder-free system (cf. Ref. ). However, for a single snap-shot (see Fig. \[fig:snapshot\]) disorder breaks the radial symmetry and induces intensity fluctuations proportional to $\sigma_P$. For scenario I and for wavevectors $|k| \lesssim \xi^{-1}$, these fluctuations decay linearly with inverse excitation power, in agreement with the expectation $\sigma^2_P \sim \kappa^2 \propto 1/P$ for $\kappa \ll 1$. We note that regions with $k \approx \xi^{-1}$ show a high ratio $\sigma_P / \sigma_{\! P_{0}}$ (peaks in Fig. \[fig:IkMeanVar\] lower left panel). In this $k$-region, the emission intensity is increasing very rapidly with excitation power (see Fig. \[fig:IkMeanVar\] upper left panel), because of the repulsive potential hill created by the finite excitation spot. [@Wouters.2008] Thus, the increase of fluctuation strengths with excitation power for $k \approx \xi^{-1}$ is really due to the increase 