$\mu_P$ (upper row) and variance $\sigma^2_P$ (lower row) of the (normalized) intensity distribution $I_P(k)$. In order to compare the fluctuations for increasing excitation powers we present the ratio $\sigma^2_P(k) / \sigma^2_{P_0}(k)$ with $P_0 = 4 P_{{\mathrm{th}}}$. The left and right column depict a quasi-equilibrium (scenario I) and a driven dissipative (scenario II) condensate, respectively. For wavevectors $|k \xi| \lesssim 1$, scenario I shows a linear reduction of fluctuations with inverse excitation power, $\sigma^2_P(k) \propto 1/P$, while scenario II exhibits a suppressed stabilization with increasing excitation power. We averaged $1560$ disorder realizations and used: ${\mathcal{L}_{\mathrm{n}}}/{L_{c}}= 1 \: (4.5)$, ${\mathcal{L}_{\phi}}/{L_{c}}= 10 \: (2.5)$, ${\mathcal{L}_{s}}/{L_{c}}= 20 \: (0.3)$ for scenario I (II). []{data-label="fig:IkMeanVar"}](./{fig_MeanVar_Ik_cP4_ckappa0.4_0.1_alpha0.5_8_beta04}.pdf){width="1\hsize"}

In order to make our analysis more quantitative, we have studied theoretically the excitation power dependence of the two-dimensional $k$-space intensity $I_P({\vec{k}}) \propto \gamma_c |\Psi_{\vec{k}}|^2$ which can be directly compared to experimental data. To this end, [Eq. ]{} was simulated for many disorder realizations (see Appendix \[sec:theory\_numsim\] for details). We have extracted the expectation value, denoted by $\mu_P(k)$, and the variance, denoted by $\sigma^2_P(k)$, of the normalized intensity $I_P(\vec{k})$ by averaging over disorder configurations. We note that for a sufficiently large number of realizations, the disorder average restores radial symmetry, 