Sec. \[sec:conclusion\].

Theoretical Predictions {#sec:theory_disorder}
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Model {#sec:theory_model}
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A phenomenological description of the dynamics of the polariton condensate wave function $\Psi(\vec{x},t)$ is given by an extended Gross Pitaevskii equation (eGPE) [@Wouters.2007; @Keeling.2008] $$\begin{aligned}
    \label{eq:eGPE}
    {\mathrm{i}}\hbar {\partial_{t}} \Psi = &\left( -\frac{\hbar^2}{2 m} \vec{\nabla}^2 + V(\vec{x}) + U \left|\Psi\right|^2 \right)\Psi \\
    \nonumber
        &+ {\mathrm{i}}\left(R(\vec{x}) - \Gamma \left|\Psi\right|^2\right) \Psi \ ,\end{aligned}$$ where $m$ is the effective mass of the lower polariton branch, $V$ an external potential and $U>0$ an onsite interaction constant. The function $R(\vec{x})$ describes the linear part of gain and loss due to inscattering from a reservoir of non-condensed polaritons and the finite lifetime of the condensate. The non-linearity $\Gamma |\Psi|^2$ implements a density dependent gain saturation with $\Gamma$ as gain depletion constant. Since the propagation of the reservoir polaritons can be neglected, the spatial shape of $R(\vec{x})$ can be related to the Gaussian profile of the excitation laser, namely $$\begin{aligned}
    \label{eq:reservoirfct}
    R(\vec{x}) = \hbar \gamma_{{\mathrm{c}}} \left(\frac{P}{P_{{\mathrm{th}}}} e^{-\vec{x}^2/\xi_P^2} - 1 \right) \ .\end{aligned}$$ The parameter $\gamma_{{\mathrm{c}}}$ is the condensate decay rate (inverse lifetime $\gamma_{{\mathrm{c}}} = 1/\tau$). The ratio $P/P_{{\mathrm{th}}}$ is the excitation power versus its value at threshold ${P_{{\mathrm{th}}}}$ at which condensation is observed first, and $\xi_P$ is the waist 