$  0.  $\lim_{\varepsilon \rightarrow 0^+} \Delta E^{H^\varepsilon}_{\eta^\varepsilon} (\gamma^\varepsilon_x) = \Delta E^{H^0}_\eta (\gamma_x)$ for every $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$.  Then each diluted model $(\nu^\varepsilon,H^\varepsilon)$ admits exactly one Gibbs measure $\mu^\varepsilon$ and as $\varepsilon \rightarrow 0^+$ $$\mu^{\varepsilon} \overset{