  The Gaussian free field (GFF) is considered in the background of random
iso-height islands which is modeled by the site percolation with the occupation
probability $p$. To realize GFF, we consider the Poisson equation in the
presence of normal distributed white-noise charges, as the stationary state of
the Edwards-Wilkinson (EW) model. The iso-potential (metallic in the
terminology of the electrostatic problem) sites are chosen over the lattice
according to the percolation problem, giving rise to some metallic islands and
some active (not metallic, nor surrounded by a metallic island) area. We see
that the dilution of the system by incorporating metallic particles (or
equivalently considering the iso-height islands) annihilates the spatial
correlations and also the potential fluctuations. Some local and global
critical exponents of the problem are reported in this work. The GFF, when
simulated on the active area show a cross over between two regimes: small (UV)
and large (IR) scales. Importantly, by analyzing the change of exponents (in
and out of the critical occupation $p_c$) under changing the system size and
the change of the cross-over points, we find two fixed points and propose that
GFF$_{p=p_c}$ is unstable towards GFF$_{p=1}$.
