Gaussian Free Field in the iso-height random islands tuned by percolation model

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$_{p=1}$.