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The Gaussian free field (GFF) is considered in the background of randomiso-height islands which is modeled by the site percolation with the occupationprobability $p$. To realize GFF, we consider the Poisson equation in thepresence of normal distributed white-noise charges, as the stationary state ofthe Edwards-Wilkinson (EW) model. The iso-potential (metallic in theterminology of the electrostatic problem) sites are chosen over the latticeaccording to the percolation problem, giving rise to some metallic islands andsome active (not metallic, nor surrounded by a metallic island) area. We seethat the dilution of the system by incorporating metallic particles (orequivalently considering the iso-height islands) annihilates the spatialcorrelations and also the potential fluctuations. Some local and globalcritical exponents of the problem are reported in this work. The GFF, whensimulated on the active area show a cross over between two regimes: small (UV)and large (IR) scales. Importantly, by analyzing the change of exponents (inand out of the critical occupation $p_c$) under changing the system size andthe change of the cross-over points, we find two fixed points and propose thatGFF$_{p=p_c}$ is unstable towards GFF$_{p=1}$.

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