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

TL;DR

This paper studies the Gaussian free field in a system with random iso-height islands modeled by percolation, revealing how metallic island dilution affects correlations, fluctuations, and critical behavior, with implications for understanding phase transitions.

Contribution

It introduces a novel analysis of GFF on percolation-based iso-height islands, identifying fixed points and the instability of the critical percolation state.

Findings

Dilution reduces spatial correlations and potential fluctuations.

Identification of two fixed points in the system's phase space.

GFF at critical percolation is unstable towards the fully metallic state.

Abstract

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$_{p=1}$.