Energy-Weighted Site Percolation in Two Dimensions

TL;DR

This paper investigates a generalized 2D site percolation model with energy-dependent bond costs, revealing how energy influences percolation thresholds, cluster structures, and critical exponents through simulations and renormalization-group analysis.

Contribution

It introduces an energy-weighted percolation model, demonstrating continuous shifts in thresholds and exponents, and explores the effects of bond energy on cluster formation and phase transitions.

Findings

Bond energy shifts percolation threshold smoothly.

Correlation length remains finite at classical threshold and decreases with energy.

Critical exponent ν varies from 1/2 to 4/3, approaching 1 at high energy.

Abstract

We study a generalization of two-dimensional site percolation by assigning an energy cost $\varepsilon$ to bonds between nearest-neighbor occupied sites. This leads to a competition between entropy-driven cluster growth and energetic suppression (or enhancement) of connectivity. Varying $\varepsilon$ continuously interpolates between dense ferromagnetic-like clusters, ordinary classical percolation, and a dilute regime of minimally connected isolated clusters. Using Monte Carlo simulations and real-space renormalization-group (RG) methods, we show that bond energy shifts the percolation threshold smoothly. We define an energy-weighted correlation length that remains finite at the classical site occupation threshold ($p_c(\varepsilon=0)$) and shrinks with increasing $\varepsilon$, capturing the energetic suppression of large-scale connectivity. The cluster size distribution exhibits an energy-dependent cutoff that drives the transition from percolation-like clusters to isolated clusters. A real-space RG with Kadanoff block recursions reveals a systematic evolution of the correlation-length exponent $\nu$ from $\nu=1/2$ (dense clusters) to $\nu=4/3$ (classical percolation), approaching $\nu=1$ (minimally connected isolated clusters), in agreement with Coulomb-gas predictions for loop models where bond energy renormalizes loop fugacity. For large values of \(\varepsilon\) (isotropic case), the suppression of nearest-neighbor bonds results in the emergence of antiferromagnetic sub-lattice ordering at high densities. Additionally, anisotropic bond energies lead to directionally selective cluster growth. Finally, we also discuss a lattice gas RG approach and scenarios where bond energy is renormalized across different scales.