Energy-Weighted Site Percolation in Two Dimensions

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.