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arxiv: 2605.18312 · v1 · pith:C4BLUMPHnew · submitted 2026-05-18 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

Energy-Weighted Site Percolation in Two Dimensions

Sayan Sircar , Kabir Ramola This is my paper

Pith reviewed 2026-05-19 23:55 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn
keywords site percolationenergy-weighted percolationrenormalization groupcorrelation lengthtwo dimensionsMonte Carlo simulationcluster size distributionloop models
0
0 comments X

The pith

Bond energy shifts the percolation threshold smoothly and changes the correlation-length exponent continuously in two dimensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors introduce an energy cost ε for bonds connecting occupied sites in two-dimensional site percolation. This creates a competition where entropy favors cluster growth while energy suppresses connectivity as ε increases. Monte Carlo simulations and real-space renormalization group methods demonstrate that the percolation threshold varies continuously with ε. They introduce an energy-weighted correlation length that remains finite even at the zero-energy percolation threshold and decreases with rising ε, explaining the suppression of large clusters. The correlation length exponent ν evolves systematically from 1/2 in dense regimes through 4/3 in classical percolation toward 1 in dilute isolated-cluster regimes.

Core claim

Assigning an energy cost ε to bonds between nearest-neighbor occupied sites leads to a smooth shift in the percolation threshold p_c(ε). An energy-weighted correlation length is defined that remains finite at the classical site occupation threshold p_c(ε=0) and shrinks with increasing ε. A real-space RG with Kadanoff block recursions reveals a systematic evolution of the correlation-length exponent ν from 1/2 for dense clusters to 4/3 for classical percolation approaching 1 for minimally connected isolated clusters, in agreement with Coulomb-gas predictions for loop models where bond energy renormalizes loop fugacity.

What carries the argument

The energy-weighted correlation length, which quantifies connectivity while incorporating the energetic cost of bonds, together with Kadanoff-block real-space renormalization group recursion relations that evolve the correlation-length exponent ν continuously.

If this is right

  • The cluster size distribution develops an energy-dependent cutoff that drives the transition from percolation-like clusters to isolated clusters.
  • For large isotropic ε the suppression of nearest-neighbor bonds produces antiferromagnetic sub-lattice ordering at high densities.
  • Anisotropic bond energies result in directionally selective cluster growth.
  • A lattice gas RG approach can be used to examine how bond energy renormalizes across different length scales.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This model offers a way to incorporate interaction energies into percolation descriptions of physical networks such as colloidal assemblies or porous media.
  • The continuous interpolation between dense, critical, and dilute regimes suggests similar energy-weighting techniques could be applied to other two-dimensional critical phenomena like loop models or Potts models.
  • The observed exponent evolution may connect to broader questions of how local energetic biases alter universality classes in statistical mechanics.

Load-bearing premise

The real-space renormalization group with Kadanoff block recursions accurately tracks the continuous evolution of the correlation-length exponent without requiring additional parameters specific to the energy term.

What would settle it

Monte Carlo simulations that measure the energy-weighted correlation length diverging at p = p_c(0) for any nonzero ε, instead of remaining finite and shrinking, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.18312 by Kabir Ramola, Sayan Sircar.

Figure 1
Figure 1. Figure 1: FIG. 1. A 4 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The left plot [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase diagram indicating percolating and non [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The plot illustrates the variation in the mean density [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The plot illustrates the density of occupied sites, de [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Panels [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The plot illustrates the scaling collapse of the wrapping probability function [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The plot illustrates the scaling collapse of the susceptibility function [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The plot illustrates the weighted cluster-size distribution for various values of the nearest-neighbor occupied-site bond [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Scaling of correlation length [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The plot shows the ratio of the mean-squared radii [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Cluster-size distributions obtained from Glauber [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Correlation length for site percolation on a square [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Schematic diagram showing two neighboring [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. RG flows around the fixed points, where both the [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Variation of the critical site percolation threshold, [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript studies a generalization of 2D site percolation in which nearest-neighbor occupied sites incur a tunable bond energy ε. Monte Carlo simulations and real-space RG with Kadanoff blocks are used to demonstrate a smooth shift of the percolation threshold with ε, to introduce an energy-weighted correlation length that remains finite at the classical p_c(ε=0) and shrinks with ε, and to report a continuous evolution of the correlation-length exponent ν from 1/2 (dense) through 4/3 (classical) to 1 (dilute), stated to agree with Coulomb-gas loop-model predictions without extra parameters.

Significance. If the RG recursions are shown to be robust and parameter-free, the work supplies a concrete interpolation between dense, percolating, and isolated-cluster regimes together with a new observable (energy-weighted length) that directly encodes energetic suppression of connectivity. The claimed match to Coulomb-gas predictions would be a clear strength provided the implementation details confirm independence from block-size artifacts or energy-specific closures.

major comments (2)
  1. [RG analysis] RG section (Kadanoff block recursions): the central claim that ν(ε) evolves continuously from 1/2 to 4/3 to 1 and matches Coulomb-gas predictions without additional parameters or energy-specific tweaks is load-bearing for the interpretation of the energy-weighted length. The manuscript must supply the explicit recursion relations (how the energy weight enters the block probabilities), the block sizes employed, and explicit checks that the flow is independent of block size; real-space RG for percolation is known to be approximate, and an energy insertion can introduce effective renormalizations whose parameter-free character is not guaranteed a priori.
  2. [Monte Carlo simulations] Monte Carlo results section: the reported smooth threshold shift and the finite value of the energy-weighted correlation length at p_c(ε=0) require quantitative support. Lattice sizes, number of independent samples, error bars, and the precise estimator used to locate the threshold (wrapping probability, Binder cumulant, etc.) must be stated so that finite-size effects and statistical significance can be assessed.
minor comments (3)
  1. [Abstract] Abstract: the phrase 'approaching ν=1' should be accompanied by the limiting value of ε or the asymptotic regime in which this occurs.
  2. [Notation] Notation: ensure that the occupation probability is uniformly denoted p and the energy parameter ε throughout the text and figures.
  3. [Figures] Figures: all plots of ν(ε) or correlation length should include uncertainty estimates derived from the RG or MC data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will revise the manuscript accordingly to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: [RG analysis] RG section (Kadanoff block recursions): the central claim that ν(ε) evolves continuously from 1/2 to 4/3 to 1 and matches Coulomb-gas predictions without additional parameters or energy-specific tweaks is load-bearing for the interpretation of the energy-weighted length. The manuscript must supply the explicit recursion relations (how the energy weight enters the block probabilities), the block sizes employed, and explicit checks that the flow is independent of block size; real-space RG for percolation is known to be approximate, and an energy insertion can introduce effective renormalizations whose parameter-free character is not guaranteed a priori.

    Authors: We agree that the RG section requires more explicit documentation. In the revised manuscript we will present the full Kadanoff block recursion relations, showing precisely how the bond energy ε enters the weighted block probabilities via the Boltzmann factor exp(−ε n_b), where n_b is the number of occupied nearest-neighbor bonds inside the block. We employed both 2×2 and 3×3 blocks; we will add a supplementary table and figure demonstrating that the extracted ν(ε) curves differ by at most 0.04 across this range of block sizes for ε ∈ [0, 3]. While real-space RG remains approximate, our closure is parameter-free once ε is fixed, and we will explicitly state the approximations and the limited range of ε over which the Coulomb-gas comparison is expected to hold. revision: yes

  2. Referee: [Monte Carlo simulations] Monte Carlo results section: the reported smooth threshold shift and the finite value of the energy-weighted correlation length at p_c(ε=0) require quantitative support. Lattice sizes, number of independent samples, error bars, and the precise estimator used to locate the threshold (wrapping probability, Binder cumulant, etc.) must be stated so that finite-size effects and statistical significance can be assessed.

    Authors: We accept that the Monte Carlo section is insufficiently detailed. The revised text will state that all simulations used square lattices with linear sizes L = 64, 128, 256 and 512, averaged over 5 000–20 000 independent samples per (p, ε) point, with uncertainties obtained by jackknife resampling. The percolation threshold p_c(ε) was located from the finite-size crossing of the wrapping probability, cross-checked with the Binder cumulant of the largest-cluster size; we will include these methodological details together with a brief finite-size scaling analysis supporting the reported smooth shift and the finite energy-weighted correlation length at the classical p_c. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent Monte Carlo and real-space RG computations

full rationale

The paper's central results are obtained via Monte Carlo simulations showing smooth threshold shift with ε and a defined energy-weighted correlation length that is computed to remain finite at p_c(ε=0). The real-space RG using Kadanoff block recursions produces the reported continuous evolution of ν(ε) from 1/2 through 4/3 toward 1, with the match to Coulomb-gas loop-model predictions presented as external validation rather than a load-bearing premise. No equations or steps reduce by construction to fitted inputs, self-definitions, or author-overlapping citations that would force the outcome. The methods are standard numerical and approximate renormalization techniques whose outputs are not tautological with the model inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claims rest on treating ε as a continuous external parameter, assuming standard nearest-neighbor site percolation on a 2D lattice, and postulating that the defined energy-weighted length and RG recursions capture the physics without additional hidden scales or interactions.

free parameters (1)
  • ε
    Continuous bond energy parameter varied to interpolate between regimes; its value is chosen by the modeler rather than derived.
axioms (2)
  • domain assumption The underlying lattice is a standard 2D square grid with nearest-neighbor connectivity and no long-range interactions beyond the assigned bond energy.
    Invoked throughout the generalization of site percolation and the RG analysis.
  • domain assumption Kadanoff block recursions in real-space RG correctly renormalize the bond energy and yield the observed exponent flow.
    Central to the reported evolution of ν and agreement with Coulomb-gas predictions.
invented entities (1)
  • energy-weighted correlation length no independent evidence
    purpose: Captures energetic suppression of connectivity and remains finite at classical p_c
    Newly defined quantity whose behavior is reported from simulations and RG; no independent falsifiable prediction outside the model is stated.

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