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arxiv: 2604.05772 · v1 · submitted 2026-04-07 · ❄️ cond-mat.stat-mech

Percolation in the three-dimensional Ising model

Jinhong Zhu , Tao Chen , Zhiyi Li , Sheng Fang , Youjin Deng This is my paper

Pith reviewed 2026-05-10 18:59 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Ising modelpercolationcritical phenomenageometric clustersMonte Carlo simulationuniversality classthree dimensions
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The pith

The three-dimensional Ising model at criticality shows only a single percolation transition for geometric spin clusters.

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

Geometric representations of the Ising model allow study of percolation by occupying bonds between parallel spins with probability p. In two dimensions, critical configurations display two consecutive percolation transitions as p increases. Monte Carlo simulations demonstrate that this double transition disappears in three dimensions, leaving only one transition. The same single-transition behavior appears in the Ising model on the complete graph. Percolation on a two-dimensional layer within the three-dimensional system shows scaling exponents that indicate a new universality class influenced by correlations in the third dimension.

Core claim

The authors establish that critical Ising configurations in three dimensions undergo a single percolation transition of geometric clusters as the bond occupation probability p is varied. This is shown through extensive Monte Carlo simulations on lattices and corroborated by theoretical analysis on the complete graph. In contrast, a two-dimensional layer embedded in the three-dimensional critical Ising model exhibits percolation exponents indicating coupling to out-of-plane correlations and a distinct universality class.

What carries the argument

Geometric spin clusters connected by occupied bonds with probability p on critical Ising spin configurations; this mechanism reveals the percolation behavior and allows measurement of scaling exponents in both the full three-dimensional system and embedded layers.

Load-bearing premise

The finite lattices and Monte Carlo sampling used are large enough and have small enough corrections to reliably detect the presence of only one percolation transition rather than missing a closely spaced second one.

What would settle it

If larger-scale simulations or more precise finite-size scaling analysis revealed two distinct percolation thresholds separated by a small interval in p, that would contradict the single-transition conclusion.

Figures

Figures reproduced from arXiv: 2604.05772 by Jinhong Zhu, Sheng Fang, Tao Chen, Youjin Deng, Zhiyi Li.

Figure 1
Figure 1. Figure 1: The phase diagrams for percolation in the Ising model on the (a) square lattice (2D), (b) cubic lattice (3D), and (c) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The phase diagrams for the 3D Ising model with percolation coordination numbers (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Binder ratio Q b s,m for minority-spin clusters versus K in the 3D bulk Ising model with the bond occupied prob￾ability p = 1 for various system sizes L. The FSS analysis gives the estimate Kb c = 0.232 38(9). The contrast between the 2D and 3D phase diagrams naturally raises the question of what happens in still higher dimensions. To address this point, we study the Ising model on the CG, where each site … view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the percolation threshold [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plots of O/LyO versus L y1 for the critical SL3D Ising model with zp = 6 at p = pc = 1, with y1 taking the correction exponent shown in Table III. (a) The largest cluster size C1 yielding the fractal dimension yh = 1.892 6(20). (b) The hull length Lhull with the estimated exponent dhull = 1.663(4). (c) The shortest-path distance sp with the estimated exponent dmin = 1.080(10). The upper and lower curves in… view at source ↗
read the original abstract

Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability $p$ between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent $y_p = 0.426(6)$ and the fractal dimensions of the largest cluster, hull, and shortest path as $d_f = 1.8926(20)$, $d_{\rm hull} = 1.663(4)$, and $d_{\rm min} = 1.080(10)$, respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.

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

3 major / 2 minor

Summary. The paper claims that the two consecutive percolation transitions observed for geometric clusters in the 2D critical Ising model do not occur in 3D, where Monte Carlo simulations on finite lattices and a complete-graph analysis both indicate only a single percolation transition as the bond-occupation probability p is varied. It further examines percolation on a 2D layer embedded in the 3D critical Ising model, reporting estimates y_p = 0.426(6), d_f = 1.8926(20), d_hull = 1.663(4), and d_min = 1.080(10) that suggest a distinct universality class arising from coupling to out-of-plane correlations.

Significance. If the single-transition conclusion is robust, the work establishes a clear dimensional crossover in the percolation behavior of Ising geometric clusters and supplies concrete exponent values for the coupled layer system that can be used to test universality-class predictions. The independent complete-graph calculation provides mean-field corroboration without relying on lattice-specific fitting.

major comments (3)
  1. [Monte Carlo results for 3D Ising] Monte Carlo section (results for 3D Ising): The manuscript does not specify the range of lattice sizes L, the number of independent samples per (L,p) point, or the exact observable (e.g., percolation probability crossing, cluster-size susceptibility peak, or wrapping probability) used to locate the transition and to rule out a second, closely spaced transition. Without these details it is impossible to quantify the effective resolution set by finite-size scaling and correlation-length corrections, which directly bears on the central claim that only one transition exists.
  2. [Finite-size scaling analysis] Finite-size scaling analysis: The paper reports that the single-transition scenario is supported by data collapse, but does not show or discuss the quality of collapse when a two-transition ansatz (with a small separation Δp) is fitted; a quantitative comparison of χ² or residual errors between one- and two-transition models would be required to substantiate the exclusion of the second transition.
  3. [Layer-system results] Layer-system exponents: The reported values y_p = 0.426(6), d_f = 1.8926(20), etc., are stated to indicate a new universality class, yet the manuscript does not compare them against existing 2D percolation or Ising-layer exponents with error bars or perform a consistency check against hyperscaling relations that would be expected in the coupled system.
minor comments (2)
  1. [Abstract and references] The abstract cites the 2D precursor as Phys. Rev. E 112, 034118 (2025); the reference list should include the full bibliographic entry once the paper is published.
  2. [Layer-system analysis] Notation for the red-bond exponent y_p is introduced without an explicit definition in terms of the bond-occupation probability derivative; a brief equation would clarify its relation to the standard percolation exponents.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we will make to improve clarity and rigor.

read point-by-point responses

  1. Referee: Monte Carlo section (results for 3D Ising): The manuscript does not specify the range of lattice sizes L, the number of independent samples per (L,p) point, or the exact observable (e.g., percolation probability crossing, cluster-size susceptibility peak, or wrapping probability) used to locate the transition and to rule out a second, closely spaced transition. Without these details it is impossible to quantify the effective resolution set by finite-size scaling and correlation-length corrections, which directly bears on the central claim that only one transition exists.

    Authors: We agree that these methodological details are essential. In the revised manuscript we will add a dedicated paragraph in the Monte Carlo section specifying the range of lattice sizes L, the number of independent samples per (L,p) point, and the precise observables (wrapping probability and largest-cluster fraction) used to locate the transition and to exclude a second, closely spaced transition. This will allow readers to assess the finite-size resolution directly. revision: yes

  2. Referee: Finite-size scaling analysis: The paper reports that the single-transition scenario is supported by data collapse, but does not show or discuss the quality of collapse when a two-transition ansatz (with a small separation Δp) is fitted; a quantitative comparison of χ² or residual errors between one- and two-transition models would be required to substantiate the exclusion of the second transition.

    Authors: We will perform and include in the revised manuscript an explicit comparison of data-collapse quality between the single-transition model and a two-transition ansatz with variable small Δp. We will report the resulting χ² values (or equivalent residual measures) for both models to provide a quantitative basis for preferring the single-transition scenario. revision: yes

  3. Referee: Layer-system exponents: The reported values y_p = 0.426(6), d_f = 1.8926(20), etc., are stated to indicate a new universality class, yet the manuscript does not compare them against existing 2D percolation or Ising-layer exponents with error bars or perform a consistency check against hyperscaling relations that would be expected in the coupled system.

    Authors: We will add to the revised manuscript a table comparing our measured exponents (with error bars) to those of ordinary 2D percolation and other relevant 2D classes. We will also include a consistency check of the appropriate hyperscaling relations for the embedded layer, discussing any deviations attributable to coupling with the out-of-plane critical correlations. These additions will strengthen the evidence for a distinct universality class. revision: yes

Circularity Check

0 steps flagged

No significant circularity; 3D results from direct Monte Carlo sampling and independent complete-graph analysis

full rationale

The paper's central claims—that only a single percolation transition occurs in 3D critical Ising configurations, unlike the two transitions in 2D—are supported by extensive Monte Carlo simulations on finite lattices and a separate theoretical analysis on the complete graph. These do not reduce by the paper's own equations to fitted parameters or self-referential definitions. The sole self-citation is to the authors' prior 2D work to establish the contrast being tested; this citation is not load-bearing for the 3D observations or the complete-graph result, which stands independently. No predictions are constructed from the target data, no ansatzes are smuggled, and no uniqueness theorems are invoked from self-citations. The estimates of exponents (y_p, d_f, etc.) for the embedded layer are direct numerical outputs from the simulations, not circular redefinitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rely on standard Monte Carlo sampling of the Ising model at criticality and percolation cluster analysis; no additional free parameters, ad-hoc axioms, or invented entities are introduced beyond the conventional lattice Ising Hamiltonian and geometric cluster definition.

axioms (1)
  • domain assumption The three-dimensional Ising model on a cubic lattice with nearest-neighbor ferromagnetic interactions exhibits a continuous phase transition at a known critical temperature.
    Invoked when sampling configurations at criticality.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations

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Reference graph

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