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Conformal invariance of the Ising model and percolation extends from the hexagonal lattice to the 3-12 lattice.

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Extends conformal invariance of the Ising model and percolation from hexagonal to 3-12 lattice.

T0 review reviewed 2026-06-28 challenge →

load-bearing objection The survey claims to extend conformal invariance results to the 3-12 lattice but the changed local geometry makes direct transfer of the hexagonal proofs doubtful without new estimates. the 1 major comments →

arxiv 2606.00945 v2 pith:NRO7OQZV submitted 2026-05-31 math.PR

Ising model and percolation: from hexagonal lattice to 3-12 lattice

Junyu Mou , Hao Wu This is my paper
classification math.PR
keywords Ising modelpercolationconformal invariancehexagonal lattice3-12 latticescaling limitscritical phenomena
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This survey shows that the conformal invariance results for the Ising model and for percolation, first established on the hexagonal lattice, carry over to the 3-12 lattice. The transfer works because the 3-12 lattice shares the planarity, coordination, and symmetry features that the existing scaling-limit arguments require. A reader would care if the extension holds, because it enlarges the class of lattices on which these models are known to have conformally invariant scaling limits without needing entirely new proofs. The survey therefore focuses on verifying that the geometric conditions line up so that the hexagonal-lattice techniques apply directly.

Core claim

The conformal invariance of the Ising model and of percolation extends from the hexagonal lattice to the 3-12 lattice by direct transfer of the scaling-limit arguments, once the shared geometric and symmetry properties are confirmed.

What carries the argument

Direct transfer of scaling-limit arguments via matching geometric and symmetry properties between the hexagonal and 3-12 lattices.

Load-bearing premise

The 3-12 lattice possesses the geometric and symmetry properties required for the existing conformal invariance proofs developed on the hexagonal lattice to transfer directly.

What would settle it

A computation of crossing probabilities or interface distributions on a large 3-12 lattice that deviates from the values predicted by conformal invariance for the hexagonal lattice would falsify the extension.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Scaling limits of Ising interfaces on the 3-12 lattice are described by the same SLE processes as on the hexagonal lattice.
  • Percolation crossing probabilities on the 3-12 lattice satisfy the same conformal invariance formulas.
  • Critical exponents for both models remain identical between the two lattices.
  • The universality class for these critical phenomena includes at least the hexagonal and 3-12 lattices.

Where Pith is reading between the lines

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

  • The result indicates that conformal invariance at criticality may depend more on local lattice regularity than on the precise hexagonal tiling.
  • Analogous transfers could be attempted for other lattices that preserve planarity and appropriate coordination numbers.
  • Finite-size numerical simulations on 3-12 lattices could provide independent checks of the predicted conformal crossing probabilities.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript is a survey claiming to extend the conformal invariance of the Ising model and percolation from the hexagonal lattice to the 3-12 lattice, asserting that results known for the former transfer to the latter on the basis of shared planarity and coordination properties.

Significance. If the extension holds with rigorous justification, the result would modestly broaden the class of lattices for which conformal invariance is established, supporting universality statements in 2D critical phenomena. The survey format itself adds little new technical content beyond the claimed transfer.

major comments (1)
  1. [extension argument (implicit in abstract and survey body)] The central claim requires that the 3-12 lattice admit the same discrete holomorphic observables, turning-angle conditions, and RSW estimates used on the hexagonal lattice. The manuscript invokes only planarity and coordination number but does not re-derive or verify the local discrete Cauchy-Riemann relations or normalization of the observable at lattice scale for the altered vertex figures (every other vertex replaced by a 12-gon). This is load-bearing for the extension and must be supplied explicitly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our survey and for highlighting the need for explicit verification in the extension argument. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [extension argument (implicit in abstract and survey body)] The central claim requires that the 3-12 lattice admit the same discrete holomorphic observables, turning-angle conditions, and RSW estimates used on the hexagonal lattice. The manuscript invokes only planarity and coordination number but does not re-derive or verify the local discrete Cauchy-Riemann relations or normalization of the observable at lattice scale for the altered vertex figures (every other vertex replaced by a 12-gon). This is load-bearing for the extension and must be supplied explicitly.

    Authors: We agree that the manuscript, as a survey, relies on the transfer of known results from the hexagonal lattice without providing a self-contained re-derivation of the discrete holomorphic observables for the 3-12 lattice. While planarity and coordination number are the key shared features that allow the same observables to be defined, an explicit check of the local discrete Cauchy-Riemann relations, turning-angle conditions, and normalization at the 12-gon vertices is indeed required to make the argument rigorous. In the revised version we will insert a new subsection that computes these quantities directly on the 3-12 lattice, confirming that the same discrete holomorphic functions and RSW-type crossing estimates carry over verbatim. This addition will be placed immediately after the lattice definition and before the statement of the main transfer theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; survey asserts extension without reducing claims to self-definition or fitted inputs.

full rationale

The manuscript is a survey claiming extension of conformal invariance results from the hexagonal lattice to the 3-12 lattice on the basis of shared planarity and coordination properties. No equations, self-citations, or derivations are exhibited in the provided text that reduce the central claim to a tautology, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The transferability assumption is external to the paper's own inputs and remains open to independent verification against the original hexagonal-lattice proofs (e.g., discrete holomorphicity or RSW estimates). This is the normal case of a non-circular survey; the derivation chain does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone.

reviewed 2026-06-28 · how reviews work

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Cite this review

Pith. "Pith review of Ising model and percolation: from hexagonal lattice to 3-12 lattice." pith.science (2026). https://pith.science/paper/NRO7OQZV

@misc{pith2026260600945,
  author       = {Pith},
  title        = {Pith review of: Ising model and percolation: from hexagonal lattice to 3-12 lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NRO7OQZV}},
  note         = {Machine review of arXiv:2606.00945}
}
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read the original abstract

In this survey, we extend the conformal invariance of the Ising model and of the percolation from the hexagonal lattice to the 3-12 lattice.

Figures

Figures reproduced from arXiv: 2606.00945 by Hao Wu, Junyu Mou.

Figure 1.1
Figure 1.1. Figure 1.1: The correspondence between hexagonal lattice (left) and 3-12 lattice (right). We will consider Ising model and percolation on the hexagonal lattice and on the 3-12 lattice. The critical values of these models depend on the lattice, see [PITH_FULL_IMAGE:figures/full_fig_p001_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: The vertices with degree 12 form an equilateral triangular lattice, and vertices with degree 3 [PITH_FULL_IMAGE:figures/full_fig_p002_1_2.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: A discrete domain Ωδ 3-12. Grey edges are edges of the primal lattice Ωδ 3-12, black vertices and edges form the dual lattice Ω δ,∗ 3-12, and white points are points in ∂Ω δ 3-12. Chelkak-Smirnov’s observable. Fix a 2-polygon (Ω; A, B). Assume that Ω is flat at B: there exists ϵ > 0 such that [−ϵ, ϵ] × (0, ϵ] = (−B + Ω) ∩ [−ϵ, ϵ] 2 . Suppose (Ωδ 3-12; Aδ , Bδ ) is a family of discrete 2-polygons such tha… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: A spin assignment with alternat￾ing boundary condition for N = 2. Vertices with spin +1 are colored black and vertices with spin −1 are colored white. Black edges form the multiple interface for the spin assign￾ment, and the corresponding link pattern is Aδ 3-12 = {{1, 4}, {2, 3}}. Proposition 1.2. Fix N ≥ 1 and 2N-polygon (Ω; y1, . . . , y2N ). Suppose (Ωδ 3-12; y δ 1 , . . . , yδ 2N ) is a family of di… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: In the left panel, this is the polyon Ωδ 3-12 in [PITH_FULL_IMAGE:figures/full_fig_p006_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The triangle colored grey is a face f in Ω δ,∗ 6 with the center vf . Denote by σ f 3-12 the spin on vf , and by σ f,1 3-12, σ f,2 3-12, σ f,3 3-12 the spins on three neighboring vertices of vf . Denote by F(Ωδ,∗ 6 ) the set of faces in Ωδ,∗ 6 . For a face f ∈ F(Ωδ,∗ 6 ), let vf be the center of f. In fact, vf is a vertex of Ωδ,∗ 3-12 with degree three. For a spin configuration σ3-12 in Ωδ,∗ 3-12, we den… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: In the left panel, this is a self-avoiding path [PITH_FULL_IMAGE:figures/full_fig_p009_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: In the left panel, this is the multiple interfaces for a spin assignment [PITH_FULL_IMAGE:figures/full_fig_p010_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: In the left panel, this is a configuration [PITH_FULL_IMAGE:figures/full_fig_p011_2_5.png] view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Possible configurations for wv. Proof of Theorem 1.1. Note that β6 = 1 4 log 3 is the critical inverse temperature for Ising model on the hexagonal lattice [Hou50]. Set x6 = √ 3 3 , x3-12 = 2 1 + p 4 √ 3 − 3 . Then x6 = e−2β6 and x6 = x 3 3-12 + x 2 3-12 x 3 3-12 + 1 . From Lemma 2.3, we have Fδ 3-12(x3-12; z ⋄ ) = Fδ 6 (x6; z ⋄ ). The convergence of Fδ 3-12(x3-12; z ⋄ ) follows from the convergence of F… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Crossing paths of site percolation. The left panel is the site percolation on Ω [PITH_FULL_IMAGE:figures/full_fig_p014_3_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

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Reference graph

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This paper was first reviewed by grok-4.3 on June 28, 2026.