Sharp phase transition for percolation with short-range dependencies

Olivier Henry; Peter M\"orters

REVIEW 3 major objections 5 minor 13 references

Sharp phase transition for percolation with short-range dependencies

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proves that percolation models on Z^d with short-range dependencies—where edge probabilities depend on independent vertex types—have sharp phase transitions: exponential decay of connection probabilities below the critical value

desk verdict Solid OSSS extension with a fixable but load-bearing typo in Definition 3.3 and an abstract that overstates the conditional finite-type theorem. read the letter →

arxiv 2607.14993 v1 pith:SAFUUEKY submitted 2026-07-16 math.PR

Olivier Henry , Peter Mörters This is my paper · ORCID

classification math.PR MSC 60K3582B43

keywords percolationsharpphasetransitionshort-rangedependenciesOSSSinequalitydecisiontreessite-bondcriticalcurvefinite-typemodel

verification ladder T0 review T1 audit T2 compute T3 formal

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 paper shows that a large class of dependent percolation models on the integer lattice, where edge probabilities are governed by independent random vertex types, has a sharp phase transition: once crossing the critical value, infinite connections appear, and before it they disappear exponentially fast. For the combined Bernoulli site-bond model, the authors construct a complete phase diagram with a critical curve separating subcritical and supercritical phases, prove exponential decay throughout the subcritical region, and give explicit bounds on the curve's slope. For the finite-type model, they prove that whenever a phase transition exists at some β_c, it is automatically sharp: the connection probability decays exponentially below β_c and is bounded below by a positive multiple of β−β_c just above it. The argument adapts the decision-tree variance method developed for the random cluster model, combining the OSSS inequality, Russo's formula, and new comparisons between pivotal vertex and edge variables. This gives a versatile template for establishing sharpness in short-range dependent percolation models beyond product measures.

What carries the argument

The key mechanism is the OSSS inequality, a variance bound for randomized decision trees: for an increasing event, the variance of its indicator is bounded by the sum, over each underlying random variable, of the reveal probability times the covariance with the event. Coupled with Russo's formula (which expresses derivatives of connection probabilities as sums of pivotal probabilities) and comparison lemmas that relate pivotal probabilities of vertex-type variables to those of edge variables, this yields a differential inequality of the form θ_n' ≥ c n θ_n (1−θ_n) / Σ θ_k. A one-dimensional bootstrap lemma converts this inequality into exponential decay below the critical point and a linear

What would settle it

For the site-bond model, one could compute or simulate the critical curve in low dimension and check the predicted slope bounds (2.46); any violation would refute the regularity claim. For the finite-type theorem, a counterexample would be a finite-type model with a genuine phase transition at some β_c for which connection probabilities below β_c decay subexponentially or fail to be at least linear above β_c.

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Extended reading notes

Core claim

The paper's central claim is that sharpness of the phase transition—exponential decay of connection probabilities below the critical point and a linear lower bound above it—holds for nearest-neighbour percolation on Z^d with edge probabilities determined by independent vertex types. In the two-parameter site-bond model, this is unconditional: there is a critical curve q_c(p), decreasing and locally Lipschitz, with exponential decay in the subcritical region and positive percolation in the supercritical region, and the transition is sharp along any admissible smooth curve. In the finite-type model, the sharpness conclusion is conditional on the existence of a critical parameter β_c; under tha

Load-bearing premise

The finite-type sharpness theorem assumes, rather than proves, that a critical value β_c exists with no percolation below and positive percolation above; if no such β_c can be established for a given model, the sharpness conclusion does not apply.

Editorial extensions

If this is right

In the site-bond model, the subcritical region {(p,q): p<p_0,c or q<q_c(p)} is open and has exponential decay; the supercritical region is open and has positive percolation.
The critical curve q_c is decreasing and locally Lipschitz with quantitative slope bounds, and its inverse p_c has the same regularity; hence the phase diagram is fully controlled.
Sharpness holds along every smooth curve crossing the critical curve in a direction consistent with the slope constraints, yielding exponential decay before the crossing and linear growth after.
For any finite-type model that satisfies the existence assumption on β_c, the phase transition is sharp, with θ(β)≥c(β−β_c) near β_c and exponential decay of θ_n below β_c.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The conditional nature of the finite-type theorem points to the natural next problem: a general construction of the critical value β_c. The paper's criteria cover many natural examples but do not characterize all finite-type models.
The slope bounds on the critical curve give a quantitative handle on the phase diagram that could be used to approximate q_c(p) numerically or to test mean-field-type predictions in high dimensions.
The Bernoulli base-variable representation may make these models accessible to other decision-tree-based tools, such as noise sensitivity or quantitative mixing estimates, beyond the sharpness question.
Because the constants in the finite-type argument depend on the number of types N, the method does not directly pass to infinite-type limits; removing this dependence would extend sharpness to continuous type distributions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take on arXiv:2607.14993. The paper does what it says: it adapts the DRT19b decision-tree proof of sharpness to two dependent percolation settings, and the main machinery is sound. The cleanest new piece is the site-bond phase diagram: a Lipschitz critical curve with explicit slope bounds and exponential decay off it. The finite-type vertex-dependent model is a broader setup, and the pivotal inequalities connecting vertex and edge variables are genuinely new and look correct. This is a solid extension of a known method rather than a new framework, and it is honestly positioned as such.

Two things to flag before relying on it. First, Definition 3.3 defines ν_x with a min, and as written that makes ν_x ≡ 0, so the Bernoulli representation is degenerate and Proposition 3.5 is false. The intended definition is clearly a max — the proof text and the probability computation only work that way. It is a typo, but it is in a load-bearing spot and needs fixing. Second, Theorem 3.11 is conditional: it assumes β_c exists with θ=0 below and θ>0 above, then proves sharpness given that. The abstract says "we show sharpness of the phase transition" without qualification, which overstates what is established. Remark 3.2 gives sufficient criteria for existence in some examples, so the theorem is not empty, but it is not a general existence proof. The conditional statement itself is internally sound.

The reliance on DRT19b as a black box is appropriate — the OSSS inequality and the sharpness lemma are standard, cited correctly, and not load-bearing self-citations. The proofs are detailed; I did not find a gap beyond the typo. Constants are sometimes not optimized, but that is not a flaw here.

Who is this for? Readers who want a template for applying OSSS to dependent percolation models, especially with vertex types, and anyone interested in the site-bond phase diagram. It is a subfield contribution, not a revolution. I would send it to a serious referee; after fixing the typo and reframing the abstract, it should be publishable. I would cite it if I were working on dependent percolation.

Referee Report

3 major / 5 minor

Summary. The paper studies percolation on Z^d with short-range dependencies: vertices carry independent types, and the probability that an edge is open depends on the types of its endpoints. In the special case of combined Bernoulli site and bond percolation, it proves monotonicity of connection probabilities along admissible curves, exponential decay in the subcritical phase, a linear lower bound near criticality, and establishes a locally Lipschitz critical curve with explicit slope bounds (Theorem 2.7). For the finite-type model, the paper introduces a Bernoulli-variable representation and proves (Theorem 3.11) that, conditional on the existence of a phase transition at some β_c, the transition is sharp: exponential decay below β_c and a linear lower bound above it. The proof follows the OSSS decision-tree method of Duminil-Copin–Raoufi–Tassion.

Significance. If the technical issues identified below are repaired, the paper makes a useful contribution by extending the OSSS-based sharpness machinery to a class of models with vertex-dependent edge probabilities. The site-bond percolation analysis is self-contained and quantitatively describes the critical curve rather than only proving its existence. The finite-type theorem is more modest than the abstract suggests, since it is conditional on an unproved existence assumption, but the conditional sharpness statement is still a non-trivial and potentially transferable result. The paper builds on external results as black boxes and shows no circularity; no fitted parameters are used.

major comments (3)

[Definition 3.3, Eq. (3.7)] The definition of ν_x as min{n∈J0,NK: ∀1≤k≤n, Z_{x,k}=1} is degenerate: the set always contains n=0, so ν_x≡0 almost surely. The proof of Proposition 3.5 computes P(ν_x≥k) as P(Z_{x,1}=...=Z_{x,k}=1), which is the distribution of the maximum of such n, not the minimum. As written, the Bernoulli representation does not match the original model and Proposition 3.5 fails, undermining the whole of Section 3. The fix is to replace 'min' by 'max'; this is a one-character correction but it is load-bearing.
[Lemma 3.8, Eq. (3.23)] The stated inequality has denominator (1-h_M)^{2d}, but the proof for interior vertices x∈Λ_{n-1}\{0} yields (3.31) with denominator (1-h_M)^{2d-1}. Since (1-h_M)<1, the RHS in (3.23) is smaller than the quantity actually bounded by the proof; the stated inequality does not follow and may be false. A valid uniform bound can be obtained by taking the maximum of the boundary and interior constants, giving a factor max(1,2d-1)/(1-h_M)^{2d-1}. The sharpness conclusion of Theorem 3.11 only needs a finite constant, so this is repairable, but the lemma statement and proof must be reconciled.
[Abstract and Theorem 3.11] The abstract claims unconditionally that 'We show sharpness of the phase transition', but Theorem 3.11 assumes the existence of β_c such that θ(β)=0 for β<β_c and θ(β)>0 for β>β_c. Remark 3.2 explicitly states that the criteria given there 'do not characterize the existence of a phase transition'. Thus the paper proves a conditional statement: if a phase transition exists, then it is sharp. The abstract and introduction should be rephrased to state this conditionality, or a general existence theorem for β_c under the assumptions of Definition 3.1 should be supplied.

minor comments (5)

[Throughout] There are several typos: 'paramater' (§1.2.1), 'seperated' (Remark 2.9), 'developping' (Introduction), 'nighboorhood' (Remark 2.6).
[Definition 1.4] The notation 'ω e /∈ A' should read 'ω^e ∉ A' (superscript e) to match the definition of pivotal events.
[Lemma 2.2 proof] The notation 'ω_x^{N(x)\{e,f\}}' is confusing and appears to mix the superscript/subscript convention from Definition 1.4. Please define it explicitly at first use.
[Theorem 2.5 proof, Eq. (2.36)] There are unmatched parentheses and a missing closing parenthesis in the displayed inequality; also the division by u(γ(t)) should be written more cleanly.
[Lemma 3.8 proof] In the display after (3.26), the notation 'ω^{(x,k)}_{I_y}' should be introduced explicitly; as written it is easy to misread whether Z_{x,k} is set to 1 or 0.

Circularity Check

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No circularity found: the sharpness results are conditional on explicit existence hypotheses and rely on external, non-self-cited OSSS/DRT19b lemmas; no fitted parameter is relabeled as a prediction.

full rationale

The derivation chain is self-contained given its stated hypotheses. The OSSS inequality (Theorem 1.1), the differential inequality lemma (Lemma 1.2), and the summation lemma (Lemma 1.3) are imported from Duminil-Copin–Raoufi–Tassion [DRT19b], an external source with no author overlap with Henry and Mörters; these are black-box external results, not self-citations. In Section 2 the new site-bond inequalities (Lemmas 2.1–2.4) are proved from Russo's formula, Lemma 1.7, and combinatorial pivotal-event estimates; the critical curve q_c in Theorem 2.7 is defined from the phase boundary, and its Lipschitz/slope bounds are derived, not assumed. In Section 3, Definition 3.3 introduces an equivalent Bernoulli-variable representation and Proposition 3.5 proves the two descriptions have identical laws; Lemmas 3.7 and 3.8 then bound covariances by pivotal probabilities, and Theorem 3.11 assembles these into exponential decay and a linear lower bound. The one caveat is that Theorem 3.11 assumes the existence of β_c with θ(β)=0 below and θ(β)>0 above; this is an input hypothesis rather than a consequence being re-labeled as a conclusion. Remark 3.2 explicitly states that its criteria 'do not characterize the existence of a phase transition,' so the paper does not claim to prove existence in general. The abstract's phrase 'we show sharpness' is broader than the conditional theorem, but that is an overstatement of scope, not circularity. No fitted parameters are used as predictions, no self-citation carries a load-bearing argument, and no uniqueness theorem is imported from the authors' own previous work.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof introduces no fitted parameters and no new physical entities. It relies on external OSSS lemmas and on model-class assumptions, the most load-bearing of which is the assumed existence of the phase transition in Theorem 3.11.

assumptions (7)

standard math OSSS inequality (Theorem 1.1) and the sharpness lemma (Lemma 1.2) from DRT19b are valid for monotone measures.
Used as black-box external theorems throughout Sections 2 and 3.
standard math FKG / monotone measure properties hold for the product measures and increasing events considered.
Used in Lemma 2.1 and to apply the OSSS framework.
domain assumption Translation invariance of Z^d allows removing the max in Lemma 1.3 and identifying P(x<->∂Λ_k(x)) with θ_k.
Used throughout, e.g. in (2.7) and (3.17).
domain assumption d≥2.
Used in Lemma 3.7 so that boundary vertices have a neighbour in the boundary; also d=1 percolation is trivial.
domain assumption Existence of a phase transition at β_c with θ=0 below and θ>0 above.
Assumed in Theorem 3.11 and not proved for the general finite-type class; Remark 3.2 only gives partial criteria.
domain assumption The functions h_m satisfy 0=h_0<h_1<...<h_M<1 and h'_m > (1-h_m)/(1-h_{m-1}) h'_{m-1}, equivalently p'_l>0.
Needed to lower-bound θ'_n in Theorem 3.11; holds for h_m(β)=1−exp(−β c_m).
domain assumption ζ is symmetric, nondecreasing in the product order, and q_ν(N)>0.
These define the model class in Definition 3.1 and are needed for the Bernoulli representation.

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Pith. "Pith review of Sharp phase transition for percolation with short-range dependencies." pith.science (2026). https://pith.science/paper/SAFUUEKY

@misc{pith2026260714993,
  author       = {Pith},
  title        = {Pith review of: Sharp phase transition for percolation with short-range dependencies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAFUUEKY}},
  note         = {Machine review of arXiv:2607.14993}
}

abstract

We show sharpness of the phase transition for a nearest-neighbour percolation model on $\mathbb Z^d$, where vertices carry independent types and the percolation probability of edges depends on the type of the adjacent vertices. Our proof uses the OSSS inequality and adapts to our setup the method developed in Duminil-Copin et al. (2017) for the random cluster model. Additionally, we provide a more extensive study of the special case of combined Bernoulli bond and site percolation featuring a phase transition with two parameters.

Figures

Figures reproduced from arXiv: 2607.14993 by the authors.

**Figure 1.** The new graph G. The vertices are white in V0 (former vertices) and black in V1 (former edges). The whole point of this transformation is that a vertex with exactly two neighbours is just an edge between them. Proof. We assume that ∀x ∈ V, µ(ω(x) = 1) > 0. Otherwise, according to the FKG inequality, the result is true. For k ∈ J1, nK we consider the decision tree Tk associated to the graph walk from ∂Lk and restri… view at source ↗

**Figure 2.** L6 (remember that boxes are twice as small) and the connected component of the origin. For the event 0 ↔ ∂L6, the pivotal edges are e1 and e2. Their endpoints are pivotal as well (provided that these edges are open). Indeed, a path going through e1 or e2 has to go through their endpoints. However, even though both endpoints of e3 are pivotal, e3 itself is not. Proof. We can notice that, for the events we consider, … view at source ↗

**Figure 3.** L6 (remember that boxes are twice as small) and the connected component of the origin. We consider the configuration on the left. For the event 0 ↔ ∂L6, the vertex x is pivotal. If x is occupied and we keep only two of the neighbouring edges, such as e1 and e2, we can make them pivotal (right). Note that several choices of e1 and e2 are possible. Therefore, P(x pivotal for An) ⩽ p q(1 − p) 2d−1 X e∈N(x) P(e pivotal … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The neighbourhood V (the shaded area) and most objects of interest. The arrowed segments have θn nondecreasing along them. The one that starts above qc(p0) has θ > 0 and is therefore above the curve q = qc(p), while the one that ends below qc(p0) has exponential decay …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

1999 , publisher =

Grimmett, Geoffrey , title =. 1999 , publisher =. doi:10.1007/978-3-662-03981-6 , keywords =

doi:10.1007/978-3-662-03981-6 1999

[2] [2]

, title =

Grimmett, Geoffrey R. , title =. 2006 , publisher =. doi:10.1007/978-3-540-32891-9 , keywords =

work page doi:10.1007/978-3-540-32891-9 2006

[3] [3]

Duminil-Copin, Hugo and Raoufi, Aran and Tassion, Vincent , title =. Ann. Math. (2) , issn =. 2019 ,. doi:10.4007/annals.2019.189.1.2 , keywords =

doi:10.4007/annals.2019.189.1.2 2019

[4] [4]

Duminil-Copin, Hugo and Tassion, Vincent , title =. Commun. Math. Phys. , issn =. 2016 ,. doi:10.1007/s00220-015-2480-z , keywords =

work page doi:10.1007/s00220-015-2480-z 2016

[5] [5]

, title =

Aizenman, Michael and Barsky, David J. , title =. Commun. Math. Phys. , issn =. 1987 ,. doi:10.1007/BF01212322 , keywords =

work page doi:10.1007/bf01212322 1987

[6] [6]

Menshikov, Mikhail Vasilyevich , title =. Sov. Math., Dokl. , issn =. 1986 ,

1986

[7] [7]

Annales Henri Lebesgue , pages =

Vanneuville, Hugo , title =. Annales Henri Lebesgue , pages =. 2025 , publisher =. doi:10.5802/ahl.230 ,

work page doi:10.5802/ahl.230 2025

[8] [8]

Phase Diagram of the Ashkin Teller Model

Aoun, Yacine and Dober, Moritz and Glazman, Alexander. Phase Diagram of the Ashkin Teller Model. Commun. Math. Phys. 2024. doi:10.1007/s00220-023-04925-0

doi:10.1007/s00220-023-04925-0 2024