{"id":"467f2c4a-bd35-4ba7-b501-da27051b5812","arxiv_id":"2607.14993","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nearest-neighbour percolation with type-dependent edge probabilities has a sharp phase transition — exponential decay below criticality and linear growth above — whenever a transition exists, with a Lipschitz critical curve in the site-bond case.","lead":"This mathematics paper proves that when lattice vertices carry random types and edges open depending on the two endpoint types, the passage from no infinite cluster to an infinite cluster is abrupt. It extends the OSSS decision-tree method to such dependent models and maps the two-parameter site-plus-bond phase diagram.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.11 is conditional on an existence assumption for β_c that the paper does not prove; the abstract's unconditional framing overstates the result.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: Theorem 3.11 assumes the existence of β_c rather than deriving it. This is not a flaw in the proof of the conditional statement—the OSSS-based differential inequality (3.36) and the pivotal-variable bounds appear correct and the logic from Lemma 3.7 to the conclusion is coherent. The issue is scope: the abstract claims to 'show sharpness of the phase transition' for the finite-type model, but the theorem only shows sharpness under an unproven existence hypothesis. Remark 3.2 explicitly disclaims a characterization of phase-transition existence, and a simple constant-h_m example shows the assumption is not automatic. Thus the paper overstates its central claim, though the underlying mathematics is credible. A secondary formal defect is the min/max typo in Definition 3.3, which would literally break the Bernoulli representation if taken at face value, but the intended meaning is recoverable from Proposition 3.5. Since the reader's CONDITIONAL verdict already reflects this limitation, no change is needed.","tokens_in":18921,"tokens_out":18674,"duration_ms":172360,"concrete_test":"Fix h_m(β)=1/2 for all m≥1 (h_0=0), choose ζ(n1,n2)=n1 n2 and any distribution q_ν on {0,...,N} with q_ν(N)>0. In this model the edge-open probability is independent of β, so θ(β) is constant in β; hence no β_c∈(0,∞) exists. This counterexample shows that Definition 3.1 does not guarantee the hypothesis of Theorem 3.11. To fully settle the concern, check whether the criteria of Remark 3.2 can be extended to prove β_c existence for every model satisfying Definition 3.1; if not, the abstract's unconditional language is unsupported and Theorem 3.11 remains a conditional sharpness result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem for the finite-type model (Theorem 3.11) assumes, rather than proves, the existence of β_c ∈ (0,∞) such that θ(β)=0 for all β<β_c and θ(β)>0 for all β>β_c. The paper provides no general construction or proof of such a β_c. Remark 3.2 offers only sufficient conditions for some examples and explicitly states that these criteria do not characterize the existence of a phase transition. Therefore, for a model satisfying all assumptions of Definition 3.1 but with no such β_c—e.g., h_m(β) constant in β—Theorem 3.11 is vacuous. The abstract's claim to 'show sharpness of the phase transition' is stronger than what is established: the paper proves that if a phase transition exists, then it is sharp, not that a phase transition always exists. This conditional nature is a genuine limitation of the central claim, even though the conditional statement itself is internally sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19162,"tokens_out":19203,"duration_ms":175559,"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":[{"comment":"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.","section":"Definition 3.3, Eq. (3.7)"},{"comment":"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.","section":"Lemma 3.8, Eq. (3.23)"},{"comment":"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.","section":"Abstract and Theorem 3.11"}],"minor_comments":[{"comment":"There are several typos: 'paramater' (§1.2.1), 'seperated' (Remark 2.9), 'developping' (Introduction), 'nighboorhood' (Remark 2.6).","section":"Throughout"},{"comment":"The notation 'ω e /∈ A' should read 'ω^e ∉ A' (superscript e) to match the definition of pivotal events.","section":"Definition 1.4"},{"comment":"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.","section":"Lemma 2.2 proof"},{"comment":"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.","section":"Theorem 2.5 proof, Eq. (2.36)"},{"comment":"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.","section":"Lemma 3.8 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a probability journal and the OSSS approach is appropriate. The two technical errors in Section 3 are local and repairable, but they currently invalidate the proof as written, and the abstract overstates the conditional nature of the finite-type result. I recommend major revision rather than rejection: the underlying strategy is sound and the site-bond section is largely convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nTwo 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.\n\nThe 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.\n\nWho 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.","headline":"Solid OSSS extension with a fixable but load-bearing typo in Definition 3.3 and an abstract that overstates the conditional finite-type theorem.","tokens_in":19595,"tokens_out":3067,"would_cite":true,"duration_ms":31219,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["percolation","sharp phase transition","short-range dependencies","OSSS inequality","decision trees","site-bond percolation","critical curve","finite-type model"],"falsifier":"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.","tokens_in":18840,"feed_emoji":"🕸️","tokens_out":7619,"duration_ms":66321,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sharp phase transition proven for dependent-edge percolation","feed_subtitle":"Exponential decay below the critical point, linear growth above, and explicit bounds on the critical curve.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["In the site-bond model, the subcritical region {(p,q): p