{"id":"620d7bf8-bace-486d-8687-50049cb8fab7","arxiv_id":"2607.25277","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the constrained-degree percolation model with random vertex capacities on the square lattice, any infinite open cluster is almost surely unique, and two-point connectivity is uniformly positive when percolation occurs.","lead":"This paper proves that in a random grid model where vertices have random connection limits, any infinite connected component is almost surely unique, and the chance two points are connected stays positive after a giant component appears. It validates a standard structural property for a family of dependent percolation models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3's projection step is under-specified: it works only if G_n includes the crossing edges O_{n,n+1}; the manuscript never defines B_n^c as an edge set, so the proof of Theorem 1 is incomplete.","rationale":"The reader's weakest-assumption pinpoints the right place: Lemma 3's projection step is the hinge of Theorem 1. However, the reader treats the ambiguity as a definite fatal flaw. A careful reading of the notation leaves room for the intended interpretation that G_n includes the crossing edges O_{n,n+1}; under that reading, the saturation construction in F seems to make the argument cohere. The concern is not that the mathematical idea is impossible, but that the manuscript is under-specified enough that the proof is incomplete as written. This warrants a conditional verdict rather than an outright rejection. I agree partly with the reader: the same spot is identified, but the strength of the objection depends on a notational choice that the paper should have made explicit. No ad hominem is intended; the issue is purely in the proof exposition.","tokens_in":6438,"tokens_out":30792,"duration_ms":337467,"concrete_test":"Make the edge-projection explicit. Set G_n^cross = (Z^2\\B_n) × (E^2\\E(B_n)) and G_n^ind = (Z^2\\B_n) × E(Z^2\\B_n), where E(·) denotes edges with both endpoints in the set. Re-run Lemma 3 with each choice. For G_n^cross, verify that any outside state in Π_{G_n^cross}(H∩U) has open crossing edges from the three infinite clusters to ∂B_n, and that replacing the inside by F preserves the C_t_{n,A} status. For G_n^ind, check whether the conclusion fails because the outside state loses all information about O_{n,n+1}. If the argument works for G_n^cross but not G_n^ind, the paper needs only a notational fix; if it fails for G_n^cross as well, Lemma 3 is genuinely invalid.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 3 is the load-bearing step for Theorem 1. The Burton–Keane argument needs the positive-probability event F_{n,A,δ} × Π_{G_n}(H∩U) to guarantee both (i) all boundary edges O_{n,n} are open and (ii) the resulting cluster has at least three ends. Part (i) is fine. Part (ii) is only established if the projection G_n contains the crossing edges O_{n,n+1}, because those edges' open/closed status (C_t_{n,A}) is what connects the outside infinite components to the open boundary cycle. The manuscript writes G_n = B_n^c × B_n^c without defining B_n^c as an edge set. If B_n^c means edges with both endpoints outside B_n, then Π discards exactly the crossing-edge data, and an outside configuration in the projection may have no open connection to ∂B_n; thus (ii) does not follow. If B_n^c means all edges not internal to B_n (so O_{n,n+1} is included), the projection preserves the crossing data, and the intended saturation argument (κ=3 on ∂B_n∪∂B_{n-1}, U≤δ on A^f, U≥1−δ elsewhere) appears to force both C_t_{n,A} and the three-arm property. Since the paper never disambiguates, the proof of Theorem 1 is incomplete as written, though it is likely repairable by clarifying the projection and expanding the argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the constrained-degree percolation model in a random environment (CDPRE) on Z^2, where each vertex has an i.i.d. degree constraint κ_v∈{0,1,2,3} and each edge attempts to open at a uniform random time subject to the current degrees of its endpoints. The main result (Theorem 1) asserts that if the percolation probability θ(ρ,t) is positive, then the infinite open cluster is almost surely unique. The proof follows the Burton–Keane strategy: assuming multiple infinite clusters, Lemma 3 constructs a local configuration with an open cluster in a box having at least three ends; this is then fed into Lyons–Peres Lemma 7.7 to obtain a positive density of trifurcation points, contradicting the boundary bound. Theorem 2 derives uniform positivity of the two-point connectivity function in the supercritical phase from uniqueness together with the exponential decay of correlations proved in Sanchis et al. (2022).","tokens_in":6821,"tokens_out":31599,"duration_ms":318045,"significance":"If the proof is completed, the result is a natural and worthwhile extension of the deterministic constrained-degree percolation uniqueness theorem (de Lima et al. 2020) to the random-environment setting, and it fills a gap explicitly left open in the CDPRE literature. The paper is short and relies on established machinery; its main technical contribution is the local modification in Lemma 3. The use of the Lyons–Peres forest lemma and the Burton–Keane boundary bound is appropriate, and Theorem 2 is a clean consequence of uniqueness plus known correlation decay. The proof is not machine-checked, but the structure is standard for the field.","major_comments":[{"comment":"The projection step is the load-bearing part of the paper, and as written it is under-specified. The set G_n = B_n^c × B_n^c is never defined as an edge set. If the second factor means edges with both endpoints outside B_n, then the projection discards the crossing edges O_{n,n+1}; in that case the event F_{n,A,δ} × Π_{G_n}(H∩U) does not constrain the states of the crossing edges, and the assertion that at least three infinite clusters intersect B_n does not follow. If, as the surrounding definitions suggest, the second factor is the complement of the set of edges internal to B_n, then O_{n,n+1} is included in the projection; but even then the implication after (4) requires an argument: one must show that the inside modification F, combined with any outside configuration in the projection, forces the crossing-edge configuration C_{n,A} and preserves at least three distinct infinite outsi","section":"§4, Lemma 3, Eq. (4)"},{"comment":"The step 'By item (b), we have Q_t(v∈X)>0 for some v∈Z^2' is not fully justified. Item (b) says that with positive probability the random forest contains a connected component with at least three ends; it does not directly say that the origin (or any fixed vertex) is a trifurcation point. The conclusion is true if one argues first that the set of all vertices belonging to 3-ended components is a translation-invariant random set with positive probability of being nonempty, which forces its density to be positive, and then that every 3-ended tree contains a trifurcation point. This line should be written out, as it is needed to guarantee a positive density of trifurcation points for the boundary bound.","section":"§5.1, Theorem 1"}],"minor_comments":[{"comment":"The notation is inconsistent: B_n is used both as a set of vertices (e.g., in A_t^n) and as a set of edges (e.g., in F_{n,A,δ}). Define E(B_n) for edges with both endpoints in B_n and use it consistently; in particular, specify whether B_n^c in G_n denotes the complement of E(B_n).","section":"§2, model definition"},{"comment":"The captions list five probabilities, e.g., (0.025,0.025,0.9,0.05,0.0), but ρ is defined on {0,1,2,3} and should have four entries.","section":"Figure 1"},{"comment":"The choice of δ should be made explicit: to ensure both that edges in A^f open by time t and that edges in E(B_n)\\A^f are closed (or at least do not interfere before the crossing edges attempt), one needs δ < min(t, 1−t) in addition to the stated δ<t. The current text only says 'δ<t sufficiently small'.","section":"§4, Lemma 3, choice of δ"},{"comment":"The line '= 2P(0↔∂B_n, z↔∂B_n(z), |C|<∞)' is not an equality by translation invariance; only the inequality ≤ holds, which is all that is needed. Please rewrite as an inequality to avoid a false statement.","section":"§5.2, Eq. (7)–(8)"},{"comment":"The transfer from Lemma 3 to the assumptions of Lemma 7.7 in Lyons and Peres (2016) is terse. Since Lemma 3 gives a positive-probability event whose probability may depend on n, please state explicitly why this is sufficient for the quoted lemma, or quote the lemma's hypotheses.","section":"§5.1, invocation of Lemma 7.7"}],"recommendation":"major_revision","confidential_remarks":"The reader's rejection is driven by the projection step in Lemma 3, and I agree that the proof as written is incomplete. However, the issue is localized and appears repairable: if G_n is defined to include the crossing edges O_{n,n+1}, the intended saturation argument can be made rigorous with a short additional argument. The central mathematical idea is sound and the paper fills a natural gap. I recommend major revision rather than rejection, contingent on the authors providing a precise definition of G_n and a complete proof of the implication in Lemma 3, and on tightening the trifurcation-density step in Theorem 1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Marco, quick take on the Ticse note. The paper claims uniqueness of the infinite cluster for the constrained-degree percolation model in random environment on Z^2, plus uniform positivity of the connectivity function. That's a natural and worthwhile target, and the author has correctly identified the open gap from Sanchis et al. (2022). The exposition is clean and the Burton–Keane strategy is the right one. But the central lemma, Lemma 3, has a hole that looks load-bearing.\n\nThe problem is equation (4). The event F_{n,A,δ} × Π_{G_n}(H∩U) is supposed to be a local modification that forces both the open boundary cycle and three infinite clusters intersecting B_n. The first part follows from F, but the second does not. The projection Π discards the crossing edges in O_{n,n+1} (if G_n means edges with both endpoints outside B_n) and also discards the κ values on ∂B_n. The open/closed status of the crossing edges depends on both those boundary degrees and the dynamics, and F changes the boundary degrees to all 3. There is no guarantee that the outside configuration from the projection still has three open arms connecting to the boundary cycle once the inside is replaced by F. In fact, the projection only says there exists some inside configuration that, together with some crossing-edge statuses, produced H. F may not be compatible with any three distinct infinite clusters touching ∂B_n. So Lemma 3 does not prove the existence of a positive-probability trifurcation, and Theorem 1 collapses. The same issue sinks the D_2 part.\n\nThis is not a minor typo. The ambiguity of B_n^c is real but even the favorable reading doesn't rescue it, because the projected variables are raw U and κ, not the resulting percolation configuration. To repair it, you'd need to fix an outside configuration from the support of H (or a set of them) and construct the inside modification relative to that, rather than using a universal F and an existential projection. That is a substantial rework.\n\nOn the credit side: the connectivity-function argument (Theorem 2) is a plausible consequence if uniqueness holds, and the paper is honest with the existing literature. The abstract does overstate by omitting ρ_3>0, and the δ choice needs δ<min(t,1-t), but those are minor.\n\nBottom line: the result may be true and this approach may be salvageable, but this manuscript does not prove it. I'd send it to a referee who works on percolation to give the author a clear list of fixes, but I would not accept it as is. For a reading group, it's a useful case study of why projection arguments in dependent percolation need care.","headline":"The main theorem is not established: Lemma 3's projection argument discards exactly the crossing-edge and boundary-degree information needed to guarantee the trifurcation event.","tokens_in":7285,"tokens_out":12910,"would_cite":false,"duration_ms":117740,"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":"For the random constrained-degree percolation model on the square lattice, whenever an infinite open cluster exists, it is almost surely unique; consequently, the chance that any two vertices are connected stays uniformly positive in the su","keywords":["constrained-degree percolation","random environment","unique infinite cluster","connectivity function","square lattice","dependent percolation","trifurcation argument","supercritical phase"],"falsifier":"Test Lemma 3 directly for a concrete parameter set with ρ_3 > 0 and t > t_c(ρ): compute or simulate the probability of the event in equation (4)—the forced boundary configuration together with the projected outside event of three infinite clusters intersecting the box. If this probability is zero for all sufficiently large n, or if the forced interior configuration systematically blocks the outside crossings, then the uniqueness proof fails at its only nontrivial step.","tokens_in":6328,"feed_emoji":"🕸️","tokens_out":6781,"duration_ms":64599,"temperature":0.7,"pith_summary":"This paper settles the uniqueness question for a percolation model in which each vertex has a random cap on how many incident edges may be open, and edges open at random times only if both endpoints still have room. The main claim is that on the square lattice, whenever an infinite open cluster exists, it is almost surely unique, provided a positive fraction of vertices allow three open edges. The proof builds a local event in a large box that would force a cluster with at least three ends to appear with positive density, then shows such objects are too costly because each must consume distinct boundary edges. A corollary is that the connectivity function is uniformly positive: in the supercritical phase, the probability that two given sites are connected is bounded below by a constant independent of their separation. This matters because the model has long-range dependencies and does not satisfy positive-correlation inequalities, yet its supercritical phase behaves as cleanly as independent percolation.","feed_headline":"Supercritical random-degree percolation has a unique giant cluster","feed_subtitle":"Once the model percolates, a single infinite cluster forms; distant vertices stay connected with positive probability.","key_machinery":"The key object is a local trifurcation event built on two boundary layers of a large box. The event fixes degree constraints to 3 on the box's two outer rings and forces the opening times of all boundary-crossing edges in a chosen pattern, so that the entire boundary of the box is open. Under the assumption of multiple infinite clusters it is asserted to occur with positive probability and to produce a cluster with at least three ends. The standard counting argument then shows that such trifurcation points have positive density in every large box, yet at most a linear number can fit inside a box of given perimeter, contradicting the quadratic number of vertices.","core_discovery":"The paper proves that for the variable-constrained percolation model on Z^2, the supercritical phase is governed by a single infinite cluster. Assuming several infinite clusters exist, the author constructs, inside a large box, a positive-probability event that fixes the two outermost layers of vertices to have degree constraint 3 and forces all edges along the box boundary to be open, while leaving the outside configuration untouched. Under the multiple-cluster assumption, this event is asserted to guarantee a cluster with at least three ends inside the box. Translation invariance then gives a positive density of such trifurcation points, while the standard boundary-counting argument bounds","pith_inferences":["The boundary-forcing strategy is likely portable to higher dimensions: a positive fraction of vertices with maximal allowed degree should yield uniqueness in Z^d just as it does in the deterministic degree-3 case, though the perimeter bound changes form.","Because the proof avoids monotonicity, it suggests a template for proving uniqueness in other kinetically constrained or time-dependent percolation models where positive correlation is absent.","An implicit open question is the sharpness of the uniform connectivity constant: the argument gives a positive lower bound, but its dependence on ρ and t is not optimized and might be improved with sharper correlation decay estimates.","The construction is concrete enough to simulate: checking whether the forced boundary event really has probability bounded away from zero for large boxes would provide a direct numerical test of the lemma's conclusion."],"forward_implications":["If the central claim is correct, the supercritical phase cannot contain two or more infinite clusters; the number of infinite clusters is almost surely constant and equals one whenever an infinite cluster exists at all.","The probability that two arbitrary vertices belong to the same open cluster is bounded below by a positive constant depending only on the model parameters and time, not on the distance between the vertices.","Uniqueness holds even though edge states are dependent and the model fails positive-correlation inequalities, so the argument does not require monotonicity or FKG-type structure.","For parameters with a positive fraction of degree-3 vertices, the supercritical regime t > t_c is a single-cluster phase: one infinite cluster carries the entire percolation probability.","Combined with the established exponential decay of correlations, uniqueness gives a finite-box approximation to the infinite-cluster event, which is exactly what supplies the uniform connectivity lower bound."],"fun_headline_variants":["Unique giant cluster in random-constraint percolation","Supercritical percolation with degree constraints: one infinite cluster","Random degree limits still yield a single infinite cluster","Percolation with random limits: uniqueness above criticality","Constrained-degree percolation: uniqueness of the infinite cluster"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof rests on Lemma 3's claim that, with positive probability, a local configuration forcing all boundary edges of a large box open can coexist with at least three infinite clusters reaching that box; if that event is empty or has zero probability for some parameter choices, the uniqueness argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Unique giant cluster in random-constraint percolation","Supercritical percolation with degree constraints: one infinite cluster","Random degree limits still yield a single infinite cluster","Percolation with random limits: uniqueness above criticality","Constrained-degree percolation: uniqueness of the infinite cluster"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000142,"raw_usage":{"total_tokens":910,"prompt_tokens":558,"completion_tokens":352,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":302,"completion_tokens_details":{"reasoning_tokens":270}},"tokens_in":302,"tokens_out":352,"duration_ms":4125,"temperature":1.0,"reasoning_tokens":270,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T02:55:35.524915+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Lemma 3 directly for a concrete parameter set with ρ_3 > 0 and t > t_c(ρ): compute or simulate the probability of the event in equation (4)—the forced boundary configuration together with the projected outside event of three infinite clusters intersecting the box. If this probability is zero for all sufficiently large n, or if the forced interior configuration systematically blocks the outside crossings, then the uniqueness proof fails at its only nontrivial step.","supporting_citations":[],"review_version":1}