A note on the random constrained-degree percolation model on $\mathbb{L}^2$

Marco Ticse

REVIEW 2 major objections 5 minor 14 references

A note on the random constrained-degree percolation model on $\mathbb{L}^2$

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2607.25277 v1 pith:TRY5WVGE submitted 2026-07-28 math.PR math-phmath.MP

Marco Ticse This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60K3582B43

keywords constrained-degreepercolationrandomenvironmentuniqueinfiniteclusterconnectivityfunctionsquarelatticedependenttrifurcationargumentsupercriticalphase

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

This 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.

On 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

Bottom 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.

Referee Report

2 major / 5 minor

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).

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 (2)

[§4, Lemma 3, Eq. (4)] 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
[§5.1, Theorem 1] 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.

minor comments (5)

[§2, model definition] 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).
[Figure 1] 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.
[§4, Lemma 3, choice of δ] 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'.
[§5.2, Eq. (7)–(8)] 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.
[§5.1, invocation of Lemma 7.7] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained modulo external results, with one non-circular proof gap in Lemma 3.

full rationale

The paper's derivation chain does not reduce to its inputs. Theorem 1 is proved by contradiction via Burton–Keane: Lemma 3 builds a local trifurcation event, and then Lyons–Peres Lemma 7.7 plus the Burton–Keane boundary bound yield the contradiction. Lemma 3 uses only the model definition, translation invariance, and the positive-probability event constructed in (2)–(4); it does not assume the target uniqueness. Theorem 2 is a consequence of Theorem 1 plus the exponential decay of correlations from Sanchis et al. (2022), an external reference with no overlapping author; the lower bound c(ρ,t) is explicit in terms of θ(ρ,t), not fitted to the target quantity. The only self-citation is Arcanjo et al. (2026) (co-authored by Ticse), cited for deterministic higher-dimensional uniqueness and positivity; it is motivational and not used in the proofs. There is a genuine manuscript-level flaw: the projection in Lemma 3, G_n = B_n^c × B_n^c, is never defined as an edge set; if B_n^c excludes the crossing edges O_{n,n+1}, then the product event F_{n,A,δ} × Π_{G_n}(H^t_{n,A} ∩ U_{n,δ}) does not by itself force the outside infinite clusters to intersect B_n. But this is an incompleteness in the proof, not a circular reduction: no equation or fitted parameter is equivalent by construction to the claimed conclusion. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof draws on standard percolation facts (Burton–Keane, Lyons–Peres forest lemma) and on the prior CDPRE correlation-decay theorem. No free parameters are fitted; ρ and t are fixed model inputs. The central gap is not an axiom but the invalid projection step in Lemma 3.

assumptions (5)

standard math Burton–Keane uniqueness theorem for translation-invariant percolation
Used in Section 5.1 to bound trifurcation points by boundary size. External classic result.
standard math Lemma 7.7 of Lyons and Peres (2016)
Invoked in Section 5.1 to extract a translation-invariant random forest with a ≥3-ended component from the percolation graph; not proven in this note.
domain assumption Exponential decay of correlations (Theorem 2 of Sanchis et al. (2022))
Used in proof of Theorem 2 to approximate P(A0,R ∩ Az,R); the stated form with dimension-only constants needs verification.
standard math Ergodicity and translation invariance of P_{ρ,t}
Used to conclude the number of infinite clusters is a.s. constant (Section 5.1).
domain assumption ρ_3 > 0
Assumed in Theorem 1; needed to force κ=3 on boundary boxes in Lemma 3.

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Pith. "Pith review of A note on the random constrained-degree percolation model on $\mathbb{L}^2$." pith.science (2026). https://pith.science/paper/TRY5WVGE

@misc{pith2026260725277,
  author       = {Pith},
  title        = {Pith review of: A note on the random constrained-degree percolation model on $\mathbbL^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRY5WVGE}},
  note         = {Machine review of arXiv:2607.25277}
}

read the original abstract

This note considers a constrained-degree percolation model in a random environment on the square lattice, where each vertex is assigned a random degree constraint, and edges attempt to open at random times. We prove that whenever an infinite open cluster exists, it is almost surely unique, implying the positivity of the connectivity function in the supercritical regime.

Figures

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Works this paper leans on

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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