Alignment percolation

Geoffrey R. Grimmett; Mark Holmes; Nicholas R. Beaton

REVIEW 2 major objections 5 minor 17 references

Alignment percolation

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that in two stochastic models of blue line segments on the integer lattice in dimension at least two, sparse occupancy almost surely gives a unique infinite blue cluster while dense occupancy almost surely gives none.

desk verdict A genuinely new pair of percolation models with a counterintuitive inverted transition; the main ideas are sound, but one theorem is explicitly left as a sketch and needs a formal proof before the claims are fully supported. read the letter →

arxiv 1908.07203 v2 pith:G67HT4YU submitted 2019-08-20 math.PR

Nicholas R. Beaton , Geoffrey R. Grimmett , Mark Holmes This is my paper · ORCID

classification math.PR MSC 60K35

keywords percolationone-choicemodelindependentphasetransitionblueclustersitelinesegmentsrandomgeometricgraph

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 treats two random models built on site percolation on $\mathbb{Z}^d$: each occupied site, or each feasible segment, chooses blue line segments connecting nearest occupied sites. Its central claim is that both models undergo a phase transition, but backwards from classical percolation: when occupied sites are sparse, an infinite blue cluster exists almost surely, and when they are dense, it disappears almost surely. For the one-choice model, the paper proves rigorous upper and lower bounds on this transition; for the independent model it proves a non-percolation region, two percolation regions, and uniqueness of the infinite cluster whenever one exists. A reader should care because the blue segments' density is essentially independent of $p$, so the transition is driven by geometric alignment and long-range correlations rather than by the number of blue edges.

What carries the argument

The argument is carried by four main mechanisms. The upper bound for the one-choice model uses a coupling in which every blue edge is contained in a 'corrupted compass model' of turquoise edges; non-percolation of that model for large $p$ is read off from the eigenvalues of an explicit $3\times 3$ matrix. The non-percolation bound for the independent model rests on a two-type branching process whose mean offspring are $(2d-1)\lambda/p$ for occupied sites and $2(d-1)\lambda/p$ for unoccupied sites, with extinction implying $\lambda < p/(2d-1)$. Uniqueness of infinite clusters is obtained through the finite-energy property for the independent model and through a surgical adaptation of the classical trifurcation argument for the one-choice model, which lacks finite energy. Percolation at small $p$ is proved by a block construction on boxes of side $6r$ in a two-dimensional sublattice, with $r\approx 1/\log_2(1/q)$; 'good' blocks dominate a one-dependent site percolation process, and an infinite cluster of good blocks forces an infinite blue cluster.

What would settle it

In the independent model on $\mathbb{Z}^2$, take $p=0.6$ and $\lambda=0.19$, just below $p/(2d-1)=0.2$; if the largest blue cluster in boxes of side $L$ grows linearly with $L$ as $L\to\infty$, the no-percolation claim of Theorem 2.4(i), and the branching-process domination behind it, would be wrong.

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Extended reading notes

Core claim

At the paper's core are Theorem 2.1 and Theorem 2.4. Place each site of $\mathbb{Z}^d$, $d\ge 2$, in the occupied state with probability $p$, and connect each occupied site to the next occupied site in each coordinate direction. In the one-choice model each occupied site makes exactly one of its $2d$ incident segments blue; in the independent model each feasible segment is blue with probability $\lambda$. The central discovery is that the one-choice model has a phase transition in $p$: a unique infinite blue cluster exists almost surely for small $p$, and no infinite blue cluster exists almost surely for $p$ close to $1$. For the independent model, the paper proves a nontrivial $(p,\lambda)$ phase diagram: no infinite blue cluster when $\lambda < p/(2d-1)$, a unique infinite blue cluster when $\lambda > c\log(1/q)$ with $q=1-p$, and a unique infinite blue cluster above a Lipschitz curve $\lambda_c(p)$ that interpolates between the site and bond percolation thresholds. In both models the marginal probability that a given edge is blue is constant in $p$, so the phase transition is not a density effect.

Load-bearing premise

The non-percolation bound for the independent model rests on the paper's assertion, not fully written out, that the blue cluster is no larger in distribution than a branching process with the stated average offspring counts; if that comparison is wrong, the bound $\lambda < p/(2d-1)$ does not follow from the proof.

Editorial extensions

If this is right

For the one-choice model on $\mathbb{Z}^d$, $d\ge 2$, there is a provable interval of low occupancy with a unique infinite blue cluster and a provable interval of high occupancy with none; the conjectured critical point for $d=2$ lies near $0.505$, and for $d=3$ near $0.862$.
For the independent model, the phase diagram contains at least three proven regions: no infinite cluster for $\lambda < p/(2d-1)$, and a unique infinite cluster for $\lambda > c\log(1/q)$ or above the Lipschitz frontier $\lambda_c(p)$.
When $p=1$, the independent model reduces to bond percolation, so the critical curve satisfies $\lambda_c(1)=p_{\mathrm{bond}}^c$ and $\lambda_c(p_{\mathrm{site}}^c)=1$; if $\lambda_c$ is continuous near $p=1$, the critical value approaches the bond percolation threshold as $p\to 1$.
The independent model on a $d$-dimensional lattice, restricted to a two-dimensional sublattice, is exactly a two-dimensional independent model; hence percolation proved in two dimensions lifts to higher dimensions.
In both models, the probability an individual edge is blue is independent of $p$, so the existence of a phase transition shows that large-scale connectivity is governed by correlations and segment alignment, not by edge density.

Reading between the lines

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

A natural next step, not taken in the paper, would be to test numerically whether the conjectured monotonicity of the origin's infinite-cluster probability in $p$ holds; if it does, the transition is a sharp order-disorder transition rather than merely two disconnected existence ranges.
The same inversion mechanism, long straight segments at low occupancy versus short aligned pieces at high occupancy, should appear in continuum line-segment models built from Poisson points, so the two lattice models can serve as a tractable proxy for those systems.
The two-type branching-process comparison suggests a sharper sufficient condition for non-percolation than $\lambda < p/(2d-1)$: the full extinction criterion for a two-type branching process gives a curve in $(p,\lambda)$ that could be checked against simulations.
Because the marginal edge density is constant in $p$, these models give a clean way to quantify alignment as an order parameter; one could ask whether a suitably defined collinearity index jumps at the percolation threshold.

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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

You should read this one for the one-choice model alone: Beaton, Grimmett, and Holmes introduce two line-segment percolation models based on site percolation, and prove that the one-choice model percolates for small p and fails for large p. That inverted transition is real and fresh; I don't know of another model with that behavior. The independent model is also natural, with a partial phase diagram mixing bond and site thresholds.

What's new: the models themselves, and the observation that blue edges become more aligned as p decreases, flipping the usual monotonicity. The proofs are mostly adaptations: the mixed percolation comparison for Theorem 2.4(iii) is elegant and rigorous; the block argument for supercriticality (Theorem 5.3) is careful and gives the right 1-dependent domination; the finite-energy computation for the independent model is solid.

The soft spot is exactly where the stress-test lands. The proof of Theorem 2.4(i) — no percolation when λ < p/(2d−1) — is not a proof. The text says the branching-process domination "may be written out formally" and that the combinatorial complications "do not interfere." That is not a theorem as written. One needs an honest coupling: conditional on the exploration history, the number of new sites discovered from an occupied or unoccupied site is stochastically bounded by the claimed two-type offspring distribution, and no double-counting can inflate the bound. That is plausible, maybe even standard, but it is not supplied. So the lower boundary of the independent model's phase diagram is currently unsupported as stated.

I'd also want the Burton-Keane adaptation in Theorem 5.2 checked closely. The outline is detailed enough that I don't suspect a hole, but the surgery on sites and directions is intricate, and "the arguments of [4] may be adapted" is doing some work there.

The self-citation to Hirsch–Holmes–Kleptsyn for the corrupted compass model is fine; that's a real paper and the reduction is clean.

Bottom line: the paper is worth a serious referee. The central phenomenon is interesting enough that the gap in 2.4(i) should not sink it, but the authors should either supply the missing formal domination or soften the claim to a conditional statement. I'd cite the one-choice theorem in my own work once the details are in place.

Recommendation: send to peer review; conditional acceptance, with the missing proof required before publication.

Referee Report

2 major / 5 minor

Summary. The paper defines two dependent percolation models on Z^d, d>=2, built over an occupied-site set with density p. In the one-choice model, each occupied site chooses one of its incident feasible segments uniformly and the chosen segments are blue; in the independent model, each feasible segment is blue with probability lambda independently. The authors prove partial phase diagrams. For the one-choice model, Theorem 2.1 asserts the existence of p0(d) <= p1(d) such that there is a unique infinite blue cluster for p < p0(d) and no infinite blue cluster for p > p1(d). For the independent model, Theorem 2.4 gives no percolation when lambda < p/(2d-1), percolation when lambda > c log(1/q), and percolation above a Lipschitz curve lambda_c(p) obtained by comparison with mixed site-bond percolation. The proofs use a coupling to the corrupted compass model, a branching-process upper bound, a finite-energy argument, a Burton-Keane-style uniqueness analysis adapted to the one-choice model, and a block construction with 1-dependent site percolation.

Significance. If the results are correct, the paper establishes rigorous, nontrivial phase transitions for two natural models of intersecting random line segments, with the counterintuitive feature that percolation occurs at low site density and fails at high density in the one-choice model. The block argument in Theorem 5.3 is quantitative and gives explicit exponential control, and the comparison with mixed percolation in Theorem 2.4(iii) is clean and parameter-free. The paper is also careful to separate proved results from conjectures and numerical estimates. The main weakness is that one of the two central lower-bound proofs is explicitly presented as a sketch rather than a formal argument, and the uniqueness proof for the one-choice model contains an abbreviated surgery step; both need to be supplied before the theorems can be regarded as fully proved.

major comments (2)

[Section 4, Proof of Theorem 2.4(i)] The proof of the non-percolation region for the independent model is not complete. The text asserts that the size of the blue cluster at the origin is stochastically bounded by a two-type branching process with mean offspring mu1 = (2d-1)lambda/p for occupied sites and mu2 = 2(d-1)lambda/p for unoccupied sites, but no formal domination or coupling is provided. The manuscript states only that 'the above argument may be written out formally' and that 'there are some complications arising from the conjunction of probability and combinatorics, but these do not interfere with the conclusion.' This is a load-bearing step: if the domination fails, the conclusion lambda < p/(2d-1) does not follow. A formal proof must specify the exploration order, show that, conditional on the exploration history, the number of newly discovered sites from a site of each type is stochastically dominated by the claimed offspring distributions, and handle the dependencies created by previously counted sites and by overlapping feasible segments. In particular, the heuristic that the 'true conditional expectations will typically be less' is not by itself an inequality. This gap is especially important because part (i) is the only proof of the lower phase boundary for the independent model.
[Section 5, Theorem 5.2, proof of statement C] The proof of uniqueness for the one-choice model is also abbreviated at a central point. After describing the construction of three paths rho1, rho2, rho3 from the segment S to three boundary sites z1,z2,z3, the text says 'It may be checked that this construction is always possible for m >= 7' and concludes that the origin is a trifurcation. This check is load-bearing for the a.s. uniqueness assertion in Theorem 2.1(i). The proof should explicitly verify that the three paths are disjoint except at their attachment points to S, that all edges on them are blue under the altered choices, that the surgery inside Dm does not create unintended blue connections, and that deleting the origin separates at least three distinct infinite components. Since the one-choice measure lacks finite energy, the usual Burton-Keane framework cannot simply be imported without this verification.

minor comments (5)

[Section 4, Proof of Theorem 2.4(iii)] The function lambda_c(p) defined in (4.3) for the mixed percolation model is denoted by the same symbol as the conjectured critical curve in Section 2.2. The authors should state explicitly that Theorem 2.4(iii) uses the mixed-percolation critical curve, not the conjectured independent-model frontier, to avoid confusion.
[Theorem 5.3 and Theorem 2.4(ii)] The theorem statements say lambda > c log(1/q) without specifying the base of the logarithm, while the proof uses log_2(1/q). Since the constant c is absolute, the base is immaterial, but the statements should either specify a base or note that the constant absorbs the choice.
[Equation (5.5)] The choice r = 1/log_2(1/q) is not generally an integer; the text says a 'small correction' is necessary and is overlooked. For a formal proof, the authors should either define r as a floor/ceiling and adjust the subsequent estimates, or include a sentence explaining that the error is absorbed by the constants c1 and c2.
[References] Reference [16] is listed as 'to appear' in the Annals of Applied Probability; please update the citation with the final publication data or a stable arXiv identifier, since Section 3 relies directly on its non-percolation criterion.
[Remark 2.6] The added remark on the work of Hilário and Ungaretti is useful, but the comparison with the hexagonal-lattice threshold p_H^c should be stated with the same notational conventions, since the reader must infer that p_H^c is a bond-percolation threshold on the hexagonal lattice rather than on Z^d.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: p and λ are model inputs, the block and comparison proofs are explicit, and the sole self-citation ([16]) is an independent theorem about a different model; the deferred branching-process argument is an omitted proof, not a circular reduction.

full rationale

After walking every load-bearing step, I find no reduction of a claimed result to its own input. The parameters p and λ are primitive model inputs; no parameter is fitted to data and then renamed a prediction. Theorem 2.1(i) and Theorem 2.4(ii) are proved by the explicit block construction of Theorem 5.3, with probabilities bounded directly in terms of p and λ (equations (5.4)-(5.10)); the supercritical 1-dependent site percolation comparison is imported from Liggett-Schonmann-Stacey, an independent theorem. Theorem 2.4(iii) follows by stochastic domination from the mixed percolation model of Chayes-Schonmann [6], whose critical curve properties are external. The only load-bearing self-citation is the corrupted compass model of [16] used for Theorem 2.1(ii); because [16] is a separately published theorem about a different model, invoked through the explicit coupling B⊆T and the spectral criterion (3.1), it is independent support and does not raise the circularity score. The text itself flags the weakest point: the branching-process domination in the proof of Theorem 2.4(i) is asserted and deferred ('The above argument may be written out formally'), with admitted 'complications arising from the conjunction of probability and combinatorics.' That is an omitted formal proof and therefore a correctness risk, but the claimed means μ1=(2d−1)λ/p and μ2=2(d−1)λ/p are model-derived bounds, not fitted values, so the step is incomplete rather than circular. Score 0.

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

The central claims rest on standard percolation theorems and one previously published model-specific criterion; no parameters are fitted to data. The parameters p and λ are model inputs rather than fitted values; the constants p0(d), p1(d), and c are existential and not assigned numerical values by fitting. The proof device r = log_2(1/q) is a mathematical choice, not a fitted parameter. No new physical entities are postulated.

assumptions (5)

domain assumption Non-percolation criterion for the corrupted compass model: the turquoise edge set T does not percolate when all eigenvalues of the matrix M in (3.1) have modulus < 1.
Used in Section 3 to prove Theorem 2.1(ii); cited from [16] (Hirsch, Holmes, Kleptsyn), with author overlap. Not re-derived in this paper.
standard math Liggett, Schonmann, Stacey domination theorem: a 1-dependent site percolation process on Z^2 with marginal density at least π has an infinite cluster almost surely.
Used in Theorem 5.3 to conclude percolation from a dense set of good blocks; see [17] and [10, Thm 7.65].
standard math Burton-Keane theorem: a translation-invariant probability measure on edge configurations with the finite energy property has at most one infinite cluster almost surely.
Adapted in Theorem 5.2 to the one-choice model, which lacks finite energy; cited as [4, 5, 9].
standard math Chayes-Schonmann mixed percolation properties: the critical curve λc(p) defined by (4.3) is Lipschitz continuous and strictly decreasing on [p_site^c, 1], with λc(p_site^c)=1 and λc(1)=p_bond^c.
Used in Theorem 2.4(iii) via stochastic domination; cited as [6, Thm 1.4].
standard math Standard extinction criterion for multi-type branching processes.
Used in the proof sketch of Theorem 2.4(i); see [2, Chap. V.3].

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Cite this review

Pith. "Pith review of Alignment percolation." pith.science (2026). https://pith.science/paper/G67HT4YU

@misc{pith2026190807203,
  author       = {Pith},
  title        = {Pith review of: Alignment percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G67HT4YU}},
  note         = {Machine review of arXiv:1908.07203}
}

abstract

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d \ge 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ${\mathbb Z}^d$ with parameter $p\in (0,1]$. For each occupied site $v$, and for each of the $2d$ possible coordinate directions, declare the entire line segment from $v$ to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the one-choice model, each occupied site declares one of its $2d$ incident segments to be blue. In the independent model, the states of different line segments are independent.

Figures

Figures reproduced from arXiv: 1908.07203 by the authors.

**Figure 1.1.** A 60 × 60 section of Z 2 with blue edges in the one-choice model of Example 1.1 indicated, with p = 0.8 (left) and p = 0.4 (right). Segments which cross the boundary of the box have been cut off for clarity. Segments which cross the left boundary, and those connected to them by blue edges, are coloured darker. Example 1.1 (The one-choice model). Let ω ∈ ΩV . Independently for each v ∈ η(ω), choose a segment (say fv… view at source ↗

**Figure 1.2.** A 60 × 60 section of Z 2 with blue edges in the independent model of Example 1.2 indicated, with p = 0.8 and with λ = 0.5 (left) and λ = 0.3 (right). Segments which cross the boundary of the box have been cut off for clarity. Segments which cross the left boundary, and those connected to them, are coloured darker. Occupied sites that are not blue have been omitted. decrease p. Similarly the probability that (o, e1… view at source ↗

**Figure 2.1.** A picture of the phase diagram for the independent model in two dimensions. We conjecture there exists an infinite blue cluster for (p, λ) above the dashed curve, and none below. The solid lines indicate the regions identified in Theorem 2.4. There is no percolation in region A where λ ≤ p/3, but there exists a unique infinite blue cluster in the two regions labelled B, and along the line C. percolation on Z d . I… view at source ↗

Figures from the paper (6 more)

**Figure 5.1.** Figure 5.1: An illustration of the first part of the proof of statement B in Theorem 5.2. The arrows indicate connections to infinity. For m < n, let Fm,n be the event that there exists no blue segment that intersects both Dm and Z d\Dn. Since all blue segments intersecting Dm a…

**Figure 5.2.** Figure 5.2: An illustration of the second part of the proof of B in Theorem 5.2. (2.ii) Path ρ2 is comprised of occupied vertices ρ2,0, ρ2,1, . . . , ρ2,l(= z2) and the unoccupied vertices between them. It starts at one of the 2d − 2 vertices s2 ± ei , i ∈ {2, . . . , d}, meetin…

**Figure 5.3.** Figure 5.3: An illustration of the proof of C in Theorem 5.2. The central vertex (indicated in red) is a trifurcation. set of feasible pairs. The configuration space is Φ(ω) = {0, 1} F(ω) , and for φ ∈ Φ(ω), we call f ∈ F(ω) (and the corresponding segment of Z d ) blue if φf =…

**Figure 5.4.** Figure 5.4: The events Ae1 and A(o). is feasible and blue. The probability of Ae1 may be calculated as follows. Let L be the line-segment [r, 2r) × {0}. For (k, 0) ∈ L, let Dk be the event that (a) when proceeding north from (k, 0), the first occupied vertex v encountered (inclu…

**Figure 5.5.** Figure 5.5: The regions Tei , 6re1 +T−e1 , and the event Ce1 . symmetry, (5.7) 1 − Pp,µ(A(o)) ≤ 4e −c1λr , where the event A(o) := Ae1 ∩ A−e2 ∩ A−e1 ∩ Ae2 is illustrated in Figure 5.4. On A(o), there exists a connected blue subgraph with large diameter in the box Λo. We turn ne…

**Figure 5.6.** Figure 5.6: The event C(o) is depicted in solid lines, and A(o) in dashed lines. (5.6) and (5.9), (5.10) Pp,µ(o is good) ≥ 1 − 4e −c1λr − 4e −c2λr . Let x ∈ Z 2 , and let τx be the translation on Z 2 by x, so that τx(y) = x + y. This induces a translation on ΩV , also denoted τx…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

R. M. Burton and M. Keane, Density and uniqueness in percolation , Commun. Math. Phys. 121 (1989), 501–505

work page 1989

[2] [1]

Aizenman and G

M. Aizenman and G. R. Grimmett, Strict monotonicity of critical points in percolation and ferromagnetic models, J. Statist. Phys. 63 (1991), 817–835

work page 1991

[3] [2]

K. B. Athreya and P. E. Ney, Branching Processes, Grundlehren der Mathe- matischen Wissenschaften, vol. 196, Springer-Verlag, Heidelberg, 1972

work page 1972

[4] [3]

D. J. Barsky, G. R. Grimmett, and C. M. Newman, Percolation in half- spaces: equality of critical densities and continuity of the percolation probability, Probab. Theory Related Fields 90 (1991), 111–148

work page 1991

[5] [5]

, Topological and metric properties of inﬁnite clusters in stationary two- dimensional site percolation, Israel J. Math. 76 (1991), 299–316

work page 1991

[6] [6]

Chayes and R

L. Chayes and R. H. Schonmann, Mixed percolation as a bridge between site and bond percolation, Ann. Appl. Probab. 10 (2000), 1182–1196

work page 2000

[7] [7]

D. J. Daley, S. Ebert, and G. Last, Two lilypond systems of ﬁnite line-segments, Probab. Math. Stat. 36 (2016), 221–246

work page 2016

[8] [8]

Gandolﬁ, M

A. Gandolﬁ, M. S. Keane, and C. M. Newman. Uniqueness of the inﬁnite com- ponent in a random graph with applications to percolation and spin glasses. Probability Theory and Related Fields, 92 (1992) 511–527

work page 1992