Exponential decay for the random connection model using asymptotic transitivity

Frankie Higgs

REVIEW 3 major objections 4 minor 13 references

Exponential decay for the random connection model using asymptotic transitivity

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that in the subcritical random connection model, the probability that the origin's cluster has size at least n decays exponentially in n, for any radial, non-increasing connection function with finite positive integral —…

desk verdict New and likely correct exponential-decay result for subcritical RCM with unbounded connections, held back by an over-stated key proposition that needs a finite-box fix. read the letter →

arxiv 2509.02310 v1 pith:HYD7BXSC submitted 2025-09-02 math.PR

Frankie Higgs This is my paper · ORCID

classification math.PR MSC 60K3560D0582B43

keywords randomconnectionmodelcontinuumpercolationexponentialdecaysubcriticalphasestochasticcomparisonasymptotictransitivityghostfieldlong-rangeconnections

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

The paper establishes exponential decay of the cluster-size distribution throughout the subcritical phase of the Poisson random connection model (RCM), under only the minimal assumption that the connection function has finite positive integral. Previously such exponential tails were known only when the connection function had bounded support; the new result covers long-range connection functions, where a cluster can have large diameter while still being very small in size. The proof works by constructing the subcritical RCM as Bernoulli site percolation on a much higher-intensity RCM, then showing that the high-intensity graph becomes 'asymptotically vertex-transitive': probabilities of events around each vertex converge to a common value uniformly over vertices. This transfers a stochastic-comparison method, originally developed for Bernoulli percolation on vertex-transitive graphs, to continuum percolation, and the author expects the method to apply to other continuum models such as the Poisson Boolean model.

What carries the argument

The central objects are the ghost field, an independent set of green vertices of intensity 1 − $e^{{-h}}$ that detects large clusters; the construction of the subcritical RCM by keeping each vertex of an RCM of intensity Nλ with probability 1/N; an exploration-and-pivotality comparison lemma showing that conditioning on the ghost field avoiding the origin's cluster changes the percolation law no more than lowering the retention probability by a factor close to 1 − m_h(λ); and the 'asymptotic vertex-transitivity' estimate (Proposition 2.13), which bounds uniformly over all vertices the conditional probability of connecting to the ghost field by m_h(λ) + δ for all large N. The uniformity is achieved by discretising space into small boxes, proving large-deviation estimates for the numbers of vertices and edges in each box, and comparing the thinned high-intensity model with a Poisson-λ reference model via the Harris–FKG inequality.

What would settle it

Simulate ψ_n(λ) for a subcritical intensity with an unbounded connection function such as g(x) = c(1 + ||x||)^{-d-ε} in dimension 2 at λ = 0.9 λ_c; if log ψ_n(λ) + cn does not stay bounded for some c > 0 as n grows, Theorem 1.3 would fail. Alternatively, test Proposition 2.13 directly: over many samples of the Nλ-intensity RCM, compute the maximum over vertices of the conditional ghost-connection probability; if this maximum exceeds m_h(λ) + δ for large N with non-negligible frequency, the uniform bound collapses.

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Extended reading notes

Core claim

Theorem 1.3 states that for every connection function g with 0 < ∫g < ∞ and every intensity λ below the critical intensity λc(g), there exist constants c, C > 0 such that the probability the origin's cluster has at least n vertices is at most C $e^{{-cn}}$ for all n. The engine is a stochastic comparison: the law of the RCM at intensity λ′ = λ(1 − m_h(λ)) is stochastically dominated by the law of an intensity-λ RCM conditioned on a 'ghost field' of green vertices avoiding the origin's cluster, which yields ψ_n(λ′) ≤ (1 − m_h(λ))^{-1} ψ_n(λ) $e^{{-hn}}$. Iterating this relation gives the exponential tail. For bounded connection functions the comparison is proved directly using the asymptotic-transitivity estimate; the general case follows by truncating g at a large radius, proving the truncated models converge to the original via a dominated-convergence argument, and then taking limits.

Load-bearing premise

The proof rests on the asymptotic-vertex-transitivity estimate: in the very high-intensity random connection model with thinning, every vertex's conditional probability of connecting to the ghost field is eventually bounded by the common limiting value m_h(λ) plus a small error, for all vertices simultaneously — this uniformity, not implied by the theorem's assumptions, is the load-bearing new step that replaces exact transitivity.

Editorial extensions

If this is right

In every subcritical phase λ < λc, the probability that a cluster contains n vertices is exponentially small, so large clusters are rare even when connection edges are unbounded.
The exponential tail holds in every dimension; in dimension one, the result is unconditional whenever the connection function has finite integral, even if the subcritical condition is trivial.
The stochastic comparison of Theorem 1.5 gives a quantitative relation between cluster-size probabilities at different intensities, which can be used to bound probabilities near criticality.
The truncation argument transfers the result from bounded-support connection functions to long-range ones, and the author notes the same approach would transfer exponential decay from finite-range to long-range percolation on lattices.

Reading between the lines

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

The uniform asymptotic-transitivity estimate likely extends to any translation-invariant point process with finite intensity, so the same two-step argument (high-intensity approximation plus uniform ghost-field control) may prove exponential cluster-volume decay for the Poisson Boolean model as well.
One could test sharpness numerically: for a heavy-tailed g with finite integral, the exponential rate c(λ) in Theorem 1.3 is expected to vanish as λ approaches λc, and the comparison inequality might give explicit bounds on how it vanishes.
The method suggests a general principle: if a continuum percolation model can be approximated by thinning a high-intensity version that becomes 'asymptotically transitive', exponential decay transfers from the transitive discrete setting without needing exact symmetry of the underlying graph.
A direct check of Proposition 2.13 would be to simulate the Nλ-intensity RCM with retention 1/N, compute the maximum over vertices of the conditional probability of connecting to the ghost field, and verify it converges to m_h(λ) as N grows; failure for some connection function would pinpoint exactly where the uniform bound breaks.

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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

Hi — quick take on Higgs, arXiv:2509.02310. The headline is that the paper proves something new: exponential decay of the cluster size in the subcritical random connection model under only integrability of the connection function. That was previously known only for bounded support (Küpper–Penrose), and the long-range case was genuinely open until now. The strategy is to adapt Vanneuville's stochastic-comparison proof by constructing the subcritical RCM as site percolation on a much denser RCM, then showing the dense graph is "asymptotically vertex-transitive". That transfer idea is the real contribution and I expect it will be reused.

The paper is mostly well argued. Section 3, where they pass from cut-off to full connection functions, is clean: Proposition 3.3 with the dominated-convergence argument is convincing, and Lemma 3.6 is a nice use of the branching-process domination. Lemma 2.16 and Lemma 2.18 are standard large-deviation estimates done carefully. The citation pattern is honest: the bounded-support prior is cited, Vanneuville is cited, and the author explicitly notes that no long-range proof exists elsewhere.

Now the soft spots. The biggest issue is Proposition 2.13. As stated, it claims a uniform bound on the supremum over all x in the infinite point process η_N^0. That is not what the proof gives and, as the stress-test note says, the infinite supremum is false as written for fixed N: with positive probability there are local configurations anywhere that make the conditional green-connection probability close to 1. The proof only establishes the bound for x in a fixed box Λ_K, with constants depending on K. Fortunately Lemma 2.8 only needs the finite-box version, so the argument can be repaired by restating the proposition with sup over x in η_N^0 ∩ Λ_K and carrying K through. Until that is made explicit, the proof of Lemma 2.8 relies on a false statement. This is a real flaw in the manuscript, but a repairable one.

A second, smaller gap: Lemma 2.6 is asserted with "little more than verifying" and no proof. I believe it, but a referee will want the verification, especially since the exploration in Definition 2.4 is a bit under-specified for random graphs. Also the transfer from the discretized graph Ĝ_s back to G_g in Proposition 2.13 is somewhat implicit; it relies on Lemma 2.12 and a monotonicity step that should be spelled out.

On balance: the central theorem is very likely true, the method is good, and the gaps are presentational/fixable. As submitted I would not accept it, but I would certainly send it to a serious referee rather than desk-reject. The right venue is a probability journal. If I were refereeing, I would ask for the finite-box reformulation of Proposition 2.13 and a proof of Lemma 2.6 before recommending acceptance.

Referee Report

3 major / 4 minor

Summary. The paper proves exponential decay of the cluster size tail for subcritical Poisson random connection models (RCM) under the minimal integrability assumption on the connection function, allowing unbounded support. The proof adapts Vanneuville's stochastic comparison method from Bernoulli percolation on vertex-transitive graphs to the RCM. The key new ingredient is an 'asymptotic vertex-transitivity' estimate: the subcritical RCM is represented as site percolation on a high-intensity RCM, and the conditional probability that a vertex connects to a ghost field is shown to be nearly uniform across vertices. Section 2 treats bounded connection functions via discretization, large deviations, and a finite-box stochastic comparison lemma. Section 3 extends to unbounded connection functions by truncation, a dominated convergence argument, and a limiting procedure. The main theorems are Theorem 1.3 (exponential decay for all subcritical intensities) and Theorem 1.5 (a quantitative stochastic comparison bound involving the ghost-field intensity).

Significance. If correct, Theorem 1.3 is a substantial advance: it gives the first exponential decay result for the RCM with long-range (unbounded-support) connection functions under essentially optimal assumptions. The proof is self-contained modulo standard external results (phase transition, branching-process domination, large deviations), and the constants are existential rather than fitted. The 'asymptotic vertex-transitivity' idea is novel and plausibly useful for transferring other discrete percolation results to continuum models. The paper is clearly written and carefully distinguishes bounded-support and unbounded-support cases. The main issues are localized to the statement and proof of Proposition 2.13, and are repairable without changing the overall strategy.

major comments (3)

[Section 2.2, Proposition 2.13] Proposition 2.13 is false as stated if η_N is the infinite Poisson process (or a process on a box larger than Λ_K). The supremum is taken over all x in η_N^0, but the proof only controls x in Λ_K, and all bounds depend on K. For a fixed N, by ergodicity of the Poisson process, local configurations in a bounded region that force the conditional connection probability close to 1 occur infinitely often almost surely, so the supremum over all x exceeds m_h(λ)+δ almost surely whenever m_h(λ)+δ<1. Lemma 2.8 only needs the estimate on the finite vertex set inside Λ_{nR}, so the proposition should be restated with the supremum over x in η_N^0∩Λ_K (with η_N a Poisson process in Λ_K, or with an explicit tail estimate outside Λ_K). This is a load-bearing gap because Lemma 2.8 relies on the estimate, but the repair is local.
[Section 2.2, proof of Proposition 2.13] The proof bounds the conditional probability that x is connected to a green vertex in the discretized graph Ĝ_s(ω_N∪{x}), but the proposition statement concerns the original graph G_g(ω_N∪{x}). The transfer between Ĝ_s and G_g via Lemma 2.12 is not written. Without an explicit argument that the two conditional probabilities differ by a quantity tending to zero in probability, the final conclusion does not follow from the displayed inequality. This is repairable by adding a short coupling argument, but it must be made explicit.
[Section 2.1, Lemma 2.6] Lemma 2.6 is the core stochastic-comparison tool, but its proof is only a one-line reference to [12], asserting that the arguments transfer to site percolation. Since the setting here involves site percolation with a ghost field, and the exploration and pivotal conditions are stated for a joint law, the adaptation is not completely immediate. Please provide a full proof, or at least a detailed statement of how the proof of [12, Lemma 8] is modified to this setting. This is important for the reader to verify the central comparison step.

minor comments (4)

[Section 2.1, proof of Lemma 2.8] In the final step of the proof, after taking N→∞, the continuity of λ↦P^n_{λ,g}(A) is invoked to let δ↓0. Please add a one-sentence justification, e.g., by writing the probability as a finite sum/integral over the bounded box and using the explicit Poisson distribution.
[Section 3] Theorem 1.3 is derived from Theorem 1.5 by referring to [12] without further detail. Since the derivation is short, please include it explicitly so the paper is self-contained.
[Section 2.2, Definition 2.15] In Definition 2.15, H_N^λ(z) is defined as |η∩Λ_s(z)| without specifying η; it should be made explicit that η is the high-intensity process η_N from the surrounding context. A small clarification would prevent ambiguity.
[Throughout] The notation η_N^0 is used in Proposition 2.13 and elsewhere without always specifying whether η_N is restricted to a box. Once the finite-box version is adopted, please define η_N^0 consistently in each statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exponential-decay result is derived from a ghost-field comparison inequality whose auxiliary quantity m_h is not fitted to the target tail, and no load-bearing self-citation appears.

full rationale

The derivation is self-contained, and no load-bearing step reduces to its own input. The target ψ_n(λ)=P[|C0|≥n] is bounded via Theorem 1.5, which states ψ_n(λ(1−m_h(λ))) ≤ (1−m_h(λ))^{-1}ψ_n(λ)e^{-hn}. Here m_h(λ)=P[C0∩G≠∅] is an auxiliary ghost-intersection probability, not the n-dependent tail probability being proved; it is determined by λ and h, not fitted to ψ_n. Lemma 2.8 and Proposition 2.13 establish the comparison by conditioning the high-intensity RCM on the ghost-free event and showing that the pivotal probability is asymptotically bounded by m_h(λ)+δ. That bound is not true by construction: the high-intensity conditional probability is a different object from m_h(λ), and the proof supplies a nontrivial discretization argument (Lemmas 2.12, 2.16, 2.18) plus a Poisson comparison. The final decay follows by choosing q∈(λ,λ_c) and h with λ=q(1−m_h(q)), so the exponential factor e^{-hn} gives the bound with constants depending on λ and g. Section 3 extends to unbounded g by a cutoff g_R and dominated convergence; no parameter is fitted to the conclusion. The method is borrowed from Vanneuville [12], but that is an external source, not a self-citation, and the paper's asymptotic-transitivity estimate is genuinely new content. A possible correctness gap in the infinite-volume formulation of Proposition 2.13 would be a mathematical flaw, not circularity, and does not change this assessment.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters and no new physical or probabilistic entities. The 'ghost field' and 'asymptotic transitivity' are tools borrowed or developed within the proof; the latter is a property, not an entity. All constants are existential. The main external inputs are the phase transition result and the branching process bound from Meester-Roy, plus large deviations theorems.

assumptions (4)

domain assumption Phase transition theorem for the RCM with finite integral connection function, giving a critical intensity lambda_c (Meester-Roy [10], Theorem 6.1).
Invoked to define the subcritical regime and to ensure that clusters are finite almost surely when lambda < lambda_c.
domain assumption Stochastic comparison lemma (Lemma 2.6) adapted from Vanneuville [12] to site percolation on finite graphs.
The entire transfer of conditional stochastic domination relies on this lemma; its proof is only sketched as a verification.
domain assumption Branching process domination of cluster size in the RCM (Meester-Roy [10], proof of Theorem 6.1).
Used in Lemma 3.6 to establish integrability of the connectivity function g_2 via domination by a Galton-Watson process.
standard math Large deviations estimates for binomial and Poisson random variables (Cramer's theorem, Le Cam's approximation).
Used in Lemma 2.16 and Lemma 2.18 to obtain exponential bounds on deviations of box counts and edge probabilities.

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Pith. "Pith review of Exponential decay for the random connection model using asymptotic transitivity." pith.science (2026). https://pith.science/paper/HYD7BXSC

@misc{pith2026250902310,
  author       = {Pith},
  title        = {Pith review of: Exponential decay for the random connection model using asymptotic transitivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HYD7BXSC}},
  note         = {Machine review of arXiv:2509.02310}
}

abstract

We prove that the probability the cluster of the origin in a subcritical Poisson random connection model (RCM) has size at least $n$ decays exponentially as $n$ increases, under minimal assumptions. We extend a recent method of Vanneuville (arXiv:2304.12110) from Bernoulli percolation on vertex-transitive graphs to the RCM. The key idea is that the subcritical RCM can be constructed by site percolation on a very high-intensity RCM. The latter RCM becomes ``almost vertex-transitive'' in a certain sense at very high intensities, which is a new method that we expect to be useful for other problems. We obtain the result for connection functions with unbounded support, a setting in which it was not previously known.

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Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages

[12]

Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison

Hugo Vanneuville. Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison. arXiv preprint, 2024. arXiv:2304.12110. 17

work page Pith review arXiv 2024
[1]

Noise sensitivity in continuum percolation

Daniel Ahlberg, Erik Broman, Simon Griffiths, and Robert Morris. Noise sensitivity in continuum percolation. Israel J. Math. , 201(2):847–899, 2014. doi:10.1007/s11856-014-1038-y . 16

work page doi:10.1007/s11856-014-1038-y 2014
[2]

Large deviations, volume 14 of Fields Institute Mono- graphs

Frank den Hollander. Large deviations, volume 14 of Fields Institute Mono- graphs. American Mathematical Society, Providence, RI, 2000

work page 2000
[3]

The emergence of a giant component in one-dimensional inhomogeneous networks with long- range effects

Peter Gracar, Lukas L¨ uchtrath, and Christian M¨ onch. The emergence of a giant component in one-dimensional inhomogeneous networks with long- range effects. In Megan Dewar, Pawe l Pra lat, Przemys law Szufel, Fran¸ cois Th´ eberge, and Ma lgorzata Wrzosek, editors, Algorithms and Models for the Web Graph , pages 19–35, Cham, 2023. Springer Nature Switze...

work page doi:10.1007/978-3-031-32296-9_2 2023
[4]

Grimmett and David R

Geoffrey R. Grimmett and David R. Stirzaker. Probability and random pro- cesses. Oxford University Press, Oxford, fourth edition, 2020

work page 2020
[5]

Lace expansion and mean-field behavior for the random connection model

Markus Heydenreich, Remco van der Hofstad, G¨ unter Last, and Kilian Matzke. Lace expansion and mean-field behavior for the random connection model. arXiv preprint, 2019. doi:10.48550/ARXIV.1908.11356

arXiv doi:10.48550/arxiv.1908.11356 2019
[6]

Power-law bounds for critical long-range percolation below the upper-critical dimension

Tom Hutchcroft. Power-law bounds for critical long-range percolation below the upper-critical dimension. Probab. Theory Related Fields, 181(1-3):533–570,

work page
[7]

Niclas K¨ upper and Mathew D. Penrose. Largest component and sharpness in subcritical continuum percolation. arXiv preprint, 2024. arXiv:2407.10715

work page Pith review arXiv 2024

Show all 13 references

[8]

Lectures on the Poisson process , volume 7 of Institute of Mathematical Statistics Textbooks

G¨ unter Last and Mathew Penrose. Lectures on the Poisson process , volume 7 of Institute of Mathematical Statistics Textbooks . Cambridge University Press, Cambridge, 2018

2018
[9]

An approximation theorem for the Poisson binomial distri- bution

Lucien Le Cam. An approximation theorem for the Poisson binomial distri- bution. Pacific J. Math. , 10:1181–1197, 1960. URL: http://projecteuclid. org/euclid.pjm/1103038058

1960
[10]

Continuum Percolation

Ronald Meester and Rahul Roy. Continuum Percolation . Cambridge Tracts in Mathematics. Cambridge University Press, 1996. doi:10.1017/ CBO9780511895357

1996
[11]

Random walk on the random connection model

Ercan S¨ onmez and Arnaud Rousselle. Random walk on the random connection model. Indagationes Mathematicae, 33(5):1049–1060, 2022. doi:10.1016/j. indag.2022.05.002

2022 doi
[2021]

doi:10.1007/s00440-021-01043-7

doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [12]

Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison

Hugo Vanneuville. Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison. arXiv preprint, 2024. arXiv:2304.12110. 17

work page Pith review arXiv 2024

[2] [1]

Noise sensitivity in continuum percolation

Daniel Ahlberg, Erik Broman, Simon Griffiths, and Robert Morris. Noise sensitivity in continuum percolation. Israel J. Math. , 201(2):847–899, 2014. doi:10.1007/s11856-014-1038-y . 16

work page doi:10.1007/s11856-014-1038-y 2014

[3] [2]

Large deviations, volume 14 of Fields Institute Mono- graphs

Frank den Hollander. Large deviations, volume 14 of Fields Institute Mono- graphs. American Mathematical Society, Providence, RI, 2000

work page 2000

[4] [3]

The emergence of a giant component in one-dimensional inhomogeneous networks with long- range effects

Peter Gracar, Lukas L¨ uchtrath, and Christian M¨ onch. The emergence of a giant component in one-dimensional inhomogeneous networks with long- range effects. In Megan Dewar, Pawe l Pra lat, Przemys law Szufel, Fran¸ cois Th´ eberge, and Ma lgorzata Wrzosek, editors, Algorithms and Models for the Web Graph , pages 19–35, Cham, 2023. Springer Nature Switze...

work page doi:10.1007/978-3-031-32296-9_2 2023

[5] [4]

Grimmett and David R

Geoffrey R. Grimmett and David R. Stirzaker. Probability and random pro- cesses. Oxford University Press, Oxford, fourth edition, 2020

work page 2020

[6] [5]

Lace expansion and mean-field behavior for the random connection model

Markus Heydenreich, Remco van der Hofstad, G¨ unter Last, and Kilian Matzke. Lace expansion and mean-field behavior for the random connection model. arXiv preprint, 2019. doi:10.48550/ARXIV.1908.11356

arXiv doi:10.48550/arxiv.1908.11356 2019

[7] [6]

Power-law bounds for critical long-range percolation below the upper-critical dimension

Tom Hutchcroft. Power-law bounds for critical long-range percolation below the upper-critical dimension. Probab. Theory Related Fields, 181(1-3):533–570,

work page

[8] [7]

Niclas K¨ upper and Mathew D. Penrose. Largest component and sharpness in subcritical continuum percolation. arXiv preprint, 2024. arXiv:2407.10715

work page Pith review arXiv 2024