pith. sign in

arxiv: 2407.10715 · v2 · pith:INIWMG3Cnew · submitted 2024-07-15 · 🧮 math.PR

Largest component and sharpness in continuum percolation

Niclas K\"upper , Mathew D. Penrose This is my paper

Pith reviewed 2026-05-23 23:12 UTC · model grok-4.3

classification 🧮 math.PR
keywords continuum percolationPoisson random connection modellargest componentsubcritical regimesupercritical regimeexponential tailsharpnessphase transition
0
0 comments X

The pith

In the Poisson random connection model the largest component in a window grows logarithmically with volume below criticality and linearly above it.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes precise growth rates for the size of the largest connected component in finite windows of the Poisson random connection model with any bounded connection function. Below the critical intensity this size grows logarithmically in the window volume while above it the growth is linear in the volume. It further shows that the probability the cluster containing the origin exceeds size n decays exponentially when the intensity is subcritical. These controls on component sizes hold uniformly away from the percolation threshold and give a continuum analogue of sharpness results known on lattices.

Core claim

We show that the asymptotic size of the largest component restricted to a window grows logarithmically in the volume of that window in the subcritical case, and linearly in the supercritical case. We also prove a sharpness result saying that the order of the cluster at the origin has an exponentially decaying tail in the subcritical regime.

What carries the argument

Poisson random connection model with bounded connection function, analyzed separately in strictly subcritical and strictly supercritical intensity regimes.

If this is right

  • Largest-component size is o(volume) subcritically and Theta(volume) supercritically.
  • The cluster-size tail at a typical point is exponentially small below criticality.
  • The logarithmic-linear distinction holds for every bounded connection function.
  • The results supply uniform control on component sizes in large but finite regions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Logarithmic growth implies that subcritical components stay localized inside any fixed large window.
  • The same tail bounds could be used to control coverage or connectivity probabilities in random geometric graphs.
  • Numerical checks of the growth exponents for the unit-disk connection function would directly test the asymptotics.

Load-bearing premise

The connection function is bounded and the intensity lies strictly away from the critical value.

What would settle it

Observing that the largest component size in a subcritical regime grows linearly with window volume for arbitrarily large windows would falsify the logarithmic claim.

Figures

Figures reproduced from arXiv: 2407.10715 by Mathew D. Penrose, Niclas K\"upper.

Figure 1
Figure 1. Figure 1: Two instances of a near critical SRGG with parameters [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The y-axis describes the intensity of each point and the x-axis represents space. For some function f and a Poisson point process η of intensity λ. The points marked in blue (below the curve) are the f-thinning of η. Theorem 3.1 (Sharpness). For any d ≥ 1 it holds that λ˜ c = λc and (I) For all λ < λc there exists some c > 0 such that for all t θt(λ, ψ) ≤ e −ct . (II) For all λ > λc we have θ(λ, ψ) ≥ λ − λ… view at source ↗
read the original abstract

We investigate the behavior of large connected components in the Poisson Random Connection model in non-critical regimes with any bounded connection function. We show that the asymptotic size of the largest component restricted to a window grows logarithmically in the volume of that window in the subcritical case, and linearly in the supercritical case. We also prove a sharpness result saying that the order of the cluster at the origin has an exponentially decaying tail in the subcritical regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript investigates the Poisson random connection model in the plane with any bounded connection function, away from criticality. It claims to establish that the largest connected component inside a large window has asymptotic size growing logarithmically with the window volume in the subcritical regime and linearly in the supercritical regime; it further claims an exponential tail bound on the order of the cluster containing the origin in the subcritical regime.

Significance. If the stated results hold, they supply the natural continuum analogues of the classical largest-component theorems for Erdős–Rényi graphs and for lattice percolation, confirming the expected logarithmic-versus-linear dichotomy and the sharpness of the subcritical tail. The bounded-connection-function hypothesis is used explicitly to obtain clean statements without additional integrability conditions.

minor comments (2)
  1. [Abstract] The abstract refers to 'a window' without specifying its shape or the precise scaling regime; a brief sentence clarifying that the window is, e.g., a large ball or square whose volume tends to infinity would improve readability.
  2. Standard references to the corresponding results in the lattice setting (e.g., the work of Grimmett–Marstrand or Bollobás–Riordan) and to earlier continuum percolation literature are absent; adding one or two such citations would help situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No major comments were provided in the report, so we have no specific points to address point-by-point. The results as stated in the abstract appear to align with the referee's description of the claims.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes its claims on logarithmic growth of the largest component in the subcritical regime and linear growth in the supercritical regime, plus the exponential tail on cluster order, via direct mathematical arguments from the Poisson point process definition and the bounded connection function. These are standard percolation techniques applied to the continuum random connection model under explicitly stated assumptions (bounded connection function, intensity strictly away from criticality). No parameter fitting, self-definitional reductions, load-bearing self-citations, or ansatz smuggling appear; the results follow from the model axioms without reducing to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard construction of the Poisson point process and the assumption that the connection function is bounded and the intensity is non-critical; no free parameters or invented entities are introduced.

axioms (2)
  • standard math The underlying point process is a homogeneous Poisson point process on Euclidean space.
    Invoked in the model definition to obtain the random connection graph.
  • domain assumption The connection function is bounded and the intensity parameter lies strictly away from criticality.
    Required for the logarithmic/linear distinction and exponential tail to hold.

pith-pipeline@v0.9.0 · 5584 in / 1300 out tokens · 22035 ms · 2026-05-23T23:12:42.971484+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    Critical exponents for marked random connection models

    Alejandro Caicedo and Matthew Dickson. Critical exponents for marked random connection models. arXiv preprint arXiv:2305.07398 , 2023. 19

    work page arXiv 2023

  2. [2]

    The triangle condition for the marked random connection model

    Matthew Dickson and Markus Heydenreich. The triangle condition for the marked random connection model. arXiv preprint arXiv:2210.07727 , 2022

    work page arXiv 2022

  3. [3]

    Expansion of the critical intensity for the random connection model

    Matthew Dickson and Markus Heydenreich. Expansion of the critical intensity for the random connection model. arXiv preprint arXiv:2309.08830 , 2023

    work page arXiv 2023

  4. [4]

    Introduction to Bernoulli percolation, 2018

    Hugo Duminil-Copin. Introduction to Bernoulli percolation, 2018

    work page 2018

  5. [5]

    A new proof of the sharpness of the phase transition for Bernoulli percolation on Zd, 2015

    Hugo Duminil-Copin and Vincent Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation on Zd, 2015

    work page 2015

  6. [6]

    The Total Progeny in a Branching Process and a Related Random Walk

    Meyer Dwass. The Total Progeny in a Branching Process and a Related Random Walk. Journal of Applied Probability , 6(3):682–686, 1969

    work page 1969

  7. [7]

    M. Fekete. ¨Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Mathematische Zeitschrift, 17(1):228–249, 1923

    work page 1923

  8. [8]

    Lace Expansion and Mean-Field Behavior for the Random Connection Model, 2019

    Markus Heydenreich, Remco van der Hofstad, G¨ unter Last, and Kilian Matzke. Lace Expansion and Mean-Field Behavior for the Random Connection Model, 2019

    work page 2019

  9. [9]

    On the Ornstein-Zernike equation for stationary cluster processes and the random connection model, 2016

    G¨ unter Last and Sebastian Ziesche. On the Ornstein-Zernike equation for stationary cluster processes and the random connection model, 2016

    work page 2016

  10. [10]

    Bernoulli percolation on the Random Geometric Graph, 2022

    Lyuben Lichev, Bas Lodewijks, Dieter Mitsche, and Bruno Schapira. Bernoulli percolation on the Random Geometric Graph, 2022

    work page 2022

  11. [11]

    Meester and R

    R. Meester and R. Roy. Continuum Percolation. Cambridge Tracts in Mathematics. Cambridge University Press, 1996

    work page 1996

  12. [12]

    Mathew D. Penrose. On a Continuum Percolation Model. Advances in Applied Probability, 23(3):536–556, 1991

    work page 1991

  13. [13]

    Mathew D. Penrose. On the Spread-Out Limit for Bond and Continuum Percola- tion. The Annals of Applied Probability , 3(1):253–276, 1993

    work page 1993

  14. [14]

    Mathew D. Penrose. Random Geometric Graphs . Oxford studies in probability. Oxford University Press, 2003

    work page 2003

  15. [15]

    Mathew D. Penrose. Connectivity of soft random geometric graphs. The Annals of Applied Probability, 26(2), 4 2016

    work page 2016

  16. [16]

    Mathew D. Penrose. Giant component of the soft random geometric graph, 2022. 20

    work page 2022