pith. sign in

arxiv: 2605.31568 · v1 · pith:E3LNZ6GKnew · submitted 2026-05-29 · 🧮 math.PR

Lipschitz continuity of the time constant for continuum percolation

Karoline Dubin , Christian Gorski This is my paper

Pith reviewed 2026-06-28 20:47 UTC · model grok-4.3

classification 🧮 math.PR
keywords continuum percolationBoolean modeltime constantLipschitz continuityPoisson point processchemical distancesupercritical regime
0
0 comments X

The pith

Above the critical intensity, the time constant in the Boolean model is Lipschitz continuous in the intensity.

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

This paper proves that in the supercritical regime of continuum percolation using the Boolean model, the time constant—which measures the asymptotic ratio of graph distance to Euclidean distance—is a Lipschitz continuous function of the Poisson point intensity. Previous work established the existence of this limit, but this result adds that small changes in intensity produce proportionally bounded changes in the time constant. A reader would care because it quantifies how sensitive the connectivity speed is to density variations in random geometric graphs.

Core claim

In the Boolean model where points are generated by a Poisson process in R^d and connected if within distance 1, the time constant is defined as the limit of the chemical distance divided by Euclidean distance for distant points. We establish that this limit, which exists above the critical intensity, is Lipschitz continuous as a function of the intensity. The proof is by adapting an argument originally developed for discrete lattices to the continuous case.

What carries the argument

Adaptation of the Can-Nakajima-Nguyen argument from the discrete lattice setting to the continuous Boolean model that preserves uniform estimates on the time constant's dependence on intensity.

Load-bearing premise

The adaptation of the discrete lattice argument to the continuous Boolean model preserves the necessary uniform estimates without introducing new technical obstructions.

What would settle it

A calculation or simulation at two intensities above critical where the time constant difference exceeds any fixed multiple of the intensity difference would disprove the Lipschitz claim.

read the original abstract

We consider the Boolean model of continuum percolation, where points are placed in $\mathbb{R}^d$ by a Poisson point process and pairs of points with distance at most 1 are connected by an edge. The time constant is the limiting ratio of the chemical distance (i.e. graph distance) to the Euclidean distance for pairs of distant connected points. Yao, Chen, and Guo established the existence of a time constant in the supercritical regime. We show that above the critical intensity, the time constant is a Lipschitz continuous function of the intensity. The proof adapts a recent argument of Can, Nakajima, and Nguyen to the continuous setting.

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

2 major / 1 minor

Summary. The manuscript proves that above the critical intensity, the time constant (limiting ratio of chemical distance to Euclidean distance) in the supercritical Boolean model driven by a Poisson point process in R^d is a Lipschitz continuous function of the intensity. The proof adapts the recent discrete-lattice argument of Can, Nakajima, and Nguyen to the continuous setting, relying on the existence result of Yao, Chen, and Guo.

Significance. If the adaptation succeeds without new obstructions, the result would establish a basic regularity property for the time constant in continuum percolation, which is relevant for parameter dependence in random geometric graphs. The contribution is incremental rather than foundational, as it transfers an existing discrete argument; no machine-checked proofs, reproducible code, or parameter-free derivations are provided.

major comments (2)
  1. [Proof of the main result (adaptation of Can-Nakajima-Nguyen argument)] The central claim requires that the uniform estimates (subadditive or concentration bounds controlling the chemical-distance increment for intensity changes of size δ) transfer from the discrete lattice to the Poisson Boolean model. The Poisson process introduces random local density and unbounded degree, which could disrupt uniformity of the tail/moment controls used in the discrete case. This issue is load-bearing and must be addressed explicitly in the adaptation section of the proof.
  2. [Section establishing the Lipschitz continuity] The manuscript relies on an external existence result and an external recent argument without indicating that the Lipschitz property is re-derived from first principles in the continuum; any loss of uniformity in the estimates would undermine the Lipschitz conclusion. A concrete verification (e.g., re-establishment of the relevant tail bounds for the random geometric graph) is needed.
minor comments (1)
  1. [Abstract] The abstract is concise but could briefly indicate the dimension d and the precise definition of the time constant to aid readers unfamiliar with the model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive feedback. Below we respond point-by-point to the major comments, agreeing that explicit verification of the continuum estimates would strengthen the presentation. We plan to incorporate these clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Proof of the main result (adaptation of Can-Nakajima-Nguyen argument)] The central claim requires that the uniform estimates (subadditive or concentration bounds controlling the chemical-distance increment for intensity changes of size δ) transfer from the discrete lattice to the Poisson Boolean model. The Poisson process introduces random local density and unbounded degree, which could disrupt uniformity of the tail/moment controls used in the discrete case. This issue is load-bearing and must be addressed explicitly in the adaptation section of the proof.

    Authors: We agree that the transfer of uniform estimates must be made fully explicit. The adaptation in the manuscript uses the complete independence of the Poisson point process together with Chernoff-type bounds on the number of points in fixed-radius balls to control local density fluctuations uniformly in the intensity parameter. Unbounded degrees are controlled by the exponential tail of the Poisson distribution for the number of neighbors within distance 1. We will add a dedicated paragraph in the adaptation section that re-derives the relevant subadditive and concentration bounds directly in the continuum setting, confirming that the same δ-uniformity holds. revision: yes

  2. Referee: [Section establishing the Lipschitz continuity] The manuscript relies on an external existence result and an external recent argument without indicating that the Lipschitz property is re-derived from first principles in the continuum; any loss of uniformity in the estimates would undermine the Lipschitz conclusion. A concrete verification (e.g., re-establishment of the relevant tail bounds for the random geometric graph) is needed.

    Authors: The Lipschitz property is obtained by carrying out the Can-Nakajima-Nguyen argument step by step in the continuum, using the existence result of Yao, Chen, and Guo only to guarantee that the time constant is well-defined and finite in the supercritical regime. To address the concern, we will insert a short but self-contained verification that re-establishes the necessary tail bounds for the chemical-distance increments in the Boolean model, employing the Mecke formula and standard Poisson tail estimates to show that the moment controls remain uniform. revision: yes

Circularity Check

0 steps flagged

No circularity; central result adapts external existence theorem and external Lipschitz argument

full rationale

The paper establishes existence of the time constant via citation to Yao-Chen-Guo (external) and obtains Lipschitz continuity by adapting the Can-Nakajima-Nguyen argument (external, recent, non-overlapping authors). No self-citations appear in the load-bearing steps, no fitted parameters are renamed as predictions, and no uniqueness theorem or ansatz is imported from the authors' own prior work. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the prior existence theorem of Yao-Chen-Guo and on the transferability of the Can-Nakajima-Nguyen technique; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence of the time constant in the supercritical regime (Yao, Chen, Guo)
    The Lipschitz result is stated conditional on this prior existence.

pith-pipeline@v0.9.1-grok · 5623 in / 1182 out tokens · 24010 ms · 2026-06-28T20:47:13.187334+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The geometry of the giant component of random geometric graphs

    math.PR 2026-06 unverdicted novelty 8.0

    Above a dimension-dependent critical average degree, the giant component of a random geometric graph on a compact Riemannian manifold converges in rescaled Gromov-Hausdorff distance to the manifold.

Reference graph

Works this paper leans on

29 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    Antal and A

    P. Antal and A. Pisztora. On the chemical distance for supercritical Bernoulli percolation.Ann. Probab., 24(2):1036–1048, 1996

    1996

  2. [2]

    Auffinger, M

    A. Auffinger, M. Damron, and J. Hanson.50 years of first-passage percolation, volume 68 ofUniversity Lecture Series. American Mathematical Society, Providence, RI, 2017

    2017

  3. [3]

    V. H. Can, S. Nakajima, and V. Q. Nguyen. Lipschitz-continuity of time constant in generalized first-passage percolation. Stochastic Process. Appl., 175:Paper No. 104402, 15, 2024. 1, 2, 3, 11

    2024

  4. [4]

    Cerf and B

    R. Cerf and B. Dembin. The time constant for Bernoulli percolation is Lipschitz continuous strictly abovep c.Ann. Probab., 50(5):1781–1812, 2022. 2

    2022

  5. [5]

    Cerf and M

    R. Cerf and M. Th´ eret. Weak shape theorem in first passage percolation with infinite passage times.Ann. Inst. Henri Poincar´ e Probab. Stat., 52(3):1351–1381, 2016

    2016

  6. [6]

    J. T. Cox. The time constant of first-passage percolation on the square lattice.Adv. in Appl. Probab., 12(4):864–879, 1980. 2

    1980

  7. [7]

    J. T. Cox and R. Durrett. Some limit theorems for percolation processes with necessary and sufficient conditions.Ann. Probab., 9(4):583–603, 1981

    1981

  8. [8]

    J. T. Cox and H. Kesten. On the continuity of the time constant of first-passage percolation.J. Appl. Probab., 18(4):809– 819, 1981. 2

    1981

  9. [9]

    Damron, J

    M. Damron, J. Hanson, and P. Sosoe. Sublinear variance in first-passage percolation for general distributions.Probab. Theory Related Fields, 163(1-2):223–258, 2015

    2015

  10. [10]

    B. Dembin. Regularity of the time constant for a supercritical Bernoulli percolation.ESAIM Probab. Stat., 25:109–132,

  11. [11]

    Garet and R

    O. Garet and R. Marchand. Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster.ESAIM Probab. Stat., 8:169–199, 2004

    2004

  12. [12]

    Garet and R

    O. Garet and R. Marchand. Large deviations for the chemical distance in supercritical Bernoulli percolation.Ann. Probab., 35(3):833–866, 2007

    2007

  13. [13]

    Garet, R

    O. Garet, R. Marchand, E. B. Procaccia, and M. Th´ eret. Continuity of the time and isoperimetric constants in supercritical percolation.Electron. J. Probab., 22:Paper No. 78, 35, 2017. 2

    2017

  14. [14]

    Gorski and E

    C. Gorski and E. B. Procaccia. Chemical distance in graphs of polynomial growth, 2025. 3

    2025

  15. [15]

    Grimmett.Percolation, volume 321 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    G. Grimmett.Percolation, volume 321 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999

    1999

  16. [16]

    J. M. Hammersley and D. J. A. Welsh. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. InProc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif., 1963, pages 61–110. Springer, New York, 1965

    1963

  17. [17]

    H. Kesten. Aspects of first passage percolation. In ´Ecole d’´ et´ e de probabilit´ es de Saint-Flour, XIV—1984, volume 1180 of Lecture Notes in Math., pages 125–264. Springer, Berlin, 1986. 2

    1984

  18. [18]

    J. F. C. Kingman. The ergodic theory of subadditive stochastic processes.J. Roy. Statist. Soc. Ser. B, 30:499–510, 1968

    1968

  19. [19]

    G. Last. Perturbation analysis of Poisson processes.Bernoulli, 20(2):486 – 513, 2014. 8

    2014

  20. [20]

    Last and M

    G. Last and M. Penrose.Lectures on the Poisson Process. Institute of Mathematical Statistics Textbooks. Cambridge University Press, 2017. 8

    2017

  21. [21]

    Last and M

    G. Last and M. Penrose.Lectures on the Poisson process, volume 7. Cambridge University Press, 2018. 3

    2018

  22. [22]

    Last and S

    G. Last and S. Zuyev. Applications of the perturbation formula for poisson processes to elementary and geometric proba- bility.arXiv preprint arXiv:1907.09552, 2019. 8

    work page arXiv 1907

  23. [23]

    T. M. Liggett. An improved subadditive ergodic theorem.Ann. Probab., 13(4):1279–1285, 1985

    1985

  24. [24]

    Meester and R

    R. Meester and R. Roy.Continuum percolation, volume 119 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996

    1996

  25. [25]

    Penrose.Random geometric graphs, volume 5 ofOxford Studies in Probability

    M. Penrose.Random geometric graphs, volume 5 ofOxford Studies in Probability. Oxford University Press, Oxford, 2003. 3, 4

    2003

  26. [26]

    Richardson

    D. Richardson. Random growth in a tessellation.Proc. Cambridge Philos. Soc., 74:515–528, 1973

    1973

  27. [27]

    C.-L. Yao, G. Chen, and T.-D. Guo. Large deviations for the graph distance in supercritical continuum percolation.J. Appl. Probab., 48(1):154–172, 2011. 1, 2, 3, 4

    2011

  28. [28]

    S. Zuyev. Russo’s formula for the poisson point processes and its applications.Discrete Math. and Applications, 3:355–366,

  29. [29]

    1− ε λ |Xλ|−1 X x∈Xλ [f(X λ)−f(X λ \ {x})] # LIPSCHITZ CONTINUITY OF THE TIME CONSTANT FOR CONTINUUM PERCOLATION 17 = 1 λ E

    8 AppendixA.A Russo-type formula Throughout this section, denote byX λ a Poisson point process of intensityλonK, whereKis a finite volume Borel subset ofR d, and letf:{F⊂K:Ffinite} →Rbe a bounded measurable function. 16 KAROLINE DUBIN AND CHRISTIAN GORSKI Proposition 21(A Russo-type formula).For everyλ >0, d dλ Ef(X λ)exists and we have d dλ Ef(X λ) = 1 λ...