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

G\"unter Last; Kilian Matzke; Markus Heydenreich; Remco van der Hofstad

REVIEW 2 major objections 4 minor 4 cited by

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

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The random connection model is proven to be mean-field in high dimensions: the infra-red bound, the triangle condition, the critical exponent gamma=1, and a continuous percolation transition.

desk verdict A strong, important paper whose only real soft spot is a load-bearing lemma with its proof omitted—worth a serious referee, and the gap looks repairable. read the letter →

arxiv 1908.11356 v4 pith:PXTCP2Y7 submitted 2019-08-29 math.PR

Markus Heydenreich , Remco van der Hofstad , Günter Last , Kilian Matzke This is my paper · ORCID

classification math.PR MSC 60K3582B4360G55

keywords randomconnectionmodelcontinuumpercolationlaceexpansionmean-fieldbehaviortriangleconditioncriticalexponentgammaOrnstein-ZernikeequationPoissonpointprocess

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 proves that the random connection model—percolation on a Poisson point cloud where two points are joined independently with probability $\varphi$(y-x)—is mean-field in high dimension. For finite-variance connection functions in dimension d>12, and for spread-out and long-range versions in dimension d>3($\alpha$ and 2) with large range parameter L, it establishes the infra-red bound $\lambda$ |tau_hat_lambda(k)| <= |phi_hat(k)| + C $\beta$ / (phi_hat(0)-phi_hat(k)) uniformly up to the critical intensity, together with the triangle condition Delta_{lambda_c}

What carries the argument

The machinery is the lace expansion in the continuum: the two-point function is expanded around a random-walk Green's function with step distribution phi, tau_lambda approximately phi star G_mu, and the error is organized into lace-expansion coefficients Pi_{$\lambda$,n} and a remainder R_{$\lambda$,n} (equation (1.15)). A new BK inequality for marked Poisson point processes (Theorem 2.1) lets the coefficients be bounded by diagrammatic integrals over products of tau_lambda, which are then controlled by triangle-type quantities Delta_lambda, Delta_lambda^circ, W_lambda(k), and H_lambda(k). A bootstrap argument comparing tau_hat_lambda with G_hat_{mu_lambda} closes the bounds: assuming a mild bound f <= 3 on normalized ratios yields the sharp f <= 2 uniformly in $\lambda$ < lambda_c once $\beta$ (equal to g(d)^{1/4} in case (H1) and $L^{{-d}}$ in cases (H2) and (H3)) is small. The load-bearing identity is the Ornstein-Zernike equation in Fourier form, tau_hat_lambda = (phi_hat + Pi_hat_lambda)/(1 - $\lambda$(phi_hat + Pi_hat_lambda)), which converts the smallness of Pi_lambda into both the infra-red bound and the divergence of chi.

What would settle it

In the Poisson blob model in dimension d=13, simulate the random connection model just below the estimated critical intensity lambda_c and estimate the ratio chi(lambda)(lambda_c-lambda); if the ratio fails to stay bounded between two positive constants as lambda approaches lambda_c, the claimed bounded-ratio gamma=1 and the underlying triangle condition are false.

Watch

Extended reading notes

Core claim

The central claim is that the lace expansion converges for the continuum random connection model, not only on lattices, under three regimes: (H1) finite-variance phi for all large d>12; (H2) spread-out phi with large L and d>6; and (H3) long-range phi with decay exponent $\alpha$, for d>3($\alpha$ and 2) and large L. In each case the two-point function tau_lambda satisfies the Ornstein-Zernike equation in Fourier space, tau_hat_lambda = (phi_hat + Pi_hat_lambda)/(1 - $\lambda$(phi_hat + Pi_hat_lambda)) with Pi_lambda = O($\beta$), which yields the infra-red bound and the triangle condition uniformly in $\lambda$ <= lambda_c. Consequently Theorem 1.3 gives the bounded-ratio divergence chi($\lambda$) between $\lambda$(lambda_c-$\lambda$)^{-1} and $\lambda$(1+C $\beta$)(lambda_c-$\lambda$)^{-1}, the identity lambda_c = 1/(1+Pi_hat_{lambda_c}(0)), and $\theta$(lambda_c)=0, so the percolation probability vanishes continuously at criticality.

Load-bearing premise

The proof rests on the connection function being spread out enough—large dimension for finite variance, large range L, or long-range decay—so that repeated self-convolutions of phi decay quickly and its Fourier transform satisfies a uniform lower bound away from 1; without those estimates every diagrammatic error term can be large.

Editorial extensions

If this is right

The infra-red bound and triangle condition hold uniformly in lambda in [0,lambda_c], so the Ornstein-Zernike equation extends to the critical point.
The expected cluster size satisfies lambda(lambda_c-lambda)^{-1} <= chi(lambda) <= lambda(1+C beta)(lambda_c-lambda)^{-1} for lambda < lambda_c, giving the critical exponent gamma = 1 in the bounded-ratio sense.
The critical intensity is controlled by 1 <= lambda_c q_phi <= 1 + C beta and is given by the explicit identity lambda_c = 1/(1+Pi_hat_{lambda_c}(0)).
The percolation probability is continuous on [0,infinity), with theta(lambda_c)=0, so the phase transition is continuous.
The lower bound on chi(lambda) is proved in all dimensions and for the general random connection model without extra assumptions on phi, implying chi(lambda_c)=infinity.

Reading between the lines

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

If the infra-red bound is the right input, standard Ornstein-Zernike theory would suggest exponential decay of tau_lambda(x) in the subcritical phase with a correlation length diverging like (lambda_c-lambda)^{-1/2}; the paper does not prove this decay, but the uniform bound is precisely the hypothesis such a derivation would need.
The dimensional condition d > 3(alpha and 2) indicates an upper critical dimension of 6 for the finite-variance and spread-out regimes and of 3 alpha for alpha < 2, making long-range continuum percolation a natural testbed for dimension-dependent mean-field behavior below d=6.
One could attempt to extract further critical exponents, such as those for the correlation length or arm events, from the triangle condition established here; the paper stops at gamma but its bounds on the two-point function are the standard sufficient input for such extensions.
A quantitative finite-size prediction of this paper is that close to criticality the ratio chi(lambda)(lambda_c-lambda) is bounded between lambda and lambda(1+C beta), with beta explicit; this is directly testable by simulation in dimension d=13 for the Boolean model.

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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

Two things to know. First, this is the first lace expansion for the general random connection model with a convergence proof that actually goes through; Tanemura's earlier attempt for the Poisson blob model is acknowledged, and the authors say they could not make it work. Second, the central bootstrap is not circular—it is the standard forbidden-region argument—but the paper does omit the proof of Lemma 4.20, and that lemma is load-bearing for the displacement bounds that feed the infrared bound.

What is genuinely good: the derivation of the expansion in the continuum with thinnings, the new BK inequality for marked Poisson point processes (a standalone tool people will use), and the careful treatment of finite-volume approximations. The three regimes (finite-variance, spread-out, long-range) are handled systematically, and the dimension thresholds match what you would expect from the lattice analogues. Once the triangle condition and infrared bound are in place, the proof of gamma = 1 in the bounded-ratio sense and continuity of theta at criticality are clean consequences. If the proof is correct, this is a major step for continuum percolation.

Where I would put my finger: Lemma 4.20 is stated with the proof omitted as 'analogous to Proposition 4.14.' That is a real gap in the written argument, and the stress-test note is right that it sits at a load-bearing point: Proposition 4.19 uses it in cases (a) and (c), and that is exactly what yields the bound on the displaced lace-expansion coefficients in Proposition 5.2, which then controls the bootstrap function f3 and the infrared bound. This is not a cosmetic omission. I do not see an actual error, and the pictorial sketch suggests the analogy is plausible—the base cases and the epsilon-splitting need to be checked term by term, but no showstopper jumps out. A referee should ask the authors to supply the full proof, or at least a detailed verification of the base cases.

The frequent use of pictorial diagrammatic arguments will slow down a referee; that is a style issue, not a correctness issue, but in a paper this tight it matters.

Bottom line: this paper deserves a serious referee. The gap in Lemma 4.20 is repairable, and the contribution is important enough that the field would benefit from the proof being fully written out. I would send it to review with a request to fix that lemma.

Referee Report

2 major / 4 minor

Summary. This paper develops a lace expansion for the random connection model, a continuum percolation model built on a stationary Poisson point process with independent edges governed by a symmetric connection function φ. The authors first derive the expansion for the two-point function with a remainder term, using a new BK-type inequality for the continuum model, and then obtain diagrammatic bounds on the lace-expansion coefficients. For three classes of connection functions—finite-variance (including the Boolean model), spread-out with large range parameter L, and long-range spread-out with parameter α—they prove the expansion converges in sufficiently high dimension, establish the triangle condition at criticality, and prove an infra-red bound uniform in λ∈[0,λ_c]. From this they deduce the mean-field value γ=1 for the susceptibility exponent in a bounded-ratio sense and continuity of the percolation function at λ_c. The main theorems are Theorem 1.2 (infra-red bound and triangle condition) and Theorem 1.3 (critical exponent and continuity), proved via a bootstrap argument on the auxiliary function f=f_1∨f_2∨f_3.

Significance. If the proof is completed as claimed, this is a substantial contribution: it provides the first full lace expansion for the random connection model in the continuum, with explicit model classes and quantitative small parameters (large dimension for H1, large L for H2/H3). The paper contains several genuinely new ingredients, notably the BK inequality for the continuum RCM (Theorem 2.1), the treatment of thinnings, and the displacement bounds needed for the bootstrap. The claimed results match the expected mean-field behavior in high dimensions and extend the discrete lace-expansion program to a continuum setting. However, one load-bearing proof is omitted (Lemma 4.20), and the central theorems depend on it; the manuscript is therefore not yet ready for acceptance in its present form.

major comments (2)

[Section 4.4, Lemma 4.20] The proof of Lemma 4.20 is omitted with the sentence that it is 'analogous to that of Proposition 4.14'. This lemma is load-bearing, not peripheral. It is used in Proposition 4.19 in cases (a) and (c) to convert diagrams with a displaced segment into factors W_λ(k) or H_λ(k), producing the bounds (4.23)–(4.24). Summing these bounds over n is precisely what yields the displacement estimate in Proposition 5.2, which controls the third bootstrap function f_3 in (5.2) and hence enters the proof of Theorem 1.2. The asserted analogy is plausible but not a formal proof: the second inequality in Lemma 4.20 involves an extra power of \bar U_λ and an ε-splitting, so the base case must include the analogues of the estimates (4.20)–(4.21) for the \bar Ψ diagrams, and the induction step must be checked for all collapsed contributions of \bar Ψ^{(0)}. A single unaccounted diagram or a missing factor of \bar U_λ would break the uniform-in-n constant and invalidate the bootstrap. Please provide a complete proof of both inequalities in Lemma 4.20.
[Section 4.4, Proposition 4.19, and Section 5.3, Proposition 5.2] The dependence of the constants in the displacement bounds needs to be made fully explicit if the paper's central claim is to be verified. Proposition 4.19 states two bounds, (4.23) and (4.24), and the proof of Proposition 5.2 only states that combining Lemma 5.4–5.8 with these bounds gives the required exponential decay in n with factor β. In particular, the passage from (4.23)/(4.24) to the displayed bound λ∫[1−cos(k·x)]Π^{(n)}_λ(x)dx ≤ [1−φ̂(k)](c'_f β)^{(n−1)∨1} is not shown term by term. Since Lemma 4.20 is the source of the \bar U_λ factors that make the n=2 case small, the omitted proof of that lemma and the constant bookkeeping in Proposition 5.2 should be presented together so that the reader can verify the uniform-in-n and uniform-in-λ constants.

minor comments (4)

[Proof of Proposition 5.9, Section 5.4] In the proof of continuity of f_3, the text refers to 'Lema 2.3'; this should be 'Lemma 2.3'.
[Section 1.3, Theorem 1.2] Equation (1.9) states the right-hand side is understood as +∞ for k=0; since the denominator also vanishes there, it would be clearer to write the bound for k≠0 and state the limiting meaning for k=0 explicitly.
[Section 4.4, Definition 4.17 and Figure 3] The pictorial proofs in Section 4.4 are useful, but several displayed diagrams are not fully defined in the text (for example the variables integrated over in the bound for H_λ). If the omitted proof of Lemma 4.20 is added, it would help to provide textual versions of each diagrammatic step so that the collapsed cases are checkable.
[Section 2.3, Theorem 2.1] In the proof of the BK inequality, the approximation argument uses the BKR inequality from [15]; the text says 'an inspection of the proof shows' the result extends to general mark spaces. A short formal statement of the extension would improve readability, though the argument given is convincing.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the bootstrap's forbidden-region argument is legitimate; self-citations to [22,23] are methodological only; the omitted proof of Lemma 4.20 is a load-bearing gap, not a circle.

full rationale

I find no circular reduction in this paper. The central claims (Theorem 1.2 infra-red bound and triangle condition, Theorem 1.3 gamma=1) are derived from in-paper estimates: the lace expansion (Prop. 3.8), the diagrammatic bounds (Props. 4.14 and 4.19), the displacement bounds (Lemmas 5.6-5.8), and the random-walk estimates (Props. A.2-A.3) proved in the appendix. The bootstrap of Section 5 is explicitly self-aware: 'In percolation theory, this (at first glance circular) argument is known as the bootstrap argument.' It is a legitimate forbidden-region argument: f(0) <= 2, f is continuous on [0, lambda_c), and f <= 3 implies f <= 2; continuity and f(0) <= 2 then exclude values in (2,3], so the 'pump' assumption f <= 3 is discharged rather than used as the conclusion. The infra-red bound is not a restatement of the bootstrap inputs f2, f3; it follows from the displacement bound |Pi_hat_lambda(0)-Pi_hat_lambda(k)| <= c_f[phi_hat(0)-phi_hat(k)] beta (Prop. 5.2 and Cor. 5.3), which is proved here for the continuum model. Self-citations are present but not load-bearing: Section 1.5 says 'As the general strategy of proof is standard, we refer to [22]', and Section 5.2 says 'The analysis of Section 5 follows the arguments in the paper by Heydenreich et al. [23], adapting them to the continuum setting.' However, every estimate actually used (Lemmas 5.4-5.8, Prop. 5.9, Appendix A) is proved in this paper, and [23] concerns lattice models, so the RCM conclusions are not imported from the cited work. The modeling assumptions (H1.2), (H2.3), (H3.3) are stated premises; Propositions A.2-A.3 derive consequences from them, with no fitted parameter renamed as a prediction. Flagged per the reviewing rule: Lemma 4.20 states the displacement-diagram bounds used in Prop. 4.19, and the paper says 'The proof of Lemma 4.20 is omitted, as it is analogous to that of Proposition 4.14.' Because Prop. 4.19 feeds (5.3) and hence the infra-red bound, this is a load-bearing completeness gap in the written proof, but an omitted proof is a correctness risk, not a circular step, and does not raise the circularity score. Overall, the reader's score of 2.0 is appropriate: minor methodological self-citation with independent central content.

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

The ledger is clean: there are no fitted free parameters in the sense of data-fitting, and the axioms are either standard results or explicit model assumptions. The thinning variables and the auxiliary bootstrap functions are technical constructions, not invented physical entities; thinnings are justified by the standard Poisson thinning theorem. The main external inputs are Meester's lambda_c = lambda_T theorem and the general framework of [22, 23].

assumptions (4)

standard math Meester's theorem [31]: lambda_T = lambda_c, so the susceptibility chi(lambda) is finite exactly when lambda < lambda_c.
Invoked throughout Section 3 to justify that integrals over tau_lambda are finite, which is needed for the lace expansion and the differential inequality.
domain assumption Connection-function assumptions (H1), (H2), or (H3): finite variance with convolution decay (H1.2), spread-out Fourier lower bounds (H2.3), or long-range Fourier lower bounds (H3.3).
These define the model regimes and are the source of the smallness parameter beta. Propositions A.2 and A.3 rely on them for every diagrammatic bound.
standard math Standard point-process and percolation tools: Mecke equation, FKG inequality, Margulis-Russo formula, and the BK inequality for the RCM.
The BK inequality is proved in the paper as Theorem 2.1 using the Gupta-Rao BKR inequality, so it is not a free postulate.
domain assumption Random-walk s-condition bounds in Propositions A.2 and A.3: integrals of powers of hat_phi over powers of 1 - mu hat_phi are O(beta).
These propositions are proved in the appendix under (H1)-(H3) and are the mechanism by which the smallness of beta propagates into the diagrammatic bounds.

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Pith. "Pith review of Lace Expansion and Mean-Field Behavior for the Random Connection Model." pith.science (2026). https://pith.science/paper/PXTCP2Y7

@misc{pith2026190811356,
  author       = {Pith},
  title        = {Pith review of: Lace Expansion and Mean-Field Behavior for the Random Connection Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PXTCP2Y7}},
  note         = {Machine review of arXiv:1908.11356}
}

abstract

We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK inequality. For our main results, we consider three versions of the connection function $\varphi$: a finite-variance version (including the Boolean model), a spread-out version, and a long-range version. For sufficiently large dimension (resp., spread-out parameter and $d>6$), we then prove the convergence of the lace expansion, derive the triangle condition, and establish an infra-red bound. From this, mean-field behavior of the model can be deduced. As an example, we show that the critical exponent $\gamma$ takes its mean-field value $\gamma=1$ and that the percolation function is continuous.

Figures

Figures reproduced from arXiv: 1908.11356 by the authors.

**Figure 1.** Diagrammatic representation of segment i, and hence of the functions ψ (j) . Factors τλ are represented by lines, factors τ ◦ λ are represented by lines endowed with a ‘◦’. The points ti , wi , zi , ui (the ones labeled by index i) are depicted as squares—in the later decomposition of the full diagram into segments, these are the ones integrated over when bounding segment i. The small diagrams in brackets indicate t… view at source ↗

**Figure 2.** The full diagrammatic events. The line with a ‘ [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗

**Figure 3.** The diagram H0 λ (a1, a2, a3; k) and the schematic Fourier diagram Hb0 λ (l1, l2, l3; k). Lemma 5.8 (Bound on Hλ). Let λ ∈ [0, λc) and let d > 12 under (H1) and d > 3(α ∧ 2) under (H2) and (H3). Then there is cf = c(f(λ)) (increasing in f) such that Hλ(k) ≤ cfβ[1 − ϕb(k)]. Proof. We define τ˜˜λ,k(x) := ϕk(x) + 2λ[(ϕk ? τλ)(x) + (ϕ ? τ˜λ,k)(x)] and note that ˜τλ,k ≤ τ˜˜λ,k by the Cosine-split Lemma 4.1. With this, by… view at source ↗

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Works this paper leans on

43 extracted references · 43 canonical work pages · cited by 4 Pith papers

[1]

Newman, Tree graph inequalities and critical behavior in perco- lation models, J

Michael Aizenman and Charles M. Newman, Tree graph inequalities and critical behavior in perco- lation models, J. Stat. Phys. 36 (1984), 107–144. 61

work page 1984
[2]

Barsky and Michael Aizenman, Percolation critical exponents under the triangle condition, Ann

David J. Barsky and Michael Aizenman, Percolation critical exponents under the triangle condition, Ann. Probab. 19 (1991), no. 4, 1520–1536

work page 1991
[3]

Vincent Beﬀara and Hugo Duminil-Copin, Planar percolation with a glimpse of Schramm-Loewner evolution, Probab. Surv. 10 (2013), 1–50

work page 2013
[4]

Jacob van den Berg, A note on disjoint-occurrence inequalities for marked Poisson point processes , J. Appl. Probab. 33 (1996), no. 2, 420–426

work page 1996
[5]

B´ ela Bollob´ as and Oliver Riordan,Percolation, Cambridge: Cambridge University Press, 2006

work page 2006
[6]

Chayes, Remco van der Hofstad, Gordon Slade, and Joel Spencer, Random subgraphs of ﬁnite graphs

Christian Borgs, Jennifer T. Chayes, Remco van der Hofstad, Gordon Slade, and Joel Spencer, Random subgraphs of ﬁnite graphs. II: The lace expansion and the triangle condition , Ann. Probab. 33 (2005), no. 5, 1886–1944

work page 2005
[7]

Broadbent and John M

Simon B. Broadbent and John M. Hammersley, Percolation processes. I: Crystals and mazes , Proc. Camb. Philos. Soc. 53 (1957), 629–641

work page 1957
[8]

Tim van de Brug, The random connection model: Construction, central limit theorem and asymptotic connectivity, Master’s thesis, Vrije Universiteit Amsterdam, 2003

work page 2003

Show all 43 references

[9]

Sudhakar Dharmadhikari and Kumar Joag-dev, Unimodality, convexity and applications , London: Academic Press, Inc., 1988

1988
[10]

Theory Relat

Robert Fitzner and Remco van der Hofstad, Generalized approach to the non-backtracking lace expansion, Probab. Theory Relat. Fields 169 (2017), no. 3-4, 1041–1119

2017
[11]

, Mean-ﬁeld behavior for nearest-neighbor percolation in d > 10, Electron. J. Probab. 22 (2017), 65, Id/No 43

2017
[12]

Gilbert, Random plane networks , J

Edgar N. Gilbert, Random plane networks , J. Soc. Ind. Appl. Math. 9 (1961), 533–543

1961
[13]

250, New York, NY: Springer, 2009

Loukas Grafakos, Modern Fourier analysis, 2nd ed., vol. 250, New York, NY: Springer, 2009

2009
[14]

321, Springer-Verlag, Berlin, 1999

Geoﬀrey Grimmett, Percolation, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fun- damental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999

1999
[15]

J. C. Gupta and Bhamidi V. Rao, Van den Berg-Kesten inequality for the Poisson Boolean model for continuum percolation, Sankhy¯ a, Ser. A61 (1999), no. 3, 337–346

1999
[16]

Theory Relat

Takashi Hara, Mean-ﬁeld critical behaviour for correlation length for percolation in high dimensions, Probab. Theory Relat. Fields 86 (1990), no. 3, 337–385

1990
[17]

, Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals, Ann. Probab. 36 (2008), no. 2, 530–593

2008
[18]

Takashi Hara and Gordon Slade, Mean-ﬁeld critical behaviour for percolation in high dimensions , Commun. Math. Phys. 128 (1990), no. 2, 333–391

1990
[19]

, The self-avoiding-walk and percolation critical points in high dimensions , Comb. Probab. Comput. 4 (1995), no. 3, 197–215

1995
[20]

Takashi Hara, Remco van der Hofstad, and Gordon Slade, Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models , Ann. Probab. 31 (2003), no. 1, 349–408

2003
[21]

Markus Heydenreich and Kilian Matzke, Expansion for the critical point of site percolation: the ﬁrst three terms, Preprint arXiv:1912.04584 [math.PR], 2019

1912 arXiv
[22]

Markus Heydenreich and Remco van der Hofstad, Progress in high-dimensional percolation and random graphs, Cham: Springer, 2017. 62

2017
[23]

Markus Heydenreich, Remco van der Hofstad, and Akira Sakai, Mean-ﬁeld behavior for long- and ﬁnite range Ising model, percolation and self-avoiding walk , J. Stat. Phys. 132 (2008), no. 6, 1001– 1049

2008
[24]

Algorithms 27 (2005), no

Remco van der Hofstad and Gordon Slade, Asymptotic expansions in n−1 for percolation critical values on the n-cube and Zn, Random Struct. Algorithms 27 (2005), no. 3, 331–357

2005
[25]

, Expansion inn−1 for percolation critical values on the n-cube andZn: the ﬁrst three terms , Comb. Probab. Comput. 15 (2006), no. 5, 695–713

2006
[26]

Gady Kozma and Asaf Nachmias, The Alexander-Orbach conjecture holds in high dimensions , In- vent. Math. 178 (2009), no. 3, 635–654

2009
[27]

, Arm exponents in high dimensional percolation , J. Am. Math. Soc. 24 (2011), no. 2, 375– 409

2011
[28]

7, Cambridge: Cambridge University Press, 2018

G¨ unter Last and Mathew Penrose,Lectures on the Poisson process , vol. 7, Cambridge: Cambridge University Press, 2018

2018
[29]

Penrose, and Sergei Zuyev,On the capacity functional of the inﬁnite cluster of a Boolean model , Ann

G¨ unter Last, Mathew D. Penrose, and Sergei Zuyev,On the capacity functional of the inﬁnite cluster of a Boolean model , Ann. Appl. Probab. 27 (2017), no. 3, 1678–1701

2017
[30]

G¨ unter Last and Sebastian Ziesche, On the Ornstein–Zernike equation for stationary cluster pro- cesses and the random connection model , Adv. Appl. Probab. 49 (2017), no. 4, 1260–1287

2017
[31]

Ronald Meester, Equality of critical densities in continuum percolation , J. Appl. Probab. 32 (1995), no. 1, 90–104

1995
[32]

Penrose, and Anish Sarkar, The random connection model in high dimensions, Stat

Ronald Meester, Mathew D. Penrose, and Anish Sarkar, The random connection model in high dimensions, Stat. Probab. Lett. 35 (1997), no. 2, 145–153

1997
[33]

119, Cambridge: Cambridge Univ

Ronald Meester and Rahul Roy, Continuum percolation, vol. 119, Cambridge: Cambridge Univ. Press, 1996

1996
[34]

Penrose, On a continuum percolation model , Adv

Mathew D. Penrose, On a continuum percolation model , Adv. Appl. Probab. 23 (1991), no. 3, 536–556

1991
[35]

, On the spread-out limit for bond and continuum percolation , Ann. Appl. Probab. 3 (1993), no. 1, 253–276

1993
[36]

5, Oxford: Oxford University Press, 2003

, Random geometric graphs, vol. 5, Oxford: Oxford University Press, 2003

2003
[37]

, Connectivity of soft random geometric graphs , Ann. Appl. Probab. 26 (2016), no. 2, 986– 1028

2016
[38]

D¨ usseldorf etc.: McGraw-Hill Book Company

Walter Rudin, Principles of mathematical analysis , International Series in Pure and Applied Math- ematics. D¨ usseldorf etc.: McGraw-Hill Book Company. X, 342 p. DM 47.80 (1976)., 1976

1976
[39]

´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour XXXIV – 2004 , vol

Gordon Slade, The lace expansion and its application. ´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour XXXIV – 2004 , vol. 1879, Berlin: Springer, 2006

2004
[40]

Proceedings of the seventh Japan-Russia symposium, Tokyo, Japan, July 26–30, 1995, Singapore: World Scientiﬁc, 1996, pp

Hideki Tanemura, Critical behavior for a continuum percolation model, Probability theory and math- ematical statistics. Proceedings of the seventh Japan-Russia symposium, Tokyo, Japan, July 26–30, 1995, Singapore: World Scientiﬁc, 1996, pp. 485–495

1995
[41]

Microstructure and macroscopic properties , vol

Salvatore Torquato, Random heterogeneous materials. Microstructure and macroscopic properties , vol. 16, New York, NY: Springer, 2002

2002
[42]

Salvatore Torquato, Eﬀect of dimensionality on the continuum percolation of overlapping hyper- spheres and hypercubes, J. Chem. Phys. 136 (2012), no. 5, 054106

2012
[43]

16, Paris: Soci´ et´ e Math´ ematique de France, 2009 (French)

Wendelin Werner, Percolation et mod` ele d’Ising, vol. 16, Paris: Soci´ et´ e Math´ ematique de France, 2009 (French). 63

2009

Pith tools

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

[1] [1]

Newman, Tree graph inequalities and critical behavior in perco- lation models, J

Michael Aizenman and Charles M. Newman, Tree graph inequalities and critical behavior in perco- lation models, J. Stat. Phys. 36 (1984), 107–144. 61

work page 1984

[2] [2]

Barsky and Michael Aizenman, Percolation critical exponents under the triangle condition, Ann

David J. Barsky and Michael Aizenman, Percolation critical exponents under the triangle condition, Ann. Probab. 19 (1991), no. 4, 1520–1536

work page 1991

[3] [3]

Vincent Beﬀara and Hugo Duminil-Copin, Planar percolation with a glimpse of Schramm-Loewner evolution, Probab. Surv. 10 (2013), 1–50

work page 2013

[4] [4]

Jacob van den Berg, A note on disjoint-occurrence inequalities for marked Poisson point processes , J. Appl. Probab. 33 (1996), no. 2, 420–426

work page 1996

[5] [5]

B´ ela Bollob´ as and Oliver Riordan,Percolation, Cambridge: Cambridge University Press, 2006

work page 2006

[6] [6]

Chayes, Remco van der Hofstad, Gordon Slade, and Joel Spencer, Random subgraphs of ﬁnite graphs

Christian Borgs, Jennifer T. Chayes, Remco van der Hofstad, Gordon Slade, and Joel Spencer, Random subgraphs of ﬁnite graphs. II: The lace expansion and the triangle condition , Ann. Probab. 33 (2005), no. 5, 1886–1944

work page 2005

[7] [7]

Broadbent and John M

Simon B. Broadbent and John M. Hammersley, Percolation processes. I: Crystals and mazes , Proc. Camb. Philos. Soc. 53 (1957), 629–641

work page 1957

[8] [8]

Tim van de Brug, The random connection model: Construction, central limit theorem and asymptotic connectivity, Master’s thesis, Vrije Universiteit Amsterdam, 2003

work page 2003