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Percolation in the marked stationary Random Connection Model for higher-dimensional simplicial complexes

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The marked stationary random connection model, lifted to simplicial complexes, has a sharp phase transition for $q$-percolation: subcritical exponential decay and supercritical linear growth of the percolation function.

desk verdict A genuinely useful unification with a fixable but load-bearing construction bug: Section 3 must specify a canonical vertex ordering before Theorem 5.4 is well-defined. read the letter →

arxiv 2506.15038 v1 pith:WFZCYCXY submitted 2025-06-18 math.PR

Dominik Pabst This is my paper · ORCID
classification math.PR MSC 60K3560D0560G5505C80
keywords percolationrandomconnectionmodelsimplicialcomplexsharpphasetransitionup-connectivityPoissonprocessBooleanVietoris-Rips
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 introduces a percolation model that lifts the classical random connection model (RCM) to random simplicial complexes: points of a Poisson process in $\mathbb{R}^d$ carry marks, and connection functions decide, level by level, which edges, triangles, and higher-dimensional simplices appear. The main result, Theorem 5.4, says that under two conditions on the connection functions—(V1), nearby vertices with marks in a positive-probability set form a $(q+1)$-simplex with at least fixed probability, and (V2), no relevant simplex has diameter larger than $D$—$q$-percolation undergoes a sharp phase transition. Below the critical intensity $\beta_c^{(q)}$ the probability that a $q$-simplex at the origin connects out to distance $r$ decays exponentially in $r$; above it, the probability of an infinite component grows at least linearly in $\beta-\beta_c^{(q)}$. The paper argues this is new even for the RCM as a random graph, and that the Vietoris-Rips, Cech, and Boolean models all fall out as special cases.

What carries the argument

The load-bearing object is the $q$-graph $G_q(K)$ of a simplicial complex $K$: its vertices are the $q$-simplices and two vertices are joined when they are both contained in a common $(q+1)$-simplex; $K$ $q$-percolates when $G_q(K)$ has an infinite component. The model is built so that all randomness lives in one marked Poisson process: each point carries an $\mathbb{M}$-valued mark of uniform decision variables, indexed by cube coordinates and lexicographic order, that determine which simplices are present. The proof of the sharp threshold then runs through an algorithm that reveals only the cubes needed to decide whether the origin connects to distance $r$; the discrete OSSS inequality bounds the influence of any one cube, and the Margulis-Russo formula for Poisson processes converts the resulting differential inequality into exponential decay below and linear growth above $\beta_c^{(q)}$.

What would settle it

For a concrete instance satisfying (V1) and (V2), for example the Boolean model in $\mathbb{R}^2$ with grains equal to balls of radius $R$, where $\beta_c$ is known, measure $\theta_r(\beta)$ at a fixed $\beta<\beta_c$. The theorem predicts $\liminf_{r\to\infty}(-\log\theta_r(\beta))/r>0$; observing polynomial decay, or any run of $-\log\theta_r(\beta)$ growing only logarithmically, would falsify part (i). Equivalently, constructing any admissible connection function below its critical intensity whose connection probability is not exponentially small would refute Theorem 5.4.

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Extended reading notes

Core claim

The central discovery is that the sharp two-sided threshold behavior familiar from lattice percolation occurs in these continuous simplicial complexes whenever the local rules are bounded-range and locally positive. Precisely: fix $q\in\{0,\ldots,\alpha-1\}$, let $\theta_r(\beta)$ be the probability that a $q$-simplex containing the origin is connected in the $q$-graph to the complement of the ball of radius $r$, and let $\theta_\infty(\beta)$ be the probability of an infinite component. If (V1) and (V2) hold, then $0<\beta_c^{(q)}<\infty$, and (i) for $\beta<\beta_c^{(q)}$ there is $c(\beta)>0$ with $\theta_r(\beta)\le e^{-c(\beta)r}$ for all $r>0$; (ii) for each $\beta_0>\beta_c^{(q)}$ there is $c(\beta_0)>0$ with $\theta_\infty(\beta)\ge c(\beta_0)(\beta-\beta_c^{(q)})$ for all $\beta\in(\beta_c^{(q)},\beta_0)$. The same statement holds for the classical RCM as a graph by taking $q=0$, and the Boolean, Vietoris-Rips, and Cech cases are recovered by concrete connection functions.

Load-bearing premise

The construction assumes that assigning each potential simplex its decision variable from the mark of the last-listed vertex, after fixing an arbitrary enumeration of cubes and a lexicographic order on $\mathbb{R}^d$, gives a well-defined random complex whose distribution does not depend on those arbitrary choices; the paper does not prove invariance under re-enumeration.

Editorial extensions

If this is right

  • For $q=0$ the theorem yields a sharp phase transition for the classical random connection model under (V1)-(V2), a two-sided statement the paper says is new in this generality.
  • The Vietoris-Rips, Cech, and Boolean percolation models satisfy (V1)-(V2), so each has a sharp $q$-percolation transition for every $q$ below its maximal simplex dimension.
  • Above the critical intensity, the existence of an infinite component in $G_q(\Delta)$ upgrades from positive probability to probability one via a zero-one law.
  • The critical intensities form a nondecreasing chain $0<\beta_c^{(0)}\le\beta_c^{(1)}\le\dots\le\beta_c^{(\alpha-1)}<\infty$, reflecting that $q$-percolation implies lower-dimensional percolation.
  • The exponential-decay half relies on the bounded-range condition (V2): the paper notes that in the Boolean model with unbounded radii, exponential decay of this type generally fails.

Reading between the lines

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

  • The pair (V1)-(V2) is a plausible general sufficient condition for sharp thresholds in other locally defined random complexes, for instance random clique complexes built from weighted edges, though the paper does not state this.
  • Because the argument uses a Euclidean cube decomposition, the sharp transition may not transfer to random connection models on hyperbolic spaces or general metric spaces; separating the metric structure from the local conditions would be a natural next test.
  • The paper leaves open whether the critical intensities are strictly ordered; a plausible conjecture consistent with its results is $\beta_c^{(0)}<\dots<\beta_c^{(\alpha-1)}$ under generic (V1)-(V2) connection functions.
  • One could test the linear lower bound numerically near criticality: the theorem gives $\theta_\infty(\beta)\ge c(\beta-\beta_c^{(q)})$, and measuring the exponent of $\theta_\infty$ near $\beta_c$ in the Boolean model would show whether the bound has the right order.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces a marked stationary Random Connection Model for higher-dimensional simplicial complexes. A Poisson process on R^d × A × M is used to construct a random simplicial complex Δ of maximal dimension α: each j-simplex is included if all its sub-simplices are present and a uniform mark associated with its vertices is below a symmetric, translation-invariant connection function φ_j. For q ∈ {0,...,α−1}, the paper studies percolation of the q-graph G_q(Δ), whose vertices are q-simplices and whose edges are induced by shared (q+1)-simplices. Under two conditions on the connection functions, (V1) (local lower bound on simplex inclusion) and (V2) (bounded support of (q+1)-simplices), the paper proves: (i) monotonicity and nontriviality of critical intensities β_c^(q) in Theorem 4.2; (ii) a sharp phase transition in Theorem 5.4, giving exponential decay of θ_r(β) below β_c^(q) and a linear lower bound for θ_∞(β) above it. The proof uses the discrete OSSS inequality, a decision-tree algorithm, and the Margulis-Russo formula for Poisson processes, following the strategy of Hirsch–Valesin [6]. Examples are given for the Boolean model, the Vietoris–Rips complex, and the Čech complex.

Significance. If the result is correct and the construction is made precise, the paper would be a meaningful contribution: it unifies and extends sharp phase transition results for several continuum percolation models, including the Boolean model and the classical RCM, and it is the first to establish such a transition for up-connectivity in random simplicial complexes with general connection functions. The proof strategy is standard and the paper provides a fairly detailed algorithm in Theorem 4.2, with explicit probability estimates. The exposition of the model as a single marked Poisson process is elegant and facilitates the use of the OSSS inequality. The paper also credits prior work appropriately and identifies limitations, such as the comparison β_c = β_T failing in general marked models. However, the current version contains a load-bearing definitional gap and an essential proof step that is deferred to [6]; these issues must be resolved before the central claim can be accepted.

major comments (3)
  1. [Section 3] The construction of Δ is not invariant under permutation of the vertices of a simplex, and no canonical ordering of the points of Ψ is specified. For a simplex σ = {(x_0,a_0),...,(x_j,a_j)}, the definition u(σ) := u^{(j)}_{m_0,l_0,...,m_{j-1},l_{j-1}} uses the u-mark of the last listed vertex and the coordinates of the other vertices, but the paper never states that the vertices of a simplex are listed in any fixed order (e.g., lexicographic order of their Euclidean coordinates). Permuting the vertices can change which component of which point's u-mark is used, and hence can change whether σ ∈ Δ. Since the events B_r, the probability θ_r(β), the critical intensity β_c^(q), and all results in Sections 4 and 5 are defined through this Δ, the model is not uniquely defined as written. This is fixable by declaring a global total order on R^d (for instance, lexicographic order, or cube index followed by lexicographic order within the cube) and listing the vertices of every simplex in that order, but the rule must be stated explicitly before the model can be claimed to be well-defined.
  2. [Theorem 5.4 (proof)] The proof of Theorem 5.4 is incomplete: the step proving β̃ = β_c^(q) is deferred entirely to [6] with the statement that the remaining proof 'proceeds in the same manner as the proof of Theorem 1 in [6] and is purely analytical in nature.' The paper does establish the analogue of Lemma 4 in [6] via the differential inequality (13) and the lower bound C_2, but it does not reproduce the analytical argument showing that the critical value defined through the limsup of T_n(β) coincides with β_c^(q). Since this equality is the core of the sharp phase transition, the author must either supply the full adaptation, or state precisely which results from [6] are being invoked and verify that all hypotheses of those results are satisfied for the present percolation function θ_r(β), including the required properties of T_n(β) with the modified summation starting at ⌈D⌉.
  3. [Equation (12), Section 5] The derivation of the bound ∑_i ζ_i ≤ 2β e^{β(2D)^d} d/dβ θ_r(β) is asserted without proof. In particular, the application of the Margulis-Russo formula to the function f(η) = E[1{η + δ_(0,V,U) ∈ B_r}], where the expectation is over the auxiliary marks V,U, requires a justification that the difference operator and the expectation can be interchanged, and that the hypotheses of the Margulis-Russo formula (Theorem 19.4 in [10]) are met for this f on the relevant bounded window W(r). A short argument using dominated convergence or the Mecke equation should be supplied, since the differential inequality (13), and hence the entire sharp phase transition, relies on this equality.
minor comments (5)
  1. [Section 2.1] The phrase 'the point process process Φ' contains a duplicated word and should read 'the point process Φ'.
  2. [Section 3] The notation u^{(j)}_{m_0,l_0,...,m_{j-1},l_{j-1}} is confusing: the components of u ∈ M(2)×...×M(2α) are indexed without a superscript, so the superscript (j) is unexplained. The definition should be written as u_{m_0,l_0,...,m_{j-1},l_{j-1}} for the component associated with z = (m_0,l_0,...,m_{j-1},l_{j-1}).
  3. [Section 3] The assertion that 'The distribution of Δ is independent of the choice of t' is not proven and is not obvious, because the coordinates (m,l) of points depend on the cube partition and on the lexicographic ordering within a cube. If the claim is true, a proof or a reference should be given; if not, the text should specify a fixed t throughout, as is done later with t = D.
  4. [Section 5, after Algorithm 5.1] In the sentence 'Since the event B_r depends only on simplices with diameter at most D are relevant', the grammar is broken and the intended meaning is unclear. It should likely read 'Since only simplices with diameter at most D are relevant for the event B_r, ...'.
  5. [Theorem 5.4 and Abstract] The claim that the sharp phase transition is 'in its generality, new even for the classical RCM as a random graph' should be stated more carefully, because the paper does not provide a survey of all existing sharp-transition results for continuum percolation. In particular, the relation to the results of [9] for the Boolean model and [18] for the classical RCM should be discussed explicitly to substantiate the novelty claim for q = 0 under general connection functions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sharp phase transition is derived from external OSSS/Margulis-Russo machinery and comparison arguments; self-citations are background only.

full rationale

The derivation chain is not circular. The model in Section 3 is constructed explicitly from the connection functions and independent uniform marks; the connection functions are inputs, not outputs of the percolation analysis. Theorem 4.2 proves 0<beta_c^(q)<infinity via a lattice-reduction algorithm whose edge-acceptance probability is bounded using conditions (V1) and (V2), with positivity deferred to the external results [1] and [4]. Theorem 5.4 is obtained by applying the OSSS inequality (Theorem 1.9 in [5]) and the Margulis-Russo formula ([10]) to Algorithm 5.1, with Lemmas 5.2 and 5.3 controlling the revealment and influence terms. The final analytic step is transferred explicitly from the external theorem of Hirsch and Valesin [6], and subcritical exponential decay for the comparison geometric graph is cited to [17] and [18]. No parameter is fitted to the percolation function, and no target conclusion is assumed in the hypotheses: conditions (V1) and (V2) are structural assumptions on the connection functions, not restatements of sharpness. The author's own prior work ([13], [14], [15]) is cited only for the earlier introduction of the marked model and for central limit theorems, and it is not used to establish the sharp phase transition. The flagged order-dependence of the vertex enumeration in Section 3 is a well-definedness/rigor concern, not a circularity: it does not make the sharp-transition conclusion equal to an input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof rests on standard external tools (OSSS inequality, Margulis-Russo formula, exponential decay of the geometric graph, the analytic scheme of [6]) and on an implicit well-definedness assumption for the model construction. No free parameters are fitted and no new entities are postulated. The two non-standard inputs are the unsupported adaptation of [6]'s final argument and the unspecified vertex ordering in the definition of u(sigma).

assumptions (6)
  • standard math The OSSS inequality for Poisson process functionals (Theorem 1.9 in [5]) applies to the stopping-set algorithm (Algorithm 5.1) that reveals the event B_r.
    This is the central tool producing inequality (9), bounding the variance of the percolation indicator by reveal probabilities and influences. It is cited from [5].
  • standard math The Margulis-Russo formula for Poisson processes (Theorem 19.4 in [10]) applies to f(eta)=P(eta+delta_{(0,V,U)} in B_r), giving d/d beta theta_r(beta).
    Used in equation (12) to turn the sum of influences zeta_i into the derivative of theta_r with respect to beta.
  • standard math The final analytic argument of Theorem 1 in [6] transfers verbatim to the percolation function theta_r, with T_n(beta) starting the sum at k=ceil(D) instead of k=1, and establishes beta~ = beta_c^(q).
    The paper explicitly states 'we refer entirely to [6] for the remainder of the proof'; this is an external argument that is asserted to adapt, not reproduced.
  • standard math The geometric graph with edge function 1_{||x-y||<=D} has exponential decay below its critical intensity (Penrose 2003, Ziesche 2018).
    Used in the proof of Theorem 5.4 to show beta~ > 0 by comparison phi~_1 <= 1_{||x-y||<=D}.
  • ad hoc to paper The construction of Delta in Section 3 is invariant under relabeling of vertices and independent of the enumeration of the Poisson process Psi.
    The decision rule for a simplex uses the u-mark of the last listed vertex, indexed by coordinates of the other vertices, but no canonical ordering of vertices is specified; the model distribution is assumed to be symmetric.
  • standard math Kolmogorov's 0-1 law applies to the event of q-percolation due to (V2) boundedness and the cube decomposition of R^d.
    Used in Proposition 4.3 to upgrade positive probability of percolation to probability one for beta > beta_c^(q).

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Pith. "Pith review of Percolation in the marked stationary Random Connection Model for higher-dimensional simplicial complexes." pith.science (2026). https://pith.science/paper/WFZCYCXY

@misc{pith2026250615038,
  author       = {Pith},
  title        = {Pith review of: Percolation in the marked stationary Random Connection Model for higher-dimensional simplicial complexes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WFZCYCXY}},
  note         = {Machine review of arXiv:2506.15038}
}
read the original abstract

We introduce a novel percolation model that generalizes the classical Random Connection Model (RCM) to a random simplicial complex, allowing for a more refined understanding of connectivity and emergence of large-scale structures in random topological spaces. Regarding percolation with respect to the notion of up-connectivity, we establish the existence of a sharp phase transition for the appearance of a giant component, akin to the well-known threshold behavior in random graphs. This sharp phase transition is, in its generality, new even for the classical RCM as a random graph. As special cases, we obtain sharp phase transitions for the Vietoris-Rips complex, the Cech complex, and the Boolean model, allowing us to identify which properties of these well-known percolation models are actually required.

Figures

Figures reproduced from arXiv: 2506.15038 by the authors.

Figure 1
Figure 1. A two-dimensional simplicial complex 𝐾 (left) and its 1-graph 𝐺1 (𝐾) (right) complex 𝐾. Here, one can immediately see that the 𝑞-graph of a connected simplicial complex does not necessarily have to be connected. The reverse implication is also not true [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. A realisation of the algorithm in the unmarked case for 𝑞 = 0, 𝜑1 (𝑥, 𝑦) = 1{‖𝑥 − 𝑦‖ ≤ 𝑟0 } with 𝑟0 = 3 5 , 𝑟 = 10, 𝑠 = 5, 𝐷 = 4 5 , and 𝛽 = 4, where the connected components of Δ 𝟎 𝑊 (𝑟) that intersect the sphere 𝜕𝐵(𝟎, 𝑠) are shown in blue. The event 𝐵𝑟 has not occurred in this realisation. with 𝛿𝑖 (𝑠) ∶= ℙ ( Ψ𝑖 is revealed by the algorithm) , 𝜁𝑖 ∶= ℙ ( 1 { Ψ𝑊 (𝑟) + 𝛿(𝟎,𝑉 ,𝑈) ∈ 𝐵𝑟 } ≠ 1 { Ψ̃ 𝑖 ∈ 𝐵𝑟 } ) , where Ψ̃ 𝑖… view at source ↗
Figure 3
Figure 3. A visualisation of the argumentation from Lemma 5.2 in the case ‖ ̃𝑧𝑖‖ > 𝑠, where we abbreviate 𝑑̃ ∶= 𝐷(1 + √ 𝑑). then Ψ𝑖 is not revealed in the first step of the algorithm. In this case, we have 𝛿𝑖 (𝑠) ≤ ℙ ( ∃(𝑥, 𝑎) ∈ Φ𝑄𝑖+𝐵(𝟎,𝐷) with 𝑥 ⇄ 𝐵(𝟎, 𝑠) ) ≤ ℙ ( ∃(𝑥, 𝑎) ∈ Φ𝑄𝑖+𝐵(𝟎,𝐷) with 𝑥 ⇄ 𝐵(̃𝑧𝑖 , |‖ ̃𝑧𝑖‖ − 𝑠|) ) ≤ ℙ ( ∃(𝑥, 𝑎) ∈ Φ𝑄𝑖+𝐵(𝟎,𝐷) with 𝑥 ⇄ 𝐵 ( 𝑥, |‖ ̃𝑧𝑖‖ − 𝑠| − 𝐷(1 + √ 𝑑) ) ) ≤ 𝔼 [ ∑ (𝑥,𝑎)∈Φ𝑄𝑖+𝐵(𝟎,𝐷) 1 { 𝑥 ⇄ 𝐵 ( … view at source ↗

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