{"id":"662b8e27-1db8-44d9-86db-ef6ab7a4682a","arxiv_id":"2506.15038","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A sharp phase transition for q-percolation is proven for the marked stationary random connection model on higher-dimensional simplicial complexes, unifying the Boolean, Vietoris-Rips, and Cech cases.","lead":"This mathematics paper proves that a general family of random geometric networks built on random points in space has a sharp all-or-nothing transition: below a critical density the network is fragmented, above it an infinite connected structure appears. The result unifies and extends known sharp-transition results for Boolean models and Vietoris-Rips complexes, and is new even for the classical random connection model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Section 3 construction is order-dependent as written: u(σ) uses the u-mark of the last listed vertex, but no canonical ordering of the vertices of Ψ is specified, so Δ and the percolation events built on it are not uniquely defined.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing gap I find: the absence of a canonical vertex ordering in Section 3. The ambiguity is not merely cosmetic: u(σ) selects a uniform mark associated with the 'last' vertex and indexes it by coordinates of the others, so different orderings can produce different complexes with positive probability. Because β_c, θ_r, and the events in Theorem 5.4 are all defined via Δ, the sharp phase transition statement is not well-defined until an ordering is fixed. The issue is readily repairable—one can order vertices by the lexicographical order already introduced for ranks—so I do not see it as a reason to reject the mathematics, but it is a genuine conditioning requirement. I also noted that the proof of Lemma 5.3 relies on a monotone coupling Δ_k ⊆ Δ_{k+1} that requires careful handling of coordinate stability when points are added, and that the final analytic step of Theorem 5.4 is deferred to [6]; both are additional reasons a revised version should include more detail, but the ordering specification is the most fundamental issue because it precedes all definitions. My recommendation therefore matches the reader's CONDITIONAL verdict.","tokens_in":23636,"tokens_out":26823,"duration_ms":286303,"concrete_test":"Take three points x<y<z in one cube with fixed u-marks. Compute membership of the triangle {x,y,z} under the Section 3 rule using the listing (x,y,z): u(σ) uses the u^{(2)}-mark of z indexed by (m_x,l_x,m_y,l_y). Then compute membership using the listing (z,y,x): u(σ) uses the u^{(2)}-mark of x indexed by (m_z,l_z,m_y,l_y). Exhibit uniform marks for which the two thresholds give different decisions, demonstrating that Δ depends on the enumeration. Then verify that adding a sentence specifying a canonical order—lexicographic on R^d, or lexicographic on (cube index, within-cube rank)—makes the inclusion of every simplex a well-defined function of Ψ and preserves the intended independent-Bernoulli structure of the model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3, for a simplex σ = {(x0,a0),...,(xj,aj)}, the paper defines u(σ) := u^{(j)}_{m0,l0,...,m_{j-1},l_{j-1}}, where (m_k,l_k) are the coordinates of (x_k,a_k) and u^{(j)} is the u-mark of the last listed point. The connection functions φ_j are symmetric, but the decision rule σ ∈ Δ ⇔ u(ρ) ≤ φ_{|ρ|-1}(x_ρ) for all ρ ⊆ σ is not invariant under permuting the vertices of σ: swapping the roles of 'last vertex' and 'other vertices' typically changes which coordinate of which point's u-mark is used, and hence can change the verdict. The paper fixes a lexicographical order on R^d only for counting ranks within a cube; it never states that the vertices of a simplex are listed in that order, nor does it specify a global total order of the points of Ψ. Thus the same realization of Ψ can yield different complexes Δ depending on how simplices' vertices are ordered. Since β_c^{(q)}, θ_r, and all events in Theorem 5.4 are defined through this Δ, the central claim is not well-defined as written. This is fixable—e.g., list vertices in lexicographic order of their Euclidean coordinates, or order by cube index then lexicographic order within the cube—but the paper must state such a rule. This concern is load-bearing because every subsequent definition and proof (OSSS, Margulis-Russo, the algorithms in Sections 4 and 5) presupposes a unique Δ.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23876,"tokens_out":5584,"duration_ms":55566,"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":[{"comment":"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.","section":"Section 3"},{"comment":"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⌉.","section":"Theorem 5.4 (proof)"},{"comment":"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.","section":"Equation (12), Section 5"}],"minor_comments":[{"comment":"The phrase 'the point process process Φ' contains a duplicated word and should read 'the point process Φ'.","section":"Section 2.1"},{"comment":"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}).","section":"Section 3"},{"comment":"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.","section":"Section 3"},{"comment":"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, ...'.","section":"Section 5, after Algorithm 5.1"},{"comment":"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.","section":"Theorem 5.4 and Abstract"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be an adaptation of the author's PhD thesis, and it leans heavily on [6] for the final analytical step. The editor may wish to ask the author to make the dependence on [6] explicit and to verify that the construction in Section 3 can be made order-invariant, as the current formulation leaves the model undefined. The novelty claim for the classical RCM (q=0) should also be checked against [9] and [18] during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious paper worth refereeing. Theorem 5.4 looks right in outline, and the unification of Rips, Cech, Boolean model, and classical RCM under (V1),(V2) is the real contribution. But the model definition in Section 3 is not well-defined as written, and the fix has to be stated explicitly before any of the percolation events make sense.\n\nWhat is new: a sharp phase transition for q-up-connectivity in the marked stationary RCM, with exponential decay below beta_c and a linear lower bound above, under two clean conditions. Special cases include Boolean model k-percolation and Rips/Cech face percolation; even q=0 for the classical RCM as a graph seems absent from the literature. The proof strategy is the established OSSS + Margulis-Russo route, and the lemmas are detailed. The example section is useful and mostly checks out.\n\nSoft spots. First, the order-dependence in Section 3. The decision rule uses u(σ)=u^{(j)}_{m0,l0,...,m_{j-1},l_{j-1}}, i.e., the u-mark of the last listed vertex indexed by coordinates of the others. The paper never states an ordering of the vertices of a simplex. Permuting the vertices changes which point's mark is used and which index tuple, so a single realization of Ψ can yield different complexes. This is not a cosmetic gap: every event in Section 5 is defined through Δ. It is easily fixable—list vertices by cube index then lexicographic rank, or by Euclidean lex order—but the author must state it. Second, the final analytic argument in Theorem 5.4 is deferred to [6]. The differential inequality and the modified T_n are given, so this is a reasonable reduction rather than a black box, but a referee will want the remaining steps written out or at least a precise statement of what is imported. Third, less important: the claim that even q=0 is new for the classical RCM is plausible but rests on a literature check; [17,18] cover the geometric graph, not general connection functions, so the claim is probably right but should be stated carefully.\n\nThe citation pattern is honest; the author's own previous work is cited for the model setup, not for the percolation conclusion. No circularity or fitted parameters. I think the central argument holds up once the model is made unambiguous.\n\nWho it is for: stochastic geometers and percolation theorists. It deserves a serious referee. My recommendation: send to peer review, with a request that the author specify the ordering in Section 3 and expand or precisely reference the deferred part of Theorem 5.4.","headline":"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.","tokens_in":24520,"tokens_out":4547,"would_cite":true,"duration_ms":49224,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60D05","60G55","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["percolation","random connection model","simplicial complex","sharp phase transition","up-connectivity","Poisson process","Boolean model","Vietoris-Rips complex"],"falsifier":"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.","tokens_in":23292,"feed_emoji":"📈","tokens_out":11580,"duration_ms":105287,"temperature":0.7,"pith_summary":"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.","feed_headline":"Random simplicial complexes percolate sharply at a critical intensity","feed_subtitle":"The same two conditions on connection functions force exponential decay below and linear growth above the threshold.","key_machinery":"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 $\\beta_c^{(q)}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the OSSS inequality for decision trees, the influence bound whose integrated form yields the sharp transition.","marker":"[12]"},{"why":"Shows how to apply the discrete OSSS inequality to Poisson-Boolean percolation and supplies subcritical exponential decay; the proof's differential-inequality template comes from here.","marker":"[5]"},{"why":"Provides the sharp-phase-transition proof for face and cycle percolation that the latter part of Theorem 5.4 follows analytically.","marker":"[6]"},{"why":"Establishes a sharp phase transition for Boolean-model $k$-percolation via decision trees; the present model is a generalization of that setting.","marker":"[9]"},{"why":"Gives the Margulis-Russo formula for Poisson processes and the point-process facts used in Lemma 5.3 and the differential inequality.","marker":"[10]"},{"why":"Introduced the random connection model as a percolation model and proved existence of a critical intensity; this is the graph-level baseline being extended.","marker":"[16]"},{"why":"Provides exponential decay for the geometric graph's percolation function, used to show the auxiliary critical value $\\tilde{\\beta}$ is positive.","marker":"[17]"},{"why":"Proves sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation, supplying the geometric-graph exponential-decay input.","marker":"[18]"},{"why":"Establishes positivity of a critical intensity for marked random connection models, used in the proof that $\\beta_c^{(q)}>0$.","marker":"[1]"},{"why":"Compares critical intensities in the marked random connection model and supports the positivity argument via an upper-bounding reflection-invariant edge function.","marker":"[4]"}],"fun_headline_variants":["Sharp percolation threshold for random simplicial complexes","Higher-dimensional random complexes percolate sharply","Exponential decay to linear growth: sharp threshold in simplicial percolation","Two local rules force sharp percolation in simplicial complexes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sharp percolation threshold for random simplicial complexes","Higher-dimensional random complexes percolate sharply","Exponential decay to linear growth: sharp threshold in simplicial percolation","Two local rules force sharp percolation in simplicial complexes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001065,"raw_usage":{"total_tokens":4470,"prompt_tokens":956,"completion_tokens":3514,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":3443}},"tokens_in":572,"tokens_out":3514,"duration_ms":24272,"temperature":1.0,"reasoning_tokens":3443,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:46:54.868420+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"O’Donnell, M","cited_arxiv_id":null,"evidence_quote":"Supplies the OSSS inequality for decision trees, the influence bound whose integrated form yields the sharp transition."},{"cited_title":"Duminil-Copin, A","cited_arxiv_id":null,"evidence_quote":"Shows how to apply the discrete OSSS inequality to Poisson-Boolean percolation and supplies subcritical exponential decay; the proof's differential-inequality template comes from here."},{"cited_title":"Hirsch and D","cited_arxiv_id":null,"evidence_quote":"Provides the sharp-phase-transition proof for face and cycle percolation that the latter part of Theorem 5.4 follows analytically."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes a sharp phase transition for Boolean-model $k$-percolation via decision trees; the present model is a generalization of that setting."},{"cited_title":"Last and M","cited_arxiv_id":null,"evidence_quote":"Gives the Margulis-Russo formula for Poisson processes and the point-process facts used in Lemma 5.3 and the differential inequality."},{"cited_title":"Penrose.Random Geometric Graphs","cited_arxiv_id":null,"evidence_quote":"Provides exponential decay for the geometric graph's percolation function, used to show the auxiliary critical value $\\tilde{\\beta}$ is positive."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation, supplying the geometric-graph exponential-decay input."},{"cited_title":"Caicedo and M","cited_arxiv_id":null,"evidence_quote":"Establishes positivity of a critical intensity for marked random connection models, used in the proof that $\\beta_c^{(q)}>0$."}],"review_version":1}