{"id":"d0df4216-ac91-45b0-8cae-3847cb1495f6","arxiv_id":"2507.19288","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"At high dimension d, the critical two-point connection probability in the random connection model decays like |x|^{-(d-2)} with an explicit anisotropic prefactor.","lead":"This paper proves that in the high-dimensional continuum percolation model, the probability that two points are connected at the critical density decays as the distance to the power minus (d minus 2), the mean-field value. A generalist should care because it removes a long-standing gap for continuum percolation and simplifies an earlier proof on the integer lattice.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main asymptotic statement is imported wholesale from an unproved external deconvolution theorem ([21, Thm 1.3]); until that theorem is verified, Theorem 1.3 is not self-contained.","rationale":"I read the paper in good faith. The organization is clear: Section 2 proves the base-case estimates, Section 3 supplies the L^p moment induction, and Section 4 plus Appendix A give the diagrammatic bounds. The internal reasoning appears coherent, and the use of L^p moments to weaken Hara's estimates is a genuine technical contribution. However, the final asymptotic formula is not derived in the paper; it is imported verbatim from [21, Theorem 1.3]. That theorem is the precise mechanism that turns the Fourier integral (1.23) into the stated |x|^{-(d-2)} decay with explicit coefficient. The present paper only verifies a list of moment and infrared conditions. If those conditions do not match [21]'s hypotheses exactly, the main theorem is unsupported. The borderline condition (iii), with an L^p moment for p<d/4 rather than an L^1 moment, is precisely where a mismatch would be most plausible, so a careful check of [21] is essential. The reader's weakest_assumption already identified this external dependency, and I agree with that assessment. I therefore keep the reader's CONDITIONAL verdict unchanged; the paper should not be accepted as fully self-contained until the deconvolution theorem is either proved, published in a refereed venue, or verified to apply verbatim. The Z^d percolation claim in the abstract is also unproved here, but it is not the load-bearing step for the main R^d result.","tokens_in":28774,"tokens_out":8051,"duration_ms":78434,"concrete_test":"Obtain [21, Theorem 1.3] and independently re-derive its statement from the arguments in [21]; verify that its assumptions are exactly (i)-(v) with no hidden extra requirement, especially that p<d/4 in (iii) is sufficient and that (1.24) follows with the stated constant and positive-definite Σ. If the theorem is correct and applicable verbatim, the concern is resolved; if any hypothesis is mismatched, Theorem 1.3 needs a proof of the needed deconvolution statement or a corrected formulation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3 is established by checking hypotheses (i)-(v) and then invoking [21, Theorem 1.3] to convert the Fourier integral (1.23) into the sharp |x|^{-(d-2)} asymptotics. That theorem is the whole deconvolution step; the new L^p moment induction only produces inputs for it. [21] is an arXiv preprint by one of the present authors and is neither proved nor restated here. If its hypotheses are not exactly the conditions verified in the proof — in particular condition (iii) with |x|^{d-2}J_{λ_c}∈L^p∩L^2 for some p<d/4, obtained via the interpolation (1.30)-(1.31) — or if its conclusion requires an extra condition such as an unweighted L^1 (d−2)-moment, then (1.23) alone gives no power-law decay and Theorem 1.3 collapses. Moreover, the dependence is not cosmetic: the explicit constant and the matrix Σ in (1.13) are taken from (1.24), i.e. from [21]. Thus the central claim rests on an unverified input. This is an external-dependency concern rather than an internal inconsistency; the diagrammatic estimates themselves appear coherent. The abstract's assertion about nearest-neighbour Z^d percolation at d≥11 is also not proved in the paper, but it is secondary to the main theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the random connection model on R^d in high dimensions (d > d_0 ≥ 8). Under Assumption 1.1 on the adjacency function φ, the authors prove that the critical two-point connection probability τ_{λ_c}(x) decays like (a_d/(λ_c √det Σ)) (x·Σ^{-1}x)^{-(d-2)/2} as |x|→∞, i.e., the critical exponent η = 0 with possibly anisotropic decay. The proof uses the lace expansion to write τ as the solution of a convolution equation with kernel J_{λ_c}, verifies moment and infrared conditions on J_{λ_c} (in particular, that |x|^{d-2}J_{λ_c}∈L^p∩L^2 for some p<d/4) via a new L^p induction for the lace function Π, and then invokes a deconvolution theorem from the preprint [21] to extract the sharp asymptotic from the Fourier integral. The abstract also claims the argument applies to nearest-neighbour Bernoulli percolation on Z^d for d≥11.","tokens_in":29042,"tokens_out":25449,"duration_ms":206741,"significance":"If the cited deconvolution theorem [21] is correct, the paper fills a gap by proving the mean-field value η=0 for the random connection model, complementing the known exponents γ=1, β=1, δ=2. The L^p moment induction is a genuine technical innovation that simplifies Hara's earlier approach and avoids the second diagrammatic estimate. The explicit verification of Assumption 1.1 for standard models (disk, Gaussian) is useful. However, the main theorem inherits its conclusion from an unproved external preprint, and the abstract's Z^d claim is not proved in the text; these dependencies must be resolved before the theorem can be considered self-contained.","major_comments":[{"comment":"The central asymptotic (1.24) is imported from [21, Theorem 1.3], which is not stated, proved, or even summarized in the present paper. The explicit constant and the matrix Σ in Theorem 1.3 are taken directly from (1.24). The authors list hypotheses (i)–(v) and say they verify them, but the reader cannot check that these are the complete and exact hypotheses of [21, Theorem 1.3]; if that theorem requires an additional condition (for instance a certain decay of Ĵ or a different unweighted moment), then the proof of Theorem 1.3 collapses. The authors should include a full statement of [21, Theorem 1.3], confirm that (i)–(v) are exactly its hypotheses, and ideally provide a proof of the deconvolution step in an appendix or cite a published version of [21].","section":"Section 1.2, Proof of Theorem 1.3 (Eq. (1.23)–(1.24))"},{"comment":"The assertion that the proof \"also applies to nearest-neighbour Bernoulli percolation on Z^d in d ≥ 11\" is not supported by any theorem, proof, or substantive sketch in the paper. The analysis is carried out only for the continuum random connection model; no Z^d analogue of the L^p induction (Proposition 1.6), the diagrammatic estimates, or the deconvolution step is presented. The paper cites [8] only for the existence of a convergent lace expansion with d0=10. The authors should either state and prove the Z^d result (or provide a detailed transfer argument), or soften the abstract to claim that the method is expected to extend to that setting.","section":"Abstract and Remark 1.4"},{"comment":"The convergence of the diagrammatic expansion, which is essential for Proposition 4.1 and hence for the bounds on Π, relies on Proposition 7.1 and Lemma 5.7 of the preprint [5], and on the definitions of U_{λ_c} and V_{λ_c} taken from that paper. These results are not proved in the present manuscript, and [5] is not published. This is a second external dependency that is load-bearing for the proof. The authors should either reproduce the statements they need from [5], or provide the proofs in an appendix, or cite a published version if one becomes available.","section":"Section 4, Lemma 4.9 and Eqs. (4.2)–(4.3)"}],"minor_comments":[{"comment":"The word \"dimensons\" in the bibliographic entry for Duminil-Copin and Panis should be \"dimensions\".","section":"Reference [6]"},{"comment":"The limit in (2.16) is written as |k|→∞, but the pointwise limit in (2.15) is taken as |k|→0; the limit in (2.16) should also be |k|→0.","section":"Proof of Lemma 2.3, Eq. (2.16)"},{"comment":"The bound ar{H}^{(a,b)}_1 ≤ (1/2 λ_c)^{-1} ar{E}^{(a)} ar{T}^{(b)} ar{S} would read more clearly with the factor λ^{-1} explained explicitly as coming from (3.36); this is a minor presentation point.","section":"Section 3.3.4, last display"}],"recommendation":"major_revision","confidential_remarks":"The key external inputs [5] (Dickson–Heydenreich) and [21] (Liu) are preprints by the authors or close collaborators and are not yet published. The main result of the manuscript cannot be independently verified without access to [21, Theorem 1.3], and the Z^d claim in the abstract is not proved. It would be appropriate to require that the essential theorems from these preprints be either included in the paper or made available in final, citable form before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good paper. The genuinely new piece is the L^p moment induction, which gives moment bounds on the lace function and bypasses the harder of Hara's two diagrammatic estimates. The diagrammatic estimates are detailed and, as far as I can tell, coherent. The reduction of Theorem 1.3 to the moment conditions (i)-(v) is clean, and the argument that the moment conditions imply the decay is outsourced to the deconvolution theorem, which is exactly the right division of labor if that theorem holds.\n\nThe paper's main theorem is therefore conditional on Liu's deconvolution preprint [21]. That theorem is not stated or proved here, and the explicit constant and matrix in (1.13) are imported from it. If its hypotheses do not match the conditions verified in the paper, or if it has an unstated extra condition, Theorem 1.3 collapses. This is not an internal inconsistency - the diagrammatic estimates stand on their own - but it is a heavy external dependency for the central result. A referee needs to check the hypothesis verification line-by-line against [21]. The same goes for [5], which supplies the bounds on U and V; though that is a 2022 preprint by one of the authors and presumably stable.\n\nThe abstract's claim that the proof 'also applies' to nearest-neighbour Z^d percolation in d>=11, and simplifies Hara, is a bit looser than what the paper shows. The paper proves the R^d theorem; the Z^d statement is plausible given the framework and [8], but no formal proof is written out. It should be either proved in a later section or downgraded to a remark.\n\nThe circularity worry some might have does not land: the target decay is not assumed; the moment conditions are proved from the lace expansion bootstrap, and the deconvolution theorem is a separate input. The self-citations are reasonable.\n\nThis is a solid, meaningful paper for the high-dimensional percolation and lace expansion community. It deserves a serious referee, not a desk rejection. I would ask the authors to restate the deconvolution theorem in enough detail to make the paper self-contained, or to cite a published version, and to tone down the Z^d claim. If those are addressed, it is a clean contribution.","headline":"The L^p moment induction is a real advance and the diagrammatic estimates look coherent, but the main theorem leans on an unverified deconvolution preprint and the Z^d claim outruns what is proved.","tokens_in":29587,"tokens_out":2722,"would_cite":true,"duration_ms":26595,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"At the critical point in the high-dimensional random connection model, two points connect with probability decaying like $|x|^{-(d-2)}$.","keywords":["random connection model","continuum percolation","critical two-point function","mean-field critical exponent","lace expansion","deconvolution","high-dimensional percolation","eta=0"],"falsifier":"For a concrete adjacency function such as the Gaussian model $\\varphi(x)=(2\\pi)^{-d/2}e^{-|x|^2/2}$ in $d=9$, numerically evaluate the Fourier integral (1.23) at the critical intensity and check whether $|x|^{d-2}\\tau_{\\lambda_c}(x)$ converges to $a_d/(\\lambda_c\\sqrt{\\det\\Sigma})$ along each coordinate axis; a different power or a non-elliptic direction-dependent prefactor would refute Theorem 1.3. A complementary check is to verify hypothesis (iii) directly for the lace function, testing whether $|x|^{d-2}\\Pi_{\\lambda_c}(x)$ lies in $L^p\\cap L^2\\cap L^\\infty$ for some $p<d/4$.","tokens_in":28548,"feed_emoji":"🕸️","tokens_out":9729,"duration_ms":84909,"temperature":0.7,"pith_summary":"The paper proves that in the random connection model on $\\mathbb{R}^d$—vertices from a Poisson process, edges drawn independently with probability given by an integrable, even adjacency function—the critical connection probability between two points decays, for all sufficiently large $d$, like a constant times $|x|^{-(d-2)}$ with possible anisotropy. That is the mean-field critical exponent $\\eta=0$ for a continuum percolation model. The proof combines the lace expansion with a deconvolution theorem that converts moment estimates on the lace kernel into an asymptotic Fourier integral, and it does so through an $L^p$ induction on the kernel's moments. The same argument is claimed to cover nearest-neighbour Bernoulli percolation on $\\mathbb{Z}^d$ in $d\\ge 11$ and to shorten the earlier 2008 proof for that model. If correct, the result supplies a standard input for studying the incipient infinite cluster and related high-dimensional questions in the continuum setting.","feed_headline":"Critical connection odds decay like distance^{-(d-2)}","feed_subtitle":"This pins the mean-field exponent eta=0 for continuum percolation and simplifies the lattice proof.","key_machinery":"The engine is the lace function $\\Pi_\\lambda$, which appears in the Ornstein-Zernike equation $\\tau_\\lambda=(\\varphi+\\Pi_\\lambda)+\\lambda(\\varphi+\\Pi_\\lambda)*\\tau_\\lambda$, rearranged as $(\\delta-J_\\lambda)*\\lambda\\tau_\\lambda=J_\\lambda$. The proof upgrades the classical moment condition by working with $L^p$ moments: Definition 1.5 calls the $a$-th moment of $\\Pi_\\lambda$ good if $|x|^a\\Pi_\\lambda(x)$ is bounded in $L^{p_a}\\cap L^2\\cap L^\\infty$ for a suitable $p_a$, and Proposition 1.6 inductively raises the good moment order from $\\phi$ to $\\phi+2$. The induction is driven by diagrammatic estimates (Proposition 2.1) that bound the weighted lace function by decorated bubble, triangle, and martini diagrams, whose finiteness is shown by Fourier-analytic derivative estimates. A deconvolution theorem then turns the final $(d-2)$-moment condition into the asymptotic formula for the inverse Fourier integral.","core_discovery":"The central claim is Theorem 1.3. Let the adjacency function $\\varphi$ obey Assumption 1.1 and take $d>d_0$ with $d_0\\ge 8$ sufficiently large. Then there exists a positive-definite diagonal matrix $\\Sigma$, given explicitly as the second-moment matrix of the lace-expansion kernel $J_{\\lambda_c}=\\lambda_c(\\varphi+\\Pi_{\\lambda_c})$, such that $\\tau_{\\lambda_c}(x)\\sim \\frac{a_d}{\\lambda_c\\sqrt{\\det\\Sigma}}\\,(x\\cdot\\Sigma^{-1}x)^{-(d-2)/2}$ as $|x|\\to\\infty$, with $a_d=\\Gamma((d-2)/2)/(2\\pi^{d/2})$. In words, the critical two-point connection probability has exact power-law decay with the random-walk exponent $d-2$, i.e. $\\eta=0$, including the correct prefactor. The paper further claims that the same proof applies to nearest-neighbour Bernoulli percolation on $\\mathbb{Z}^d$ for $d\\ge 11$. The route is to derive the Fourier integral $\\lambda_c\\tau_{\\lambda_c}(x)=J_{\\lambda_c}(x)+\\int \\frac{\\hat J_{\\lambda_c}(k)^2}{1-\\hat J_{\\lambda_c}(k)}e^{-ik\\cdot x}\\frac{dk}{(2\\pi)^d}$, verify five moment and infrared conditions on $J_{\\lambda_c}$, and invoke a deconvolution theorem that outputs the asymptotic form.","pith_inferences":["If the deconvolution theorem cited as [21] holds, the same $L^p$-moment route could be transferred to other lace-expansion models in $\\mathbb{R}^d$—for example self-avoiding walks or massive models—where moment bounds are cheaper than pointwise decay bounds; the paper does not make this extension.","The $d>8$ restriction is not intrinsic to the argument: the square diagram is used only for convenience, and the authors note that replacing it with a $b<1$ triangle should handle $d=7$ or $8$. A testable extension would be to run the same induction for spread-out models in those dimensions.","One could numerically test the prefactor, not just the exponent, by Monte Carlo simulation of the Palm connection probability in $d=9$ for the disk or Gaussian model and comparing $|x|^{d-2}\\tau_{\\lambda_c}(x)$ with $a_d/(\\lambda_c\\sqrt{\\det\\Sigma})$.","The paper's method requires $|x|^{d-2}\\varphi(x)\\in L^p$ with $p<d/4$; a natural open direction is to determine whether the same $\\eta=0$ asymptotic survives for adjacency functions with heavier tails, where only an $L^1$ version of the $(d-2)$-moment is available."],"forward_implications":["In high-dimensional random connection models at criticality, the probability that two distant points are connected decays as $|x|^{-(d-2)}$ along every direction, with an elliptically anisotropic prefactor governed by the second moments of the lace kernel.","The mean-field critical exponent $\\eta$ takes the value $0$ for the continuum model, matching the exponents $\\gamma=1$, $\\beta=1$, and $\\delta=2$ already known in this setting.","The method supplies the same $\\eta=0$ result for nearest-neighbour Bernoulli percolation on $\\mathbb{Z}^d$ in $d\\ge 11$, with a proof the authors describe as considerably simpler than the 2008 argument.","The established two-point decay is the type of input used to construct the incipient infinite cluster and to study one-arm exponents, half-space percolation, and torus problems; the authors state the result is expected to be useful for those continuum analogues.","Because the proof is not tied to perturbative small parameters beyond convergence of the lace expansion, the same moment-induction scheme can be applied to any sufficiently spread-out random connection model in $d>8$ once the expansion converges."],"supporting_citations":[{"why":"Supplies the deconvolution theorem that converts the verified moment conditions (i)-(v) into the asymptotic formula (1.24); this is the load-bearing step.","marker":"[21]"},{"why":"Provides the lace expansion for the random connection model, including convergence, bootstrap bounds, and the diagrammatic estimates used in Section 2.","marker":"[15]"},{"why":"Is the earlier proof of $\\eta=0$ for nearest-neighbour percolation and the source of the induction strategy; the paper claims to simplify it.","marker":"[10]"},{"why":"Establishes the deconvolution strategy on the lattice that [21] extends to $\\mathbb{R}^d$; motivates the $L^p$ moment conditions.","marker":"[23]"},{"why":"Gives lace expansion convergence for nearest-neighbour Bernoulli percolation in $d>10$, enabling the $d\\ge 11$ claim.","marker":"[8]"},{"why":"Provides the $U$ and $V$ diagram bounds used in Section 4 to control the lace expansion terms.","marker":"[5]"},{"why":"Earlier work realising that $L^p$ moment estimates suffice for the deconvolution approach, cited as the source of this insight.","marker":"[22]"}],"fun_headline_variants":["Exact power-law decay in continuum percolation","Mean-field exponent confirmed for high-d percolation","Bernoulli percolation proof simplified via lace expansion","Critical two-point function decays like |x|^{-(d-2)}","Percolation decay: exponent d-2 nailed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof imports, without proving it here, a deconvolution theorem from a companion preprint (cited as [21, Theorem 1.3]) that turns verified moment bounds on the lace kernel into the exact asymptotic decay formula; if that theorem is false or cannot be applied at $\\lambda_c$, the main conclusion does not follow from this paper.","fun_headline_variants_meta":{"raw":{"variants":["Exact power-law decay in continuum percolation","Mean-field exponent confirmed for high-d percolation","Bernoulli percolation proof simplified via lace expansion","Critical two-point function decays like |x|^{-(d-2)}","Percolation decay: exponent d-2 nailed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000279,"raw_usage":{"total_tokens":1667,"prompt_tokens":967,"completion_tokens":700,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":619}},"tokens_in":583,"tokens_out":700,"duration_ms":7543,"temperature":1.0,"reasoning_tokens":619,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:56:22.340457+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete adjacency function such as the Gaussian model $\\varphi(x)=(2\\pi)^{-d/2}e^{-|x|^2/2}$ in $d=9$, numerically evaluate the Fourier integral (1.23) at the critical intensity and check whether $|x|^{d-2}\\tau_{\\lambda_c}(x)$ converges to $a_d/(\\lambda_c\\sqrt{\\det\\Sigma})$ along each coordinate axis; a different power or a non-elliptic direction-dependent prefactor would refute Theorem 1.3. A complementary check is to verify hypothesis (iii) directly for the lace function, testing whether $|x|^{d-2}\\Pi_{\\lambda_c}(x)$ lies in $L^p\\cap L^2\\cap L^\\infty$ for some $p<d/4$.","supporting_citations":[{"cited_title":"Gaussian deconvolution on $\\mathbb R^d$ with application to self-repellent Brownian motion","cited_arxiv_id":"2411.16058","evidence_quote":"Supplies the deconvolution theorem that converts the verified moment conditions (i)-(v) into the asymptotic formula (1.24); this is the load-bearing step."},{"cited_title":"Heydenreich, R","cited_arxiv_id":null,"evidence_quote":"Provides the lace expansion for the random connection model, including convergence, bootstrap bounds, and the diagrammatic estimates used in Section 2."},{"cited_title":"Fitzner and R","cited_arxiv_id":null,"evidence_quote":"Gives lace expansion convergence for nearest-neighbour Bernoulli percolation in $d>10$, enabling the $d\\ge 11$ claim."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier work realising that $L^p$ moment estimates suffice for the deconvolution approach, cited as the source of this insight."}],"review_version":1}