{"id":"553ec59e-41a4-4000-9054-9c9cb026075e","arxiv_id":"2608.12954","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The continuum random connection model at critical inverse-square decay percolates for λβ≥31 and fragments for λβ<1, while its integer-lattice version never percolates at all.","lead":"This paper proves that a random graph on the line with heavy-tailed connection radii has a sharp phase transition: finite components for small density, an infinite component for large density. The result supplies the first rigorous bounds for this dependent, critical-decay model and shows its lattice analog never percolates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The work is an intricate but internally consistent proof of a genuine phase transition. The subcritical argument's load-bearing step, the balanced-rainbow construction, is explicitly verified with probability bounded below uniformly in the geometric scale, so the reader's flagged assumption is not a defect. The supercritical renormalization is independent and rests on a standard Peierls argument; its only external dependency is the cited boundary-connectivity lemma, whose stated hypothesis appears to be satisfied. No circular dependence on the numerical appendix, no fitted parameters, and no post-hoc exclusions were found. The proofs are long and could contain hidden slips, but I found no concrete gap. Hence the ACCEPT verdict stands.","tokens_in":37817,"tokens_out":56892,"duration_ms":516573,"concrete_test":"Worth running as a verification: re-derive the application of Timár's boundary-connectivity lemma to the floored binary tiling H by confirming that the cycle space of H is generated by the bounded faces, each of which is either a triangle or a quadrilateral whose two diagonals are present in H*; then check against Lemma 2 of [26] that this 'generating set of chordal cycles' hypothesis is exactly what the lemma requires. If the lemma instead requires every cycle of H to have a chord in H*, test a finite patch of H* for an induced cycle of length at least 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. I checked the subcritical proof: the balanced-rainbow construction (Definition 3.1, Proposition 3.1) is carried out at every scale with uniform positive probability, the marked cut-certificates (Lemma 3.3) and the confinement lemma (Lemma 3.4) are logically consistent, and the Kochen–Stone plus zero–one-law assembly (Propositions 3.3 and 3.4) is valid. The continuum transfer in Section 4 correctly replaces the exact cap products by exact void probabilities of Poisson regions. The supercritical dyadic-band renormalization is also coherent: the independent site events have failure probability at most 2^{-7} for β≥31, Lemma 5.4's deterministic linkage is correct, and the Peierls contour bound on the floored binary tiling is sound, given the cited Timár boundary-connectivity lemma as stated. The numerical study is clearly separated from the proofs. The remaining uncertainty is the sheer length of the proof, which I could not machine-check line by line; this matches the reader's low (not negligible) correctness risk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a weight-dependent random connection model on a Poisson point process of intensity λ on R×(0,1), with the deterministic edge rule (t∨s)|x−y|≤β, equivalently min-rule connection of Pareto radii with scale β. The main results are Theorem 1 (for λβ<1, almost surely every connected component is finite) and Theorem 2 (for λβ≥31, almost surely an infinite component exists), giving β_c∈[1,31] at intensity one. The lower bound is proved through a discrete skeleton on Z with i.i.d. Pareto radii, for which Theorem 3 establishes total fragmentation for every admissible scale x_m∈(0,1); the proof uses balanced rainbow events at independent scales, cut-point certificates with exact product probabilities, a confinement lemma, and a Kochen–Stone plus zero–one-law assembly. The continuum transfer replaces exact products by exact void probabilities of Poisson regions. The supercritical proof is a dyadic-band renormalization: independent band processes are glued deterministically by the min-rule, and the resulting site-percolation problem on a floored binary tiling is solved by a Peierls contour argument. A numerical appendix, clearly separated from the proofs, estimates β_c near 2.","tokens_in":37926,"tokens_out":22250,"duration_ms":219567,"significance":"If the results hold, this is a substantial contribution to dependent long-range percolation at the critical inverse-square decay. The paper gives a genuine phase transition in a model where edges through a common vertex are strongly dependent and where classical independent-edge results do not apply. The proofs are largely parameter-free and deliver explicit constants: the construction fixes 1 and 31 as rigorous bounds, with no fitted parameter entering the proofs. The discrete skeleton result is of independent interest and is surprising: total fragmentation coexists with almost surely infinitely many edges crossing every site. The supercritical argument uses only exact products over disjoint regions and deterministic gluing, avoiding correlation inequalities. The numerical study is clearly labeled as non-rigorous, and the ancillary code, seeds, and manifests make it reproducible; it is not used in any proof. The paper is careful to separate proven statements from numerical extrapolations, and its claims are falsifiable.","major_comments":[],"minor_comments":[{"comment":"In the displayed computation of E[X] after Eq. (1), the expression β^2/2+β^2∫_β^∞ dd/d appears to contain a typesetting artifact; it should read β^2/2+β^2∫_β^∞ d^{-1} dd.","section":"Section 2, proof of Proposition 2.1"},{"comment":"The glossary omits several later central objects, including the graph H and H⋆ of Section 5.3, the site events O_{k,i}, and the events D_k and E_k used throughout Sections 3 and 4; adding these would improve usability.","section":"Notation section"},{"comment":"The caption states that the spanning threshold extrapolates to about 1.08, below β_c, without explaining that the per-sample spanning threshold is not a consistent estimator of β_c; the surrounding text explains this, but the caption alone is misleading.","section":"Figure 12, left panel"},{"comment":"The assertion that the bounded face boundaries generate the cycle space of H is stated without proof; since the application of Timár's lemma depends on it, a one-sentence justification for locally finite planar graphs, or a reference, would make the argument self-contained.","section":"Section 5.3, Lemma 5.6"},{"comment":"The phrase 'the dependence costs an exponent' could be misread: since x_m>x_m^2, the certificate exponent 1−x_m is smaller than the independent-pairs exponent 1−x_m^2, so the cost is a decrease of the exponent; the intended meaning is clear but the wording is terse.","section":"Remark 1.1(iv)"}],"recommendation":"accept","confidential_remarks":"I agree with the reader's high confidence. The lower-bound proof is long and intricate; I verified the overall architecture and the key probabilistic estimates, but I did not machine-check every line. This is the only residual risk I see, and it is small. No citation or scope concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper settles the boundary case that Yukich explicitly excluded (p=1 on Z) and proves a real phase transition in the continuum model. I read the main theorems as proved. The lattice skeleton is totally fragmented for every x_m in (0,1); the continuum model is subcritical for lambda*beta<1 and supercritical for lambda*beta>=31. The proofs are built from first principles, no fitted constants anywhere, and the numerical appendix is clearly separated from the theorems.\n\nWhat is genuinely new: the model sits exactly at the critical 1/d^2 decay with dependent edges (shared marks), so classical independent-edge results do not apply. The paper develops two substantial tools: a rainbow/cut-point certificate for the subcritical side, and a dyadic-band renormalisation on a floored binary tiling for the supercritical side. The lattice/continuum dichotomy is real and the explanation via per-band density is convincing.\n\nThe soft spots are proportional. The proof is very long and intricate; I could not machine-check it line by line, and at this level of detail that is a genuine referee burden. The upper bound constant 31 is admittedly crude, and the authors do not pretend otherwise. The numerical study is honestly labelled as heuristic and plays no role in the proofs. I checked the conditioning in the subcritical argument and the balanced-rainbow construction is indeed carried out at every scale with uniform probability; the continuum transfer via void probabilities is sound. The only place that made me pause is the application of Timar's boundary-connectivity lemma: it looks like the right tool, and the chordality claim appears correct, but since connectivity of Gamma in H* is load-bearing for the Peierls contour bound, I would like the authors to spell out that verification more fully.\n\nThis paper is for probabilists working on long-range percolation, spatial random graphs, and weight-dependent connection models. It deserves a serious referee, and I would accept it for peer review despite the length. I would also bring it to a reading group and would cite it in my own work.","headline":"A genuine resolution of the excluded p=1 endpoint in one-dimensional long-range percolation, with a credible proof that the continuum model has beta_c in [1,31] while its lattice skeleton never percolates; deserves serious refereeing.","tokens_in":38531,"tokens_out":1810,"would_cite":true,"duration_ms":20005,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60D05","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"A dependent long-range percolation model on the line has a genuine phase transition, with β_c between 1 and 31.","keywords":["long-range percolation","critical decay","weight-dependent random connection model","Pareto radii","phase transition","Poisson point process","percolation threshold","rainbow construction"],"falsifier":"Simulate the model at $\\lambda=1$, $\\beta=0.9$ on a window of length $10^9$: Theorem 1 predicts the largest component fraction decays to zero with window size, so a reproducible spanning component would falsify it; symmetrically, at $\\beta=31$ the construction predicts a percolating cluster, so observing none on large windows would falsify Theorem 2. A sharper check is to measure the per-scale probability of the balanced-rainbow event of Definition 3.1: it should stay uniformly positive in the scale, and if it decays to zero the lower-bound mechanism is broken.","tokens_in":37551,"feed_emoji":"🔗","tokens_out":8362,"duration_ms":78298,"temperature":0.7,"pith_summary":"The paper studies a one-dimensional random graph on a Poisson point process of intensity $\\lambda$, where a point with mark $t$ behaves as a ball of radius $\\beta/t$; two points are joined when each lies inside the other's ball. The connection probability between points at distance $d$ is $\\min(1,\\beta/d)^2$, the critical decay of one-dimensional long-range percolation, but edges are dependent because they share radii. The paper proves that this model has a genuine phase transition: for $\\lambda\\beta<1$ all components are finite almost surely, while for $\\lambda\\beta\\ge 31$ an infinite component exists almost surely, so at unit intensity the critical value satisfies $\\beta_c\\in[1,31]$; a numerical study places it near 2. In contrast, the discrete version on $\\mathbb{Z}$, obtained by pinning vertices to integer sites, is totally fragmented for every admissible parameter and has only a degenerate transition—evidence that the supercritical phase is a purely continuum effect driven by point density. This matters because the model sits at a boundary case of weight-dependent random connection models where prior criteria are silent, and the proof introduces two portable mechanisms: balanced 'rainbows' to certify disconnection and a dyadic hyperbolic renormalisation to certify percolation.","feed_headline":"Infinite cluster appears: critical parameter is between 1 and 31","feed_subtitle":"At unit intensity, β_c sits in [1,31] — settling a boundary case of dependent long-range percolation.","key_machinery":"The rainbow is a pair of nested edges $\\{a,b\\}$ and $\\{c,d\\}$ with $a<c<d<b$ whose diagonals $\\{a,c\\}$ and $\\{d,b\\}$ are missing and whose overhangs $\\ell=c-a$, $m=d-c$, $r=b-d$ satisfy the balance condition $|\\ell-r|\\le m$. Balance ensures that the outer arch is the only edge allowed to cross a cut, so on a containment event the component of every vertex between two cut positions is trapped (Lemma 3.4); cut-point certificates are cap events whose probabilities are exact products, and Kochen–Stone plus the zero–one law turn uniform per-scale probability into almost sure confinement at infinitely many scales. For the supercritical direction, the mark space is split into dyadic bands $X_k$ with radii in $(\\beta 2^k,\\beta 2^{k+1}]$, each band being a dilated, dense bounded-range graph; the min-rule makes a chain of band-$k$ points absorb every larger-radius point in its span, and independent site percolation on the floored binary tiling of the hyperbolic half-plane, controlled by a Peierls contour bound using a boundary-connectivity lemma, yields the infinite cluster.","core_discovery":"The central claim is that the weight-dependent random connection model on $\\mathbb{R}\\times(0,1)$ with edge condition $(t\\vee s)|x-y|\\le\\beta$ undergoes a non-degenerate phase transition. Theorem 1 states that when $\\lambda\\beta<1$, almost surely every connected component is finite; Theorem 2 states that when $\\lambda\\beta\\ge 31$, an infinite connected component exists almost surely; together with the scaling to intensity one these give $1\\le\\beta_c\\le 31$. Theorem 3 states that the discrete skeleton on $\\mathbb{Z}$—vertices at integer sites carrying independent Pareto radii of scale $x_m$, joined when $|i-j|\\le R_i\\wedge R_j$—has no supercritical phase: for every $x_m\\in(0,1)$ all components are finite, even though almost surely infinitely many edges cross every site, and the only transition is the degenerate one to full connectivity at $x_m\\ge 1$. The paper also proves that infinitely many edges cross every point of the line almost surely, so fragmentation is achieved not by avoiding long edges but by confining every component between the cuts of rainbows.","pith_inferences":["The balance condition $|\\ell-r|\\le m$ appears to be the load-bearing geometric constraint; one could test whether relaxing it to $|\\ell-r|\\le cm$ for a large constant $c$ preserves a uniform per-scale confinement probability, which would pinpoint what the lattice skeleton's geometry contributes beyond the radius law.","The sharp contrast between the discrete and continuum models suggests that the same percolation mechanism should appear on any vertex set with unbounded local density per connection range, such as a Poisson process, but not on lattices; this is an editorial extrapolation, since the paper only proves the $\\mathbb{Z}$ and $\\mathbb{R}$ cases.","If the reported jump in the percolation density is confirmed rigorously, the model would extend the Aizenman–Newman discontinuity phenomenon from independent to dependent edges; proving or disproving that jump is a natural next step.","The dyadic renormalisation may carry to other one-dimensional models with scale-invariant connection kernels, since only the min-rule and the band density enter; testing it on a kernel with a different boundary exponent would show whether the mechanism is specific to Pareto radii."],"forward_implications":["At unit intensity every $\\beta<1$ gives only finite components, while every $\\beta\\ge 31$ gives an infinite component; the critical point is genuinely between 1 and 31, with numerics near 2.","The discrete skeleton shows that infinite components are not forced by the heavy-tailed radii alone: on $\\mathbb{Z}$, with $x_m<1$, all components are finite and yet their diameters are unbounded, and infinitely many edges cross every site.","The boundary case $\\gamma=0$ of the weight-dependent random connection model in one dimension is decided: a supercritical phase exists, closing a case where percolation-threshold criteria were silent.","The supercritical proof is quantitative: the site-open probability $(1-e^{-\\lambda\\beta/4})^{16}$ controls the failure rate, and optimising the contour constants would lower the 31 to about 15, still above the numerical critical value.","The numerical curves indicate a jump in percolation density at the threshold, with crossing heights 0.7–0.9, consistent with the known discontinuity for independent $1/|x-y|^2$ percolation."],"supporting_citations":[{"why":"Supplies the classical baseline: for independent $1/d^2$ bonds percolation depends on the multiplicative constant, with $\\beta_{\\text{eff}}\\le 1$ subcritical; Theorem 1 reproduces that constant for dependent edges.","marker":"[2]"},{"why":"Provides the cut-point and gap method in one-dimensional long-range percolation that the rainbow disconnection certificates generalise.","marker":"[23]"},{"why":"Defines the weight-dependent random connection model whose one-dimensional boundary case $\\gamma=0$ with indicator profile is the model studied here and was left open.","marker":"[17]"},{"why":"Supplies the Poisson-process tools (restriction, marking, Harris–FKG, ergodicity) used to make certificate probabilities exact exponentials.","marker":"[22]"},{"why":"Provides the Kochen–Stone lemma used to upgrade per-scale positive probability to almost-sure occurrence of infinitely many confinement scales.","marker":"[21]"},{"why":"Provides the boundary-connectivity lemma used in the Peierls contour argument on the floored binary tiling.","marker":"[26]"},{"why":"Introduces the min-rule with Pareto radii: the discrete skeleton is his $G_{1,x_m}$ on $\\mathbb{Z}$ and the continuum model is the Poisson version he anticipated.","marker":"[27]"},{"why":"Supplies the Harris inequality used to bound probabilities of decreasing events in the subcritical reduction and in the supercritical glueing.","marker":"[18]"}],"fun_headline_variants":["Rainbow percolation: βc between 1 and 31","Critical beta for rainbow percolation: 1 to 31","Infinite cluster for β≥31, none for β<1 in rainbow percolation","Degenerate transition in discrete skeleton: no supercritical phase","Rainbow percolation: sharp bounds, degenerate limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The subcritical bound rests on the assertion that balanced rainbows—nested edge pairs whose left and right overhangs differ by at most the inner span—form at every geometric scale with probability bounded below uniformly; if that block construction ever fails at a scale, the confinement argument and the lower bound $\\beta_c\\ge 1$ collapse.","fun_headline_variants_meta":{"raw":{"variants":["Rainbow percolation: βc between 1 and 31","Critical beta for rainbow percolation: 1 to 31","Infinite cluster for β≥31, none for β<1 in rainbow percolation","Degenerate transition in discrete skeleton: no supercritical phase","Rainbow percolation: sharp bounds, degenerate limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000744,"raw_usage":{"total_tokens":3340,"prompt_tokens":988,"completion_tokens":2352,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":2259}},"tokens_in":604,"tokens_out":2352,"duration_ms":16940,"temperature":1.0,"reasoning_tokens":2259,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:48:43.015809+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the model at $\\lambda=1$, $\\beta=0.9$ on a window of length $10^9$: Theorem 1 predicts the largest component fraction decays to zero with window size, so a reproducible spanning component would falsify it; symmetrically, at $\\beta=31$ the construction predicts a percolating cluster, so observing none on large windows would falsify Theorem 2. A sharper check is to measure the per-scale probability of the balanced-rainbow event of Definition 3.1: it should stay uniformly positive in the scale, and if it decays to zero the lower-bound mechanism is broken.","supporting_citations":[{"cited_title":"Discontinuity of the percolation density in one dimensional 1/|x−y| 2 percolation models","cited_arxiv_id":null,"evidence_quote":"Supplies the classical baseline: for independent $1/d^2$ bonds percolation depends on the multiplicative constant, with $\\beta_{\\text{eff}}\\le 1$ subcritical; Theorem 1 reproduces that constant for dependent edges."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Poisson-process tools (restriction, marking, Harris–FKG, ergodicity) used to make certificate probabilities exact exponentials."},{"cited_title":"A lower bound for the critical probability in a certain percolation pro- cess","cited_arxiv_id":null,"evidence_quote":"Supplies the Harris inequality used to bound probabilities of decreasing events in the subcritical reduction and in the supercritical glueing."}],"review_version":1}