{"id":"7a9eb49d-bc21-4c3e-84bd-31f2bb00cb57","arxiv_id":"2607.24561","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In high-dimensional critical percolation, the rescaled cluster measure converges to the super-Brownian occupation measure, the rescaled cluster converges to its range in Hausdorff distance, and the one-arm probability has exact r^{-2} asymptotics.","lead":"Critical percolation clusters in dimensions 11 and higher are shown to converge, after rescaling, to the random mass distribution described by super-Brownian motion, and the cluster itself converges to the range of that continuum object. The paper also derives the sharp one-arm asymptotics r^2 P(0↔∂B_r) → θ_1, pinning down the constant first bounded by Kozma and Nachmias.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof is internally coherent, and the main residual risk is the external k-point convergence input (1.3), as the reader noted.","rationale":"The reader's ACCEPT verdict is appropriate. The paper is careful and explicitly identifies its external inputs. The most delicate internal points — moment convergence under the sigma-finite canonical measure, tightness under size-biasing, the support-convergence lemma, and the thin-region passage from measure to geometry — are all argued in detail with correct hypotheses (d>6, (1.2), (1.3)). I did not find a place where the argument silently assumes something beyond what is stated. The only concern that could shift a verdict is the dependence on the very recent and not-yet-independently-refereed results [21] and [9]; this is exactly the reader's weakest_assumption. Because this is an external-input risk rather than an internal inconsistency, the verdict should remain unchanged, and the concrete test above would settle the residual risk.","tokens_in":57842,"tokens_out":46153,"duration_ms":409743,"concrete_test":"Independently verify the normalization dictionary in Appendix A: derive the m=1 and m=2 cases of (1.3) from [21, Theorem 1] and confirm that γ=λ/a equals 2daβρ, then check that Proposition 3.11's dominated convergence uses exactly the pointwise convergence stated in (1.3) with no additional uniformity beyond the domination provided by Lemma 3.9. If the k-point convergence has the stated normalization, the measure-convergence theorem follows and the geometric consequences are supported by the internal arguments.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the central chain. Section 3's moment convergence is an exact lattice-to-continuum rewriting (3.13)-(3.14) plus dominated convergence with the Aizenman-Newman tree-graph domination (Lemma 3.9); size-biasing converts the sigma-finite limit into a finite measure with finite exponential moments (Corollary 3.5), and tightness is supplied by the one-arm bound (Lemma 3.15). Section 4's range convergence depends only on the measure convergence, the thin-region estimate from Section 2, and the deterministic Athreya-Lohr-Winter support lemma, all of which are checked. The one-arm proof then follows from the conditional range convergence on the event μ(φ0)>ε and a thin-region omission estimate. I found no internal inconsistency or missing step that would invalidate the central claim. The genuinely load-bearing condition is external: Theorem 1.1, and hence Theorems 1.3 and 1.4, rest on the k-point convergence (1.3) of [21] and on the analogue estimate (2.7) adapted from [9]. Both are recent preprints, and the paper does not re-prove them; if either had a hidden uniformity or normalization error, the conclusions would fail. This is a refereeing risk, not an internal defect.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves, for critical nearest-neighbour bond percolation on Z^d with d ≥ 11, that the rescaled empirical cluster measure (1.1) converges in a σ-finite sense to the total occupation measure of super-Brownian motion, conditional on the total mass exceeding ε. It also proves a uniform lower mass bound for the critical cluster in the Euclidean metric, derives Hausdorff convergence of the rescaled cluster to the range of super-Brownian motion, and obtains the sharp one-arm asymptotics r^2 P(0 ↔ ∂B_r) → θ_1 ∈ (0,∞). The derivation is carried out under the explicit external hypotheses (1.2) and (1.3), the latter being the k-point convergence result of [21]; the paper is transparent that d ≥ 11 serves only to guarantee these inputs.","tokens_in":58111,"tokens_out":33709,"duration_ms":289686,"significance":"If the external inputs are accepted, this is a major step: it upgrades the moment-level k-point convergence of [21] to law-level convergence of the cluster measure in the high-dimensional nearest-neighbour setting, and it refines the Kozma–Nachmias one-arm bound to an exact asymptotic with an identified constant. The lower mass bound is of independent interest for random-walk and Gromov–Hausdorff-type scaling programmes. The paper is carefully structured: the size-biasing mechanism converts the infinite canonical measure into a finite measure, the moment-convergence step is an exact lattice-to-continuum rewriting rather than an approximation, and the deterministic support-convergence lemma cleanly isolates the role of the lower mass bound. The dependence on the recent preprints [21] and [9] is clearly flagged; there is no sign of circularity or of fitted constants manufacturing the conclusion.","major_comments":[{"comment":"In the proof of Theorem 2.1, the displayed estimate ∑_{i=ε^{-1/2}}^{ε^{-1}} [r^{d+1}e^{-c(log αr^2)^4} + α/(iε)^2 + r^2 e^{-δr^2/2}] ≲_Λ ε^{-1}[r^{d+1}e^{-c(log αr^2)^4} + α + r^2 e^{-δr^2/2}] is not correct for the middle term: ∑_{i=ε^{-1/2}}^{ε^{-1}} (iε)^{-2} ≍ ε^{-3/2}, not ε^{-1}. With the stated choices α ≍ ε^2 and ε ≍ η^{1/4}, the correctly bounded first term is of order η^{1/8}, not η^{1/4}. Thus the proof as written establishes at most limsup ≤ C η^{1/8}, and Theorem 2.1's stated η^{1/4} rate is not justified. The later uses of Theorem 2.1 only require the bound to vanish as η↓0, so the main scaling-limit theorems are not endangered, but the theorem statement and Lemma 4.2 should be adjusted to the weaker rate or the argument revised.","section":"§2.4, display after (2.20)"},{"comment":"Lemma 2.6 relies on the exterior-geometry estimate (2.7), which is described only as an analogue of [9, Claim 5.5] with the assertion that 'the same argument remains valid.' This estimate is load-bearing for Lemmas 2.6 and 2.7 and hence for Theorem 2.1. Since [9] is a recent preprint rather than a published theorem, the authors should either prove (2.7) in the present paper or state it as a standalone lemma with full hypotheses and a sufficiently detailed proof sketch, so that the lower mass bound is not contingent on an unverified adaptation.","section":"§2.2, Eq. (2.7)"}],"minor_comments":[{"comment":"The symbol C(K) denotes different quantities in Lemma 2.7 and in the proof of Lemma 2.10; renaming one of them would avoid confusion.","section":"§2.2–§2.3"},{"comment":"There is a stray extra parenthesis in the sentence 'with a, λ as in (1.3)), jointly in M_F(R^d) × K(R^d)'; the closing parenthesis after (1.3) should be removed.","section":"Theorem 1.3 statement"},{"comment":"In the definition of C(K) = K^d / p_c^{2dK}, please check whether the exponent should be dK rather than 2dK, to match the binomial domination parameter p_c^{dK} used a few lines later in the proof.","section":"Lemma 2.7"},{"comment":"The one-arm estimate (2.1) is introduced as a known input at the start of Section 2, but it is also proved/later stated as Proposition 3.8; adding a forward cross-reference would clarify the logical dependence of Section 2.","section":"§2.1 and Proposition 3.8"}],"recommendation":"major_revision","confidential_remarks":"The main technical issue I found is the incorrect summation estimate in the proof of Theorem 2.1. It is local and fixable: the proof yields a slightly worse power of η, and the rest of the paper only needs the bound tend to zero as η↓0. I would not reject on this basis. A second concern is that the lower mass bound depends on an unproved adaptation of an estimate from the preprint [9]; I would ask the authors to make that dependence explicit and self-contained. The editor should also note that Theorems 1.1, 1.3 and 1.4 rest on the k-point convergence result of [21], another recent preprint; if that input is verified, the contribution is substantial. No circularity or hidden assumption was found."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper does the real thing. It upgrades the known k-point convergence [21] to law-level convergence of the rescaled cluster measure, and then to Hausdorff convergence of the cluster to the SBM range and sharp one-arm asymptotics. The lower mass bound is a genuinely self-contained result of independent interest. The authors are explicit about what is theirs and what is imported; the central chain in Sections 2-4 is coherent and I did not find a load-bearing gap.\n\nThe main risk is external. Theorem 1.1, and hence the geometric corollaries, rest on the k-point convergence hypothesis (1.3) proved in [21], a recent preprint, plus the uniform two-point bound (1.2). The paper does not reprove either. If either had a hidden uniformity or normalization error, everything downstream fails. That is a refereeing risk, not a defect in the manuscript. Also, Lemma 2.7 uses an estimate (2.7) adapted from [9]; the authors say the same argument works for the exterior-of-a-box geometry. That is plausible but should be checked line by line. Minor; the proof of Lemma 2.7 also leaves constants implicit.\n\nThe paper is honest about the parallel independent work of Blanc-Renaudie and Hutchcroft [18], and explains the difference in inputs and coverage. The dimension handling is clean: results stated under (1.2)+(1.3) with d>6, applied to nearest-neighbour d>=11. No invented free parameters, no fitting.\n\nWho is this for? Researchers in high-dimensional percolation and scaling limits. It is a serious contribution and should go to a serious referee. I would accept it for review and expect it to be published after a careful check of the external inputs.","headline":"Upgrades k-point convergence to law, range, and sharp one-arm asymptotics; main risk is reliance on external preprints.","tokens_in":58628,"tokens_out":1641,"would_cite":true,"duration_ms":15836,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60J68","60F17","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"Critical percolation clusters converge to super-Brownian range in d≥11.","keywords":["critical percolation","high-dimensional percolation","super-Brownian motion","canonical measure","one-arm exponent","range convergence","k-point functions","lower mass bound"],"falsifier":"Run high-precision simulations of critical bond percolation on $\\mathbb{Z}^{11}$ and record $r^2\\mathbb{P}(0\\leftrightarrow\\partial B_r)$ for $r$ a sequence of dyadic scales; if the sequence does not converge to a positive finite constant, or if the constant disagrees with $\\theta_1$ evaluated numerically from the radial PDE $\\tfrac12 v''+\\tfrac{d-1}{2r}v'=\\tfrac{\\lambda}{2a}v^2$ with $v\\to\\infty$ at $1$, the central claim fails. Equivalently, simulate the conditioned cluster $\\{|\\mathcal C|\\ge R^4\\}$ and test the lower mass bound: if with positive probability some ball of radius $\\delta R$ met by the cluster contains $o(R^4)$ sites as $R\\to\\infty$ at fixed $\\delta$, Theorem 1.2 and the range convergence collapse.","tokens_in":57676,"feed_emoji":"🎲","tokens_out":9567,"duration_ms":80720,"temperature":0.7,"pith_summary":"The paper proves the super-Brownian scaling limit for high-dimensional critical percolation at the level of laws. For critical nearest-neighbour bond percolation on $\\mathbb{Z}^d$ with $d\\ge 11$, the rescaled empirical measure of the cluster of the origin, conditioned to carry macroscopic mass, converges in a $\\sigma$-finite sense to the canonical total occupation measure of super-Brownian motion. A new uniform lower mass bound—any macroscopic region the cluster reaches contains of order $R^4$ points—then upgrades this to convergence of the rescaled cluster as a compact set, in the Hausdorff metric, to the range of super-Brownian motion. The same machinery yields the sharp one-arm asymptotics $r^2\\mathbb{P}(0\\leftrightarrow \\partial B_r)\\to\\theta_1\\in(0,\\infty)$, replacing the previously known order-only bound.","feed_headline":"Critical percolation clusters converge to super-Brownian range","feed_subtitle":"Measure and Hausdorff range convergence pin down the r² one-arm constant in dimensions 11 and up.","key_machinery":"The load-bearing identity is the exact moment dictionary between lattice connection functions and tree integrals. The rescaled $(m+1)$-point functions of percolation converge to sums over binary trees of integrals $I_T$ of Brownian Green kernels, and the same tree integrals are the moment densities of the total occupation measure of super-Brownian motion under its canonical measure. Around this identity, the proof uses size-biasing by $\\mu(\\varphi_0)$ to turn the infinite $\\sigma$-finite limit into finite measures with convergent moments, exponential-moment determinacy to identify limits, and the one-arm bound for tightness; a separate pioneer-point regularity argument over annuli produces the lower mass bound, and a deterministic support-convergence lemma (weak convergence plus uniform lower mass bound implies Hausdorff convergence of supports) closes the geometric step.","core_discovery":"On the paper's own terms, the discovery is that moment-level convergence of connection functions can be promoted to law-level convergence of the cluster and its geometry. With $\\mu_n=(an^4)^{-1}\\sum_{x\\in\\mathcal C}\\delta_{x/n}$ and $\\nu_n=n^2\\mathbb{P}(\\mu_n\\in\\cdot)$, Theorem 1.1 states that for every $\\varepsilon>0$, $\\nu_n(\\cdot;\\mu_n(\\mathbb{R}^d)>\\varepsilon)$ converges weakly to $N_{0,\\lambda/a}(\\cdot;\\mu(\\mathbb{R}^d)>\\varepsilon)$, the canonical measure of a $(\\tfrac12\\Delta,\\lambda/a)$-super-Brownian motion restricted to total mass above $\\varepsilon$. Theorem 1.2 states a uniform lower mass bound: conditionally on the cluster reaching scale $R$, every ball of radius $\\delta R$ that the cluster meets contains at least order $R^4$ sites, up to an arbitrarily small failure probability as $\\delta\\to0$. Combining these, Theorem 1.3 gives joint convergence of $(\\mu_n,n^{-1}\\mathcal C)$ to $(\\mu,\\operatorname{supp}\\mu)$ under the conditioned canonical measure, and Theorem 1.4 identifies $\\lim_r r^2\\mathbb{P}(0\\leftrightarrow\\partial B_r)=\\theta_1=(a/\\lambda)N_{0,1}(\\mathcal R\\not\\subset B_1)\\in(0,\\infty)$.","pith_inferences":["If the expected extension of the $k$-point convergence to spread-out models in $d>6$ materializes, the same proof would give measure, range, and one-arm convergence there without modification, since the argument is hypothesis-driven.","The constant $\\theta_1$ is characterized by the radial boundary blow-up problem $\\tfrac12 v'' + \\tfrac{d-1}{2r}v' = \\tfrac{\\gamma}{2}v^2$ with $v\\to\\infty$ at $r=1$; numerically solving this ODE would yield a quantitative prediction that lattice simulations could test.","The lower mass bound supplies exactly the volume-growth condition demanded by resistance-form scaling-limit criteria, so a random-walk ('ant in the labyrinth') scaling limit on the high-dimensional critical cluster is a plausible next application.","The size-biasing and un-biasing template may transfer to other $\\sigma$-finite scaling limits where only moment convergence is known."],"forward_implications":["The one-arm exponent is exactly $2$ with an identified constant: $r^2\\mathbb{P}(0\\leftrightarrow\\partial B_r)\\to\\theta_1$, so the cluster's reach has the same tail as the super-Brownian range.","Conditioned on carrying macroscopic mass, the rescaled cluster converges as a compact set, so macroscopic geometric quantities that are continuous functions of the range—such as diameter—converge to the corresponding super-Brownian quantities.","The uniform lower mass bound holds under the two-point bound alone for $d>6$, giving a ready-made no-thin-region input for other scaling-limit programs on critical structures.","The measure convergence recovers the known cluster-size tail $\\mathbb{P}(|\\mathcal C|\\ge N)\\asymp N^{-1/2}$ with the explicit constant forced by the normalization, confirming consistency of the constants $a,\\lambda$.","Because Theorem 1.1 factors through the two-point bound and the $k$-point convergence, any lattice model verified to satisfy those two hypotheses inherits the same three conclusions."],"supporting_citations":[{"why":"Supplies the rescaled $k$-point convergence hypothesis (1.3) that the paper upgrades from moments to laws.","marker":"[21]"},{"why":"Supplies the one-arm upper bound used for tightness, far-field control, and diameter tightness.","marker":"[50]"},{"why":"Provides the tree-graph inequality that turns lattice connection functions into tree-integral sums for the domination bound.","marker":"[3]"},{"why":"Establishes the two-point asymptotic for nearest-neighbour percolation in $d\\ge 11$, verifying the two-point bound (1.2) for the model.","marker":"[33]"},{"why":"Provides the two-point function and convolution estimates for spread-out models used in the probability bounds.","marker":"[36]"},{"why":"Supplies the restricted-cluster two-point estimate used in the second-moment volume arguments.","marker":"[22]"},{"why":"Provides the pioneer-point regularity framework and local good-event estimates that make the lower mass bound quantitative.","marker":"[9]"},{"why":"Gives the graphical superprocess moment formula from which the occupation-moment tree integrals are derived.","marker":"[1]"},{"why":"Provides the canonical measure, snake representation, and range properties used for the super-Brownian side.","marker":"[54]"},{"why":"Supplies the deterministic principle that weak convergence plus a uniform lower mass bound gives Hausdorff convergence of supports.","marker":"[10]"}],"fun_headline_variants":["Sharp one-arm law from percolation-to-SBM convergence","Range convergence yields sharp one-arm exponent","Cluster geometry limit fixes percolation one-arm constant","High-dim percolation: sharp r^2 one-arm asymptotics","Super-Brownian range emerges from critical percolation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusions rest on two load-bearing inputs from outside this paper: the uniform two-point bound (1.2) and the $k$-point convergence (1.3); if either failed at the needed uniformity, the measure convergence, range convergence, and one-arm constant would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Sharp one-arm law from percolation-to-SBM convergence","Range convergence yields sharp one-arm exponent","Cluster geometry limit fixes percolation one-arm constant","High-dim percolation: sharp r^2 one-arm asymptotics","Super-Brownian range emerges from critical percolation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000928,"raw_usage":{"total_tokens":3997,"prompt_tokens":988,"completion_tokens":3009,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":2939}},"tokens_in":604,"tokens_out":3009,"duration_ms":21374,"temperature":1.0,"reasoning_tokens":2939,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:26:25.160798+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run high-precision simulations of critical bond percolation on $\\mathbb{Z}^{11}$ and record $r^2\\mathbb{P}(0\\leftrightarrow\\partial B_r)$ for $r$ a sequence of dyadic scales; if the sequence does not converge to a positive finite constant, or if the constant disagrees with $\\theta_1$ evaluated numerically from the radial PDE $\\tfrac12 v''+\\tfrac{d-1}{2r}v'=\\tfrac{\\lambda}{2a}v^2$ with $v\\to\\infty$ at $1$, the central claim fails. Equivalently, simulate the conditioned cluster $\\{|\\mathcal C|\\ge R^4\\}$ and test the lower mass bound: if with positive probability some ball of radius $\\delta R$ met by the cluster contains $o(R^4)$ sites as $R\\to\\infty$ at fixed $\\delta$, Theorem 1.2 and the range convergence collapse.","supporting_citations":[{"cited_title":"Kozma and A","cited_arxiv_id":null,"evidence_quote":"Supplies the one-arm upper bound used for tightness, far-field control, and diameter tightness."},{"cited_title":"Aizenman and C","cited_arxiv_id":null,"evidence_quote":"Provides the tree-graph inequality that turns lattice connection functions into tree-integral sums for the domination bound."},{"cited_title":"Fitzner and R","cited_arxiv_id":null,"evidence_quote":"Establishes the two-point asymptotic for nearest-neighbour percolation in $d\\ge 11$, verifying the two-point bound (1.2) for the model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the two-point function and convolution estimates for spread-out models used in the probability bounds."},{"cited_title":"Chatterjee and J","cited_arxiv_id":null,"evidence_quote":"Supplies the restricted-cluster two-point estimate used in the second-moment volume arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the graphical superprocess moment formula from which the occupation-moment tree integrals are derived."},{"cited_title":"Le Gall.Spatial Branching Processes, Random Snakes and Partial Differential Equa- tions","cited_arxiv_id":null,"evidence_quote":"Provides the canonical measure, snake representation, and range properties used for the super-Brownian side."},{"cited_title":"Athreya, W","cited_arxiv_id":null,"evidence_quote":"Supplies the deterministic principle that weak convergence plus a uniform lower mass bound gives Hausdorff convergence of supports."}],"review_version":2}