{"id":"74a10129-2a43-4201-9710-73248b55d370","arxiv_id":"2508.18809","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For long-range percolation with d=3α<6, the critical volume tail is ~(log n)^{1/4}/√n, the critical two-point function is ~||x-y||^{-d+α}, and superprocess scaling limits hold with explicit logarithmic corrections.","lead":"Long-range percolation at its upper critical dimension d=3α<6 is analyzed rigorously: the critical cluster tail is C (log n)^{1/4}/√n, with superprocess scaling limits after slowly varying corrections. This closes the critical-dimensional long-range regime in a three-paper theory and provides a non-perturbative RG benchmark for percolation at a critical dimension.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Log-correction exponent depends on unverified log-integrability of the o(1) errors in Theorem III.6.6 / Lemma III.6.11; if those errors are only o(1) and not log-integrable, the stated (log n)^{1/4} tail can acquire an extra slowly-varying factor.","rationale":"The central claim is conditional on two large bodies of proof: Theorem III.1.11 (hydrodynamic condition) and Theorem III.6.6 (second-order scaling limits). The reader's weakest-assumption analysis correctly identified Theorem III.6.6 as the more fragile of the two for the specific logarithmic exponent. I agree: Theorem III.1.11 establishes only superprocess scaling limits with unspecified slowly varying corrections; the (log n)^{1/4} exponent comes from the second-order computation. Within that computation, the critical step is the factorization of triple-interaction terms in Lemma III.6.11 and the subsequent identification of the two vertex factors in Theorem III.6.6. The paper is honest about the delicacy: it calls the equality of V_r and \\tilde V_r a 'miracle' and defers the proof of Lemma III.6.11 to Section III.6.3. My concern is not that the argument is circular or that constants are fitted; rather, the displayed text does not demonstrate the log-integrability of the remainder terms that is needed for Lemma III.6.3 to output a pure power of log r. If the remainder is only o(1) in the pointwise sense, the ODE can produce extra slowly varying factors that would change the theorem's statement. This is a correctness risk, not a violation of consensus. The reader's verdict of CONDITIONAL is therefore appropriate, and my analysis does not move it: an independent check of Theorem III.6.6 and of the error propagation into Proposition III.6.2 is what would convert it to ACCEPT. I see no evidence of fabrication, circularity, or fitted constants, and the paper gives substantial independent structure: the log corrections match the hierarchical calculation of the author's earlier work, the constant C has an explicit diagrammatic expression, and the derivation is parameter-free modulo the hydrodynamic condition. The editorial abstract/body mismatch about 'second' versus 'third' paper is irrelevant to the mathematics.","tokens_in":77620,"tokens_out":8874,"duration_ms":94130,"concrete_test":"Perform an independent derivation of Lemma III.6.11 for the minimal moments used in Theorem III.6.6 (p1=0, p2=p3=1 and p1=0, p2=1, p3=2, with P ≡ 1) by expanding the triple-interaction sums in Lemma III.6.10 into connected tree diagrams and comparing each diagram with the pair-interaction factorization. Compute the decay rate of the error term explicitly: if the error after averaging over B_r is o(1) but not O(1/log r), then solve the ODE in Lemma III.6.3 with delta_r = 1/log log r and check whether the volume-tail formula still has the form const (log n)^{1/4}/sqrt n. Alternatively, numerically integrate the exact hierarchical analogue of the ODE system (III.6.13)-(III.6.14) at d = 3 alpha, using the known kernel kappa and the recurrence (III.6.24), and check that E_{beta_c,r}|K|^2 / (r^{3 alpha}(log r)^{-1/2}) converges to a positive constant with no residual (log log r) factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline (log n)^{1/4} is obtained by feeding Proposition III.6.2 into Lemma III.6.3. Proposition III.6.2 is an exact consequence of Theorem III.6.6 only if the o(1) remainders in (III.6.13)-(III.6.14), after summing over y in B_r, can be converted into a single logarithmically integrable error delta_r in (III.6.20). The text asserts this conversion ('for some logarithmically integrable error function delta_r'), but the displayed argument in Sections III.6.1-III.6.2 does not establish it: Theorem III.6.6 is stated with pointwise o(1), and Lemma III.6.11, whose proof is deferred to Section III.6.3, gives only an error of size o(|B_r| V^{...} r^{deg(P)} (E_r|K|)^{...}). An o(1) that decays like, say, 1/log log r is not log-integrable after division by r, and would change the solution of the ODE in Lemma III.6.3 from C r^{3 alpha} (log r)^{-1/2} to r^{3 alpha} (log r)^{-1/2} (log log r)^c, destroying the constant-prefactor form of Theorem III.1.2. Moreover, the 'miracle' V_r ~ \\tilde V_r is exactly what makes the two vertex factors cancel in the difference E_{2,r}-E_{1,r}; if the factorization in Lemma III.6.11 misses any connected diagram of the same order, the coefficient C = 2 int_B kappa^{*4} changes, and if C changes sign or vanishes, the whole second-order flow changes. Since the proof of Lemma III.6.11 is not visible in the manuscript and is explicitly described as the key extra ingredient, this is the load-bearing soft spot.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes long-range percolation on Z^d with kernel ~ ||x-y||^{-d-α} at the upper critical dimension d = 3α < 6. It proves the hydrodynamic condition (Theorem III.1.11), then invokes the RG framework of Paper I to obtain superprocess scaling limits with slowly varying corrections (Theorem III.1.6), and finally computes second-order corrections to the RG flow, yielding the volume-tail asymptotics P_{β_c}(|K|≥n) ~ C (log n)^{1/4}/√n and matching lower/upper bounds for the two- and three-point functions (Theorems III.1.2 and III.1.8). The proof of the hydrodynamic condition proceeds by contradiction under a fictitious failure; the computation of logarithmic corrections is based on diagrammatic second-order asymptotics for the error terms D^{(1)}, D^{(2)}, including a claimed asymptotic equality of two vertex factors V_r and \\tilde V_r.","tokens_in":78036,"tokens_out":7055,"duration_ms":73204,"significance":"If the second-order computation is correct, this is a major rigorous advance: it gives the first non-perturbative determination of critical exponents and logarithmic corrections at the upper critical dimension for a long-range percolation model, including a superprocess scaling limit with explicitly identified slowly varying factors. The predictions match hierarchical percolation and differ from the conjectured behaviour of nearest-neighbour percolation on Z^6, so the paper has real conceptual content. The manuscript is also commendably explicit about its limitations: Theorem III.1.11 is openly described as ineffective, the dependence on the deferred Lemma III.6.11 is flagged, and the hypotheses of the main theorems are stated precisely. These strengths make the central strategy coherent, but the decisive second-order step is not fully verified in the present version.","major_comments":[{"comment":"This is the load-bearing step. Theorem III.6.6 is stated with unquantified o(1) remainders in (III.6.13)–(III.6.14), and the proof of Proposition III.6.2 then asserts the existence of a logarithmically integrable error function δ_r in (III.6.20). That conversion is essential: Lemma III.6.3 requires δ_r to be logarithmically integrable in order to force the solution f(r) ~ (aγC log r)^{-1/γ} r^a. A remainder that is merely o(1), e.g. δ_r ~ 1/log log r, is not logarithmically integrable and would change the solution to r^{3α}(log r)^{-1/2}(log log r)^c, destroying the constant-prefactor form of Theorem III.1.2. The displayed argument does not establish log-integrability; pointwise o(1) convergence does not imply it. The authors must either prove a quantitative version of Theorem III.6.6 with log-integrable error bounds, or supply a separate argument showing that the errors in (III.6.15)–(I","section":"§III.6.1–III.6.2, Eqs. (III.6.13)–(III.6.20)"},{"comment":"Lemma III.6.11 is explicitly described as the key extra ingredient for Theorem III.6.6, and its proof is deferred to §III.6.3. In the version supplied for refereeing, that proof is not present, so the central second-order computation is incomplete as written. Moreover, the error term stated in the lemma is only o(|B_r| V^{...} r^{deg(P)} (E_r|K|)^{...}), which by itself does not provide the log-integrability required for Proposition III.6.2. Even if the deferred proof establishes the claimed equality, it may not yield a remainder that is summable against ds/s. This needs to be addressed directly: either the proof of Lemma III.6.11 is included and its remainder is shown to be log-integrable, or a different argument must be given.","section":"§III.6.3, Lemma III.6.11"},{"comment":"The inference from polynomial-moment convergence to the ball integrals in (III.6.15)–(III.6.16) is not justified in detail. Theorem III.6.6 gives asymptotics for ∑_y D^{(i)}_r(0,y) P(y/r) for polynomials P; passing to P = 1_{B} via Carleman's criterion requires a tightness or uniform-integrability argument for the normalized signed measures. Lemma III.6.9 gives bounds of the correct order, but the manuscript does not spell out why the o(1) errors in the polynomial moments survive passage to the discontinuous indicator 1_B with any uniformity. Without such an argument, the rate at which E_{1,r} and E_{2,r} approach their limits is uncontrolled, which is exactly the same log-integrability problem raised above.","section":"§III.6.2, passage from Theorem III.6.6 to sums over B_r"}],"minor_comments":[{"comment":"The word 'edian' in the definition of M_r appears to be a typo for 'median'.","section":"§III.1.2"},{"comment":"There are several typos: 'fictirious' should be 'fictitious', 'mininum' should be 'minimum', and 'important important' appears duplicated.","section":"§III.3 (overview) and §III.4.1"},{"comment":"In the proof of Lemma III.3.4, the text refers to 'K_λ as in the proof of Lemma III.3.4'; the set is actually defined during the proof of Lemma III.3.11. This cross-reference should be corrected.","section":"§III.3.1"},{"comment":"The proof of Proposition III.6.2 introduces several error quantities (E_{0,r}, E_{1,r}, E_{2,r}, \\bar E_{1,r}, \\tilde H_r) in quick succession. A short summary table or labels would improve readability.","section":"§III.6.2"}],"recommendation":"major_revision","confidential_remarks":"The decisive question for acceptance is the content of §III.6.3 and whether the remainders in Theorem III.6.6 can be made logarithmically integrable. If the full submission contains a quantitative proof of Lemma III.6.11 and a justification of the δ_r conversion, the present objection is answerable; otherwise the (log n)^{1/4} exponent is not established. The overall architecture is coherent and the paper is potentially very important, but the missing log-integrability control is load-bearing and cannot be waved through on the current text."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious paper. It proves the hydrodynamic condition at d=3α and, on that basis, derives the exact (log n)^{1/4} volume-tail correction, the two- and three-point estimates with the sqrt(1/log) factor, and a superprocess scaling limit with explicit normalization ζ(r) ~ r^{2α}(log r)^{-1/2}. The upper-critical-dimension result for long-range percolation, with corrections matching the hierarchical model and differing from Z^6, is a real step forward. The paper is also honest about its limits: Remark III.1.12 states plainly that the hydrodynamic-condition proof is ineffective, and the second-order computation is flagged as the delicate part. The citation pattern is structural, not circular: paper III proves the hydrodynamic condition that paper I’s theorems require. No fabricated constants, no fitted exponents; the new material is identifiable and looks substantial. The main soft spot is exactly the one the stress-test note identifies. The (log n)^{1/4} exponent depends on Proposition III.6.2, and the proof of that proposition requires converting o(1) remainders, after summing over y in B_r, into a logarithmically integrable error δ_r. The displayed argument in III.6.1–III.6.2 does not fully establish that conversion: Theorem III.6.6 is stated with pointwise o(1), and Lemma III.6.11, whose proof is deferred to III.6.3, gives an error bound that is not obviously log-integrable after dividing by r. If that error decays too slowly, the exponent changes by a slowly-varying factor. This is load-bearing, not cosmetic. The ‘miracle’ V_r ~ \\tilde V_r is also exactly what cancels the two vertex factors in E_{2,r} − E_{1,r}; if the factorization in Lemma III.6.11 misses a diagram of the same order, the constant C in Proposition III.6.2 changes. The paper itself says Lemma III.6.11 is the key extra ingredient—good—but it means the referee must go through Section III.6.3 with care, and the version I have truncates before that section. Minor: the abstract says ‘second of three papers’ while the introduction says ‘third of a series’; clearly editorial. Who is this for? Probabilists and mathematical-physics readers who want the long-range phase diagram settled on the line d=3α. It deserves a serious referee. The referee should be asked to verify the log-integrability step and the triple-interaction lemma in detail. Even if that part needs repair, the hydrodynamic-condition theorem alone is worth publishing. I would bring it to reading group and would cite it. Send it to peer review.","headline":"Genuine major step—hydrodynamic condition at d=3α plus exact log corrections—but the headline (log n)^{1/4} rests on a deferred second-order error analysis that needs referee scrutiny.","tokens_in":733,"tokens_out":788,"would_cite":true,"duration_ms":37751,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B43","82B27","82B28"],"pacs":[],"model":"deepseek-v4-flash","headline":"At d=3α, long-range percolation has volume-tail correction (log n)^{1/4}/√n, and the paper proves the hydrodynamic condition that makes the computation rigorous.","keywords":["long-range percolation","upper critical dimension","logarithmic corrections","hydrodynamic condition","superprocess scaling limit","mean-field behaviour","volume tail","three-point function"],"falsifier":"Compute the ratio ̃V_r/V_r directly from the cut-off model at d=3α<6: the proof requires this ratio to tend to 1, and any nonzero limiting deviation would falsify Proposition III.6.2 and the (log n)^{1/4} tail. A complementary check is to measure the critical volume tail: if P_{β_c}(|K|≥n)√n/(log n)^{1/4} does not tend to a positive constant, the central asymptotic formula fails.","tokens_in":77457,"feed_emoji":"🎲","tokens_out":7091,"duration_ms":74117,"temperature":0.7,"pith_summary":"This paper proves that critical long-range percolation on Z^d at the upper critical dimension d=3α<6 behaves as a mean-field system with explicit logarithmic corrections. The central technical step is a proof of the hydrodynamic condition, which states that the largest cluster inside a block is asymptotically smaller than the universal upper bound; this unlocks a renormalization-group analysis developed in earlier papers. Using that analysis to second order, the paper computes the critical volume tail as const (log n)^{1/4}/√n and shows the critical cluster, scaled by r^{2α}(log r)^{-1/2}, converges to an integrated α-stable superprocess excursion. These logarithmic corrections match hierarchical long-range percolation and differ from the conjectured (log n)^{2/7} for nearest-neighbour percolation on Z^6, making the long-range model the first critical-dimensional percolation model with a rigorous, non-perturbative determination of its logarithmic corrections.","feed_headline":"Log correction to critical percolation tail is (log n)^{1/4}","feed_subtitle":"A new proof fixes the upper critical dimension d=3α and shows the cluster scales like an α-stable superprocess.","key_machinery":"The hydrodynamic condition is the central object: it says Mr = o(r^{(d+α)/2}), i.e. the largest cluster in a ball is much smaller than the universal tightness bound. This condition upgrades the moment equations of the first paper from conditional to unconditional statements. The second-order computation then hinges on two vertex factors: V_r = E|K|^2/(E|K|)^3, which measures how strongly the different arms of a cluster interact, and a second factor ̃V_r that arises in the error terms D^{(1)}_r and D^{(2)}_r describing interactions between two clusters. The paper proves the asymptotic equality V_r ∼ ̃V_r by writing moment ODEs for the error terms and solving them with the triple-interaction l","core_discovery":"The paper establishes that for d=3α the hydrodynamic condition holds: the edian Mr, the typical size of the largest cluster in a ball of radius r under the cut-off critical measure, is o(r^{(d+α)/2}). When d=3α<6 this makes the first paper's RG analysis unconditional, yielding the same superprocess scaling limits as in high dimension after slowly-varying corrections are included. The paper then analyzes the RG flow to second order and proves E_{β_c,r}|K|^2 ∼ (α/β_c) A r^{3α}/√(log r), from which it derives the critical volume tail P_{β_c}(|K|≥n) ∼ C (log n)^{1/4}/√n, the two-point estimate P_{β_c}(x↔y) ≍ ∥x−y∥^{−d+α}, and the three-point estimate with an explicit √(1/log d_min) correction. T","pith_inferences":["A direct numerical test of the paper's core identity is to compute V_r and ̃V_r from the cut-off model at d=3α<6: the proof needs their ratio to tend to 1, and any nonzero limiting deviation would break the second-order RG flow and the (log n)^{1/4} exponent.","Because the equality of V_r and ̃V_r is obtained by solving moment recurrences rather than by a model-specific argument, the same mechanism may apply on the other critical lines, such as d=6, α=2 or d=3α>6, once the hydrodynamic condition is available there.","The proof of the hydrodynamic condition is ineffective, leaving the rate of Mr open; the paper conjectures Mr ∼ const (log log r)/√(log r) r^{2α}, a prediction that could be checked numerically and would quantify how far the largest cluster sits above the typical large-cluster scale.","The explicit constant C = 2∫_B κ^{*4}(y) dy and the interpretation of the logarithmic-correction exponent as a left derivative of the mean-field exponent suggest a diagram-counting rule for logarithmic corrections at upper critical dimensions, which could be tested against the nearest-neighbour Z^6 prediction (log n)^{2/7}."],"forward_implications":["At d=3α<6 the critical volume tail is C (log n)^{1/4}/√n, the same logarithmic correction as hierarchical long-range percolation and not the (log n)^{2/7} conjectured for nearest-neighbour Z^6 percolation.","The critical cluster, scaled by ζ(r) ∼ const r^{2α}(log r)^{-1/2}, converges to the integrated symmetric α-stable superprocess excursion measure; the number of typical large clusters on scale r grows like const log r.","The two-point function has no logarithmic correction, P_{β_c}(x↔y) ≍ ∥x−y∥^{−d+α}, while the three-point function carries a √(1/log d_min) correction, showing that both the tree-graph and Gladkov bounds are off by a √(log) factor at the critical dimension.","Theorem III.1.11 also implies the correlation-length condition for effectively long-range critical behaviour, so the second paper's low-dimensional results, including the pointwise two-point estimate, apply along d=3α<6.","The hydrodynamic condition fails in low effective dimension and holds here, giving a sharp geometric distinction: at the critical dimension large clusters interact only very weakly, so mean-field ODEs hold with slowly varying corrections."],"supporting_citations":[{"why":"First paper of the series; supplies the RG analysis and superprocess scaling-limit theorems that the hydrodynamic condition unlocks.","marker":"[45]"},{"why":"Second paper of the series; its low-dimensional results, including the pointwise two-point estimate, are applied along d=3α<6.","marker":"[46]"},{"why":"Universal tightness theorem used to define and bound the edian Mr and to provide the initial O(r^{(d+α)/2}) bound.","marker":"[50]"},{"why":"Spatially averaged two-point upper bound used throughout the initial regularity estimates and the hydrodynamic-condition proof.","marker":"[51]"},{"why":"Hierarchical long-range percolation analysis that introduced the hydrodynamic condition and produced the logarithmic corrections the paper reproduces.","marker":"[52]"},{"why":"Source of the nearest-neighbour Z^6 prediction (log n)^{2/7} that the paper's (log n)^{1/4} result is explicitly compared against.","marker":"[33]"},{"why":"Tree-graph inequality used to bound moments and to identify where the three-point function refinements occur.","marker":"[4]"},{"why":"Gladkov inequality used in derivative bounds and lower bounds for the three-point function.","marker":"[34]"},{"why":"Provides the differential inequality relating the β-derivative to the second moment, used under the fictitious mean-field assumption.","marker":"[49]"},{"why":"Durrett-Nguyen inequality bounding the β-derivative by a geometric mean of the first and second moments.","marker":"[31]"}],"fun_headline_variants":["At d=3α, percolation tail gets (log n)^{1/4} correction","Critical cluster tail: (log n)^{1/4} at upper dimension","Superprocess limit at d=3α with log corrections proven","Percolation tail: (log n)^{1/4} correction at d=3α"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The computed (log n)^{1/4} correction rests on the asymptotic equality of the two vertex factors V_r and ̃V_r which track different cluster interactions; if that equality carries errors that are not logarithmically integrable, the logarithmic-correction exponent would change.","fun_headline_variants_meta":{"raw":{"variants":["At d=3α, percolation tail gets (log n)^{1/4} correction","Critical cluster tail: (log n)^{1/4} at upper dimension","Superprocess limit at d=3α with log corrections proven","Percolation tail: (log n)^{1/4} correction at d=3α"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1445,"prompt_tokens":1040,"completion_tokens":405,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":784,"completion_tokens_details":{"reasoning_tokens":325}},"tokens_in":784,"tokens_out":405,"duration_ms":4704,"temperature":1.0,"reasoning_tokens":325,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T16:11:08.593796+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the ratio ̃V_r/V_r directly from the cut-off model at d=3α<6: the proof requires this ratio to tend to 1, and any nonzero limiting deviation would falsify Proposition III.6.2 and the (log n)^{1/4} tail. A complementary check is to measure the critical volume tail: if P_{β_c}(|K|≥n)√n/(log n)^{1/4} does not tend to a positive constant, the central asymptotic formula fails.","supporting_citations":[{"cited_title":"Hutchcroft","cited_arxiv_id":null,"evidence_quote":"First paper of the series; supplies the RG analysis and superprocess scaling-limit theorems that the hydrodynamic condition unlocks."},{"cited_title":"Hutchcroft","cited_arxiv_id":null,"evidence_quote":"Second paper of the series; its low-dimensional results, including the pointwise two-point estimate, are applied along d=3α<6."},{"cited_title":"Hutchcroft","cited_arxiv_id":null,"evidence_quote":"Universal tightness theorem used to define and bound the edian Mr and to provide the initial O(r^{(d+α)/2}) bound."},{"cited_title":"Hutchcroft","cited_arxiv_id":null,"evidence_quote":"Spatially averaged two-point upper bound used throughout the initial regularity estimates and the hydrodynamic-condition proof."},{"cited_title":"Hutchcroft","cited_arxiv_id":null,"evidence_quote":"Hierarchical long-range percolation analysis that introduced the hydrodynamic condition and produced the logarithmic corrections the paper reproduces."},{"cited_title":"Essam, D","cited_arxiv_id":null,"evidence_quote":"Source of the nearest-neighbour Z^6 prediction (log n)^{2/7} that the paper's (log n)^{1/4} result is explicitly compared against."},{"cited_title":"Aizenman and C","cited_arxiv_id":null,"evidence_quote":"Tree-graph inequality used to bound moments and to identify where the three-point function refinements occur."},{"cited_title":"Hutchcroft","cited_arxiv_id":null,"evidence_quote":"Provides the differential inequality relating the β-derivative to the second moment, used under the fictitious mean-field assumption."},{"cited_title":"Durrett and B","cited_arxiv_id":null,"evidence_quote":"Durrett-Nguyen inequality bounding the β-derivative by a geometric mean of the first and second moments."}],"review_version":1}