{"id":"b9461369-3b44-4190-a6d3-e5a817e428f3","arxiv_id":"2607.18011","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Long-range percolation on polynomial-growth transitive graphs is local for α∈(0,2), and a new Voronoi-tile renormalization yields giant-cluster, cluster-decay, isoperimetric and transience results.","lead":"This paper proves that in long-range percolation on transitive graphs of polynomial growth, the critical threshold is continuous in the local topology of the graph and kernel, together with a family of sharp supercritical estimates. The engine is a new multiscale renormalization scheme that proves local existence and uniqueness of a linear-sized giant cluster.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniform integrability condition (1.6) is too weak: it does not imply the uniform tail decay used in Proposition 6.2, and without that, Theorem 1.1 is false as stated.","rationale":"The reader's weakest assumption was Proposition 4.2, an external structural statement about A-controlled nets. That is a reasonable point, but the proof of Theorem 1.1 has a more serious internal gap: the notion of uniform integrability in (1.6) is not the standard uniform-integrability/tail-decay condition, and the argument in Proposition 6.2 explicitly relies on the stronger property. The counterexample above satisfies the stated definition and local convergence yet violates the conclusion of Theorem 1.1, so the theorem is false as written. The paper's main technical machinery (the giant component, net construction, natural coupling) may be sound for the intended class of examples, and Lemma 3.6 shows the natural kernels satisfy the stronger tail condition. The central claim is therefore repairable, but only if the definition is strengthened and the proofs in Sections 6 are updated. This warrants a conditional rather than unconditional acceptance of the current preprint.","tokens_in":61162,"tokens_out":32507,"duration_ms":287644,"concrete_test":"Take G=Z^d, d≥2, α∈(1,2), and J(x,y)=d(x,y)^{-dα}. Fix c>1/β_c(G,J) and define, for each n, J_n(x,y)=J(x,y)+ c/#S(n) if d(x,y)=n and J_n=J otherwise. Verify that (i) sup_n ∑_{x≠o} J_n(o,x)≤‖J‖+c<∞, so (1.6) holds; (ii) for every fixed R, ∑_{B(R)}|J_n−J|=0 for n>2R, so local convergence holds; (iii) the long-edge graph has mean degree βc, so for β with 1/c<β<β_c(G,J), the origin percolates in (G,J_n) while it does not in (G,J). This directly contradicts Theorem 1.1. If the authors intend uniform tail decay, the definition should be revised to: for every ε>0 there exists R such that sup_n ∑_{d(o_n,x)≥R} J_n(o_n,x)<ε.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper defines (J_n) as uniformly integrable if sup_n ∑_{x≠o_n} J_n(o_n,x) ≤ M ((1.6)). But Proposition 6.2, which proves lower semi-continuity of β_c, needs a uniform tail estimate: it asserts 'By the uniform integrability (1.6) there exists R_* such that ∑_{y∈G_n\\B_n(R_*)} J_n(o,y) ≤ ε/(2β) for all n sufficiently large.' This does not follow from (1.6): a sequence of kernels can have uniformly bounded total mass while a positive amount of mass moves to distance n→∞. A concrete such sequence on Z^d, d≥2, α∈(1,2), is J_n(x,y)=J(x,y)+ c/#S(n) for d(x,y)=n, and J_n=J otherwise, with c>1/β_c(J). Then (1.6) holds with M=‖J‖+c, and (Z^d,J_n) converges locally to (Z^d,J) because the extra term is supported outside every fixed ball. However, the extra edges alone give each vertex a Poisson number of long edges with mean βc; for βc>1 this long-edge graph is supercritical. Choosing β with 1/c < β < β_c(G,J) yields θ(β,G,J)=0 but θ(β,G_n,J_n)>0 for all large n, so β_c(G_n,J_n) ≤ 1/c < β_c(G,J). Hence the conclusion of Theorem 1.1 fails for the stated definition of uniform integrability. The proof of Proposition 6.5 for upper semi-continuity of θ uses the same unjustified tail-decay step. The paper's own Lemma 3.6 proves the stronger, standard uniform-tail condition for the natural examples, but the definition in Section 1.2 is not that condition.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies long-range percolation on transitive graphs of polynomial growth. Its main results are: (1) locality of the critical value under the local topology for kernels decaying as distance^{-d α} with α∈(0,2), together with joint continuity of the percolation probability θ; (2) a suite of supercritical sharpness results — stretched-exponential finite-cluster decay, truncated one-arm bounds, anchored isoperimetric dimension, transience/recurrence of the infinite cluster, and smoothness of percolation characters — and (3) the underlying technical engine, a multiscale renormalisation scheme based on iteratively merged Voronoi tiles and scale-invariant nets, which yields local existence and uniqueness of a linear-sized giant component. The giant-component theorems (Theorems 1.10 and 1.11) are proved in the text with detailed estimates, and most of the corollaries are derived from them. The paper is written in a clear and structured way, and the sphere calculus of Section 3 is a genuinely useful technical contribution.","tokens_in":61657,"tokens_out":9395,"duration_ms":83378,"significance":"If the main theorems are correct, this is a substantial advance: it gives the long-range analogue of Schramm's locality conjecture in a broad class of graphs, answers a special case of a question of Nekrashevych and Pete, and provides new proofs of transience and recurrence for long-range percolation on general transitive graphs of polynomial growth. The paper's explicit multi-scale renormalisation, with uniform constants and quantitative error bounds, is a methodological strength. The dependence on the companion paper [MAK26b] for Corollary 1.7 and Theorem 1.8 is a significant caveat, and the treatment of the uniform-integrability hypothesis contains a load-bearing gap. With those two points addressed, the paper would be a strong contribution.","major_comments":[{"comment":"The assumption (1.6) is too weak for the proof: it bounds the total mass but not the tail. Proposition 6.2 asserts: \"By the uniform integrability (1.6) there exists R_* such that Σ_{y∉B_n(R_*)}J_n(o,y) ≤ ε/(2β)\" — this is exactly the missing implication. Example on Z^d, α∈(1,2): take J_n=J + c/#S(n) on pairs at distance n. Then (1.6) holds and (G_n,J_n)→(Z^d,J) locally, but for c>1/β_c(J) and 1/c<β<β_c(J) the extra edges make the model supercritical, giving β_c(J_n)≤1/c<β_c(J), so Theorem 1.1 is false as stated. The same unjustified tail step appears in Proposition 6.5. Fix: replace (1.6) by the uniform tail condition sup_n Σ_{d(o_n,y)>R}J_n(o_n,y)→0 as R→∞ (or add hypotheses implying it), and adjust Theorems 1.1 and 1.4 accordingly.","section":"§1.2, Eq. (1.6); Prop. 6.2, Eq. (6.9); Prop. 6.5, Eq. (6.23)"},{"comment":"These results are not proved in this manuscript. Corollary 1.7 is dismissed with one line, and Theorem 1.8 imports [MAK26b, Theorems 1.5 and 1.8]. These are load-bearing for the anchored-isoperimetric-dimension claims and for the proof of Theorem 1.14 in the range α∈(1,1+1/d) via Corollary 1.7. In a journal submission, the relevant companion results should either be proved, or the corollaries should be explicitly stated as conditional on an available companion paper. The present text does not give the reader enough to verify this part of the programme.","section":"§12.2, Cor. 1.7 and Thm. 1.8"}],"minor_comments":[{"comment":"The d=1 case is handled by a terse sentence: \"the bound in (5.5) is enough to initialise the induction in the proof of Proposition 9.1.\" Since Theorem 1.3 for d=1 is already covered by known results (Berger; Hutchcroft), a citation would be clearer; otherwise the induction for d=1 should be spelled out.","section":"§5, proof of Thm. 1.3"},{"comment":"Minor typos: \"strcuture theory\" in §1.5; \"calclus\" in the proof of Lemma 3.5. These do not affect the mathematics.","section":"§1.5 and §3"},{"comment":"The exponent notation \\(\\log(k)^{1(\\alpha=1+1/d)}\\) is hard to parse; please write it with an explicit indicator function, e.g. \\(\\log(k)\\cdot \\mathbf{1}_{\\alpha=1+1/d}\\).","section":"Thm. 1.6"},{"comment":"In the final paragraph of the proof of Theorem 1.1, \"uniformly decaying\" should presumably read \"uniformly integrable\".","section":"§6.1, end of proof of Thm. 1.1"}],"recommendation":"major_revision","confidential_remarks":"The main technical gap is the uniform-integrability issue in Propositions 6.2 and 6.5. I believe it is fixable by replacing (1.6) with the standard uniform tail condition, which the authors' own Lemma 3.6 supplies for the natural examples; if the authors insist on keeping (1.6), the theorems appear to be false. Please also ensure that the companion paper [MAK26b] is publicly available and that the imported theorems are stable, since Corollary 1.7 and Theorem 1.8 depend on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The stress-test note is right, and it lands on the central claim. The paper defines uniform integrability in (1.6) as a uniform bound on the total kernel mass, but Proposition 6.2 needs uniform vanishing tail mass: the step where they assert the existence of R_* such that the tail sum is at most epsilon/(2β) does not follow from (1.6). The counterexample is valid: add c/#S(n) to the kernel at distance n. The total mass stays bounded, local convergence holds, and for β with 1/c < β < β_c(G,J) the approximating models have an infinite cluster while the limit does not. So Theorem 1.1 and also Theorem 1.4 fail under the stated hypotheses. Lemma 3.6 proves the stronger uniform tail condition for the natural examples, so the repair is reasonably clear: redefine uniform integrability as uniform tail decay, or add that condition explicitly to the theorems. But as written, the headline results are false, not just under-proved.\n\nThat said, the paper is not a throwaway. The renormalization scheme built from iteratively merged Voronoi tiles and scale-invariant nets is a genuine technical contribution, and the sphere calculus is a nice way around the missing sharp sphere-volume bounds. The giant existence and uniqueness theorems, Theorems 1.10 and 1.11, are proved in detail and look internally consistent; the consequences—cluster-size decay, the law of large numbers, transience/recurrence—are natural and mostly well-sourced. The d=1 continuity argument is indeed only a sketch, and Corollaries 1.7 and 1.8 lean on the companion paper [MAK26b], but those are secondary concerns next to the uniform-integrability issue.\n\nMy bottom line: this deserves a serious referee, but not in its current form. The authors need to fix the definition and re-check every place the tail condition is used. Once the definition is corrected, the main arguments have a good chance of going through.","headline":"The paper's main locality theorem is false as stated because their uniform integrability is too weak, but the flaw is a fixable definition gap and the rest of the package is substantial.","tokens_in":62137,"tokens_out":3163,"would_cite":false,"duration_ms":33849,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B43","20F65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The critical threshold of long-range percolation is a local quantity on transitive graphs of polynomial growth of dimension at least two, for connection kernels decaying like distance^{-dα} with 0<α<2.","keywords":["long-range percolation","locality","critical parameter","giant component","renormalization","transitive graphs of polynomial growth","supercritical sharpness","percolation probability"],"falsifier":"Find β>β_c(G,J) on some transitive polynomial-growth graph with d≥2, and for every radius R a finite ball event: a linear-sized cluster exists in B(R) and any two large sets inside B(10AR) at distance ≥2R are connected with probability ≥1-δ, yet there is no infinite cluster at parameter β. The paper's finite-size criterion says such a configuration is impossible; exhibiting one would falsify the renormalization core.","tokens_in":61056,"feed_emoji":"🔗","tokens_out":6056,"duration_ms":52324,"temperature":0.7,"pith_summary":"This paper proves the long-range analogue of the locality conjecture for Bernoulli percolation: for long-range percolation on transitive graphs of polynomial growth with dimension at least two and connection kernel decaying as distance^{-dα} with 0<α<2, the critical percolation parameter β_c is determined by the local geometry and the local kernel. If two approximating graphs and kernels agree on larger and larger balls, their thresholds converge to the threshold of the limit. The paper also proves that the percolation probability θ is jointly continuous in β, the graph, and the kernel, including at criticality, and derives a batch of sharp supercritical results: continuity of the phase transition, stretched-exponential decay of finite clusters, truncation of long edges, anchored isoperimetric dimension, transience and recurrence of the infinite cluster, and smoothness of percolation characters. The engine is a new multiscale renormalization scheme proving local existence and uniqueness of a linear-sized giant cluster, with a law of large numbers as a corollary.","feed_headline":"Long-range percolation thresholds are local","feed_subtitle":"Graphs that look alike on large balls have nearly identical percolation thresholds, a new proof shows.","key_machinery":"The central mechanism is a multi-scale renormalization scheme built on nets and iteratively merged Voronoi tiles. A net is a subset of vertices that is separated and dense at a given scale; taking nets at increasing scales and redefining each higher-level tile as the union of lower-level tiles whose centers lie in a Voronoi cell produces a nested partition with controlled volume fluctuations. The workhorse estimate is the bulk-to-bulk connection probability: two sets of size about r^d at distance about r are connected with probability 1 - exp(-Θ(r^{d(2-α)})), which is overwhelming when α<2. Combining this with the nested-tile coarse-graining yields local existence and uniqueness of the giant","core_discovery":"On its own terms, the paper establishes that for α∈(0,2), long-range percolation on a transitive graph of polynomial growth with dimension d≥2 is a local model: whenever (G_n,J_n) converges to (G,J) in the sense that balls of radius R are isomorphic and kernels converge in L^1 on those balls, then β_c(G_n,J_n)→β_c(G,J), provided the kernels are uniformly integrable when α>1 (for α≤1 the threshold is automatically 0). It goes further: the percolation probability θ(β,G,J) varies continuously with β, the graph, and the kernel for every β≥0. The structural reason is that the supercritical giant is locally detectable: in every large ball there is, with overwhelming probability, a unique cluster o","pith_inferences":["Editorial inference: the finite-size criterion suggests a practical numerical method: estimate β_c by finding the smallest β at which a ball of radius R develops a linear-sized cluster and bulk-to-bulk connections; the theorem gives explicit error bounds in R.","Editorial inference: the same nested-tile renormalization should apply to other long-range spatial random graph models on polynomial-growth spaces, such as spread-out percolation or random connection models, whenever α<2 ensures bulk-to-bulk connections.","Editorial inference: the one-dimensional failure of locality highlights that the dimension assumption is not technical: for α=2 on the line the phase transition is discontinuous, so locality cannot hold; hence the d≥2 and α<2 regime is a natural boundary."],"forward_implications":["Approximate thresholds from local data: if a graph and kernel match the limit on balls of radius R, then β_c is within a controllable error of β_c(G,J), making finite-box simulations rigorous.","The phase transition is continuous: θ(β_c)=0 for this class, so the percolation density does not jump at criticality.","Truncation: for every β>β_c, some finite-range truncation of the kernel is still supercritical; long edges are not load-bearing for the existence of percolation.","Finite clusters have stretched-exponential tails: P(k ≤ |K|<∞) ≤ exp(-c k^{min(2-α,1)}), giving quantitative control on the supercritical phase.","The infinite cluster has anchored isoperimetric dimension max(1/(α-1), d) for α∈(1,2), and is transient for α∈(1,2) but recurrent when d=2 and α≥2."],"fun_headline_variants":["Percolation thresholds are local—even for long-range","Locality extends to long-range percolation thresholds","For percolation, local similarity means same threshold","Long-range percolation: the threshold sees only locally","Local giants prove percolation thresholds are local"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The weak point is the structural input, imported from earlier work, that every graph in the approximating sequence—not just the limit—admits nets, uniform in scale, whose net graph contains a two-dimensional grid-like subgraph; the locality proofs and the finite-size criterion collapse if this uniformity fails.","fun_headline_variants_meta":{"raw":{"variants":["Percolation thresholds are local—even for long-range","Locality extends to long-range percolation thresholds","For percolation, local similarity means same threshold","Long-range percolation: the threshold sees only locally","Local giants prove percolation thresholds are local"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00036,"raw_usage":{"total_tokens":1819,"prompt_tokens":813,"completion_tokens":1006,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":939}},"tokens_in":557,"tokens_out":1006,"duration_ms":9137,"temperature":1.0,"reasoning_tokens":939,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T16:22:49.760507+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find β>β_c(G,J) on some transitive polynomial-growth graph with d≥2, and for every radius R a finite ball event: a linear-sized cluster exists in B(R) and any two large sets inside B(10AR) at distance ≥2R are connected with probability ≥1-δ, yet there is no infinite cluster at parameter β. The paper's finite-size criterion says such a configuration is impossible; exhibiting one would falsify the renormalization core.","supporting_citations":[],"review_version":1}