REVIEW 2 major objections 4 minor 12 references
From local giants to locality in long-range percolation
T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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
What would settle it
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.
Extended reading notes
Core claim
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
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [§1.2, Eq. (1.6); Prop. 6.2, Eq. (6.9); Prop. 6.5, Eq. (6.23)] 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.
- [§12.2, Cor. 1.7 and Thm. 1.8] 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.
minor comments (4)
- [§5, proof of Thm. 1.3] 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.
- [§1.5 and §3] Minor typos: "strcuture theory" in §1.5; "calclus" in the proof of Lemma 3.5. These do not affect the mathematics.
- [Thm. 1.6] The exponent notation \(\log(k)^{1(\alpha=1+1/d)}\) is hard to parse; please write it with an explicit indicator function, e.g. \(\log(k)\cdot \mathbf{1}_{\alpha=1+1/d}\).
- [§6.1, end of proof of Thm. 1.1] In the final paragraph of the proof of Theorem 1.1, "uniformly decaying" should presumably read "uniformly integrable".
Circularity Check
No circularity: the central locality and giant theorems are derived self-containedly from external structure theory and the paper's own renormalisation scheme.
full rationale
The paper's central derivation chain is not circular. Theorem 1.1 is proved from Proposition 6.2 (lower semi-continuity of β_c) and Proposition 6.4 (upper semi-continuity), neither of which reduces to the theorem's conclusion by construction: Proposition 6.2 uses the Duminil-Copin–Tassion φ_S characterization of β_c, the definition of local convergence, and uniform integrability; Proposition 6.4 uses the paper's own Theorem 1.10, the finite-size criterion Proposition 4.3, and the external structural input Proposition 4.2 quoted from [CMT23, Proposition 1.4]. The giant existence and uniqueness theorems (Theorems 1.10 and 1.11) are obtained through an explicit multi-scale renormalization scheme whose base case is ergodicity and uniqueness of the infinite cluster and whose inductive step uses bulk-to-bulk estimates such as (1.18); no fitted parameter is renamed as a prediction. Theorems 1.2–1.6, 1.9, and 1.14 are derived from Theorems 1.10/1.11, Proposition 4.3, and standard external results, so their inputs are not equivalent to their outputs by definition. The few same-author citations ([MAK26a], [MAK26b]) support ancillary statements such as Corollary 1.7 and Theorem 1.8, and those statements are not used as load-bearing inputs for the central locality or continuity theorems; the manuscript does not assert that [MAK26b] derives from the present renormalisation scheme, so no reduction to a self-citation chain can be exhibited from the text. The proof of Proposition 6.2 does contain a potentially unjustified step: the text moves from the uniform integrability bound (1.6) to a uniform tail estimate 'there exists R_* such that ∑_{y∈G_n\B_n(R_*)} J_n(o,y) ≤ ε/(2β)', which is a correctness gap rather than a circular equivalence between the theorem's assumptions and conclusion. Accordingly, under the stated hard rules, no circular step is identified and the circularity score is 0.
Assumptions & free parameters
free parameters (1)
- Renormalisation tuning constants (ξ, δ, s, r0, r̄0, M, ν, A)
assumptions (10)
- standard math Gromov–Trofimov–Bass–Guivarc'h polynomial volume bounds #B(r) ≍ r^d
- standard math Finitary structure theory of transitive graphs of polynomial growth, packaged as Proposition 2.1 and Proposition 4.2 ([CMT23, Prop 1.4], [TT24])
- standard math Uniqueness of the infinite cluster for transitive graphs of subexponential growth [GKN92]
- standard math Duminil-Copin–Tassion characterization β_c = sup{β: φ_β(S)<1}
- standard math Ergodicity of the long-range percolation measure and Følner property of metric balls
- standard math O'Connell large deviations for the giant in Erdős–Rényi random graphs [O'C98]
- standard math Thomassen transience criterion, finite-energy flow criterion of Lyons, and Nash-Williams recurrence criterion
- domain assumption Kernel assumption J(x,y)=Ω/Θ(d_G(x,y)^{-dα}) with α∈(0,2), and uniform integrability of (J_n) when α>1
- domain assumption G and G_n are transitive graphs of polynomial growth with the stated dimension constraints
- standard math Classical phase-transition facts: β_c>0 exactly when d≥2 and α>1, or d=1 and 1<α≤2 [Sch83, NS86]
Cite this review
Pith. "Pith review of From local giants to locality in long-range percolation." pith.science (2026). https://pith.science/paper/4LFEKQPW
@misc{pith2026260718011,
author = {Pith},
title = {Pith review of: From local giants to locality in long-range percolation},
year = {2026},
howpublished = {\url{https://pith.science/paper/4LFEKQPW}},
note = {Machine review of arXiv:2607.18011}
}
abstract
We prove the analogue of Schramm's locality conjecture for long-range percolation on transitive graphs of polynomial growth with $\alpha \in (0,2)$. In this setting, we also prove the joint continuity of the percolation probability $\theta$ with respect to three parameters: the underlying graph with respect to the local topology, the connectivity kernel, and the percolation parameter $\beta$ for all values of $\beta \in \mathbf{R}_+$, including the critical parameter $\beta_c$. We also prove a number of results related to the supercritical sharpness of long-range percolation: the long-range order decay of the distribution of finite clusters, the truncation problem, the anchored isoperimetric dimension and the transience of the infinite percolation cluster, and the smoothness of the percolation characters. We obtain these results from proving the local existence-and-uniqueness of the linear-sized (giant) cluster. As an immediate corollary of the local existence-and-uniqueness of the giant we obtain the law of large numbers, which answers a special case of a question of Nekrashevych and Pete \cite[Question 1.3]{nekrashevych_scale-invariant_2011}. The main technical contribution is the construction of a renormalisation scheme combining iteratively merged Voronoi tiles with scale-invariant nets, related to the scale-invariant groups of Benjamini.
Figures
Reference graph
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Reviewed August 1, 2026 · model on record in the stance chip above.
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