The polynomial growth of effective resistances in one-dimensional critical long-range percolation

Jian Ding; Lu-Jing Huang; Zherui Fan

REVIEW 2 major objections 6 minor 1 cited by

The polynomial growth of effective resistances in one-dimensional critical long-range percolation

T0 review · 2 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Resistances grow as n^δ in critical long-range percolation

desk verdict The upper bound and the comparison machinery are genuinely new, but the lower-tail recursion in Proposition 5.8 relies on a false independence claim that currently breaks the supermultiplicativity argument. read the letter →

arxiv 2504.21378 v2 pith:MLBGQ55F submitted 2025-04-30 math.PR

Jian Ding , Zherui Fan , Lu-Jing Huang This is my paper · ORCID

classification math.PR MSC 60K3582B2782B43

keywords long-rangepercolationeffectiveresistancecriticalpolynomialgrowthrandomelectricnetworkone-dimensionalrenormalizationmulti-scalecoarsegraining

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that in the one-dimensional critical long-range percolation model, where an edge between $i$ and $j$ is present with probability roughly $\beta|i-j|^{-2}$, the effective electrical resistance from the origin to the complement of $[-n,n]$ grows like $n^{\delta(\beta)}$ for a constant $\delta(\beta)\in(0,1)$ depending only on $\beta$. The same power law holds for the resistance between the intervals $[-n,n]$ and $[-2n,2n]^c$, conditioned on there being no edge directly joining them. A sympathetic reader should care because effective resistance is the natural metric for random walks on a random network, so this pins down how the critical model conducts current and, by extension, how a random walk spreads on it.

What carries the argument

The load-bearing object is $\Lambda(n)=\max_{i,j\in[0,n)}\mathbb{E}[R_{[0,n)}(i,j)]$, the largest expected point-to-point resistance inside an interval of length $n$ when edges leaving the interval are ignored. The argument shows several resistances of interest are comparable to $\Lambda(n)$, and then proves that $\Lambda$ is almost multiplicative: $\Lambda(mn)\asymp \Lambda(m)\Lambda(n)$ up to constants depending only on $\beta$. Submultiplicativity comes from a renormalization of the line into blocks together with a comparison lemma stating that raising the conductance of one edge incident to a vertex cannot increase the current through any other incident edge. Supermultiplicativity is the harder direction: the paper introduces 'very good' intervals that force any unit flow to spend at least $\alpha m^\delta$ of energy, then uses a multi-scale coarse-graining argument to show that, with high probability, the rare bad intervals can be covered by 'good red animals' whose energy cost is negligible compared with the surrounding regions. Once $\Lambda$ is multiplicative up to constants, a standard subadditivity limit gives $\delta=\lim_n \log\Lambda(n)/\log n\in(0,1)$.

What would settle it

For a fixed $\beta$ and large $n=2^k$, compute $\mathbb{E}[R(0,[-n,n]^c)]$ (or estimate it by simulation over many samples) and check whether the ratio $\mathbb{E}[R(0,[-n,n]^c)]/n^{\delta}$ stays bounded away from $0$ and $\infty$ as $k\to\infty$, where $\delta$ is the slope of $\log\mathbb{E}[R(0,[-n,n]^c)]$ against $\log n$. If that slope fails to converge to a value in $(0,1)$, or if the ratio diverges along a subsequence, Theorem 1.1 is false.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for every $\beta>0$ there is an exponent $\delta=\delta(\beta)\in(0,1)$, depending only on $\beta$, such that for all $n$ both $\mathbb{E}[R(0,[-n,n]^c)]\asymp_P R(0,[-n,n]^c)\asymp_P n^\delta$ and, conditionally on there being no edge between $[-n,n]$ and $[-2n,2n]^c$, $\mathbb{E}[R([-n,n],[-2n,2n]^c)\mid \text{no such edge}]\asymp_P R([-n,n],[-2n,2n]^c)\asymp_P n^\delta$. Here $\asymp_P$ means that for every $\varepsilon>0$ the ratio is bounded between constants with probability at least $1-\varepsilon$. The proof obtains $\delta$ by proving that the maximal expected point-to-point resistance $\Lambda(n)$ inside an interval of length $n$ is both submultiplicative and supermultiplicative up to constants, so a standard subadditivity argument yields $\delta=\lim_n \log\Lambda(n)/\log n$. The inequalities $0<\delta$ and $\delta<1$ are imported from earlier polynomial lower bounds and from the known chemical-distance exponent of the same critical model; the present contribution is showing that one well-defined exponent governs all these resistances and that point-to-box and conditioned box-to-box resistances are comparable to $\Lambda(n)$.

Load-bearing premise

The proof relies on earlier results giving a uniform high-probability polynomial lower bound on resistances and a separate bound showing the growth exponent is below 1; if either of those imported bounds failed for some value of $\beta$, the new exponent would not be guaranteed to lie in $(0,1)$.

Editorial extensions

If this is right

For every $\beta>0$ the point-to-box resistance and the conditioned box-to-box resistance share one polynomial growth exponent $\delta(\beta)\in(0,1)$, so no separate exponent is needed for different resistance types.
The lower-tail estimate of Corollary 1.2 holds: for sufficiently small $\varepsilon>0$, $P(R(0,[-n,n]^c)\ge\varepsilon n^\delta)\ge 1-\varepsilon^q$, so the resistance is bounded away from zero with overwhelming probability.
The paper's resistance estimates place the critical one-dimensional model in position to support random-walk heat-kernel and spectral-dimension bounds by established resistance-form techniques, although the authors deliberately leave those derivations for later work.
The internal exponents constructed in the proofs can be chosen non-increasing in $\beta$, as recorded in Remarks 3.4 and 4.2, so stronger coupling does not make the resistance grow faster in $n$ within the constructed family of exponents.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural testable extension is whether $\delta(\beta)$ is strictly decreasing and, in particular, approaches $1$ as $\beta\to0$; the paper only records monotonicity, not limits.
The same coarse-graining machinery, with $\Lambda(n)$ in place of the graph distance, may be adaptable to the two-dimensional critical model, where the analogous resistance exponent is not known; the paper does not make this claim.
Because the model is exactly scale-invariant, the theorem suggests that the resistance metric itself may admit a scaling limit with a Hausdorff-type exponent $\delta(\beta)$, even though the paper notes that deriving such a limit remains a major open challenge.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper has a real new core: the comparison lemma for electric flows, the submultiplicativity proof, and the coarse-graining scheme adapted to resistance rather than graph distance. These are not imports from [3] and they hold up on inspection. The comparability between point-to-point, point-to-box, and box-to-box resistances is also a substantial step forward. I came away believing the upper bound is solid and the overall strategy is the right one.

The soft spot is where the reader landed, and it is worse than 'details omitted.' In the proof of Proposition 5.8, after equation (5.38), the paper asserts independence of the annulus resistances R(A_{MN+kN,MN+(k-1)N}, A_{MN+(k+2)N,MN+(k+1)N}; E\setminus C_{\ge N}) for odd k. For k and k+2, the target annulus of one is exactly the source annulus of the other. Both resistances are decreasing functions of the shared edge set inside that annulus, so the FKG inequality makes them positively correlated. The intersection probability is not the square of the marginal; it can be as large as the smaller marginal. The bound by M^2 max_i P(R_i < x)^2 is therefore unjustified. The recursion (5.40) and the lower tail for the box-to-box resistance do not follow. Since Proposition 5.8 feeds Proposition 6.3, which feeds Proposition 6.1, the second half of Theorem 1.1 is currently unsupported.

A separate, smaller concern: Proposition 6.1 is explicitly sketched rather than proved in detail. That would be acceptable if the rest were airtight, but it is not.

The external inputs — [11] for the positive lower bound and [3] for δ<1 — are prior results, not restatements of the conclusion. The citation pattern is honest, and the paper is clearly written despite its length.

Who should read this: anyone working on critical long-range percolation, effective resistance, or random walk on such clusters. The techniques in Sections 2–4 will be useful even if the current version is not the last word. But a serious referee should not pass it in this state. The dependence issue in Proposition 5.8 needs a genuine fix, possibly by decoupling the annuli through a gap or by a more careful martingale or negative-association argument. I would not cite the main theorem yet, but I would send it to a strong referee and ask for a major revision.

Referee Report

2 major / 6 minor

Summary. The paper studies critical long-range percolation on Z with edge probability 1-exp(-β∫∫|u-v|^{-2}dudv) and unit conductances, and proves that the effective resistance from 0 to [-n,n]^c, as well as the conditioned box-to-box resistance from [-n,n] to [-2n,2n]^c, both grow like n^{δ(β)} for some δ(β)∈(0,1). The proof defines Λ(n) as the maximum expected point-to-point resistance in [0,n), establishes submultiplicativity (Prop. 2.1), a weak form of supermultiplicativity (Prop. 4.1), a second-moment bound (Prop. 5.1), comparability of point-to-point, point-to-box, and box-to-box resistances (Props. 5.4 and 5.8), and finally full supermultiplicativity (Prop. 6.1). The exponent δ is then obtained via Fekete's lemma. The lower endpoint δ>0 is imported from a strengthened lower-bound lemma (Lemma 3.3) based on [11], and the upper endpoint δ<1 is imported from [3].

Significance. If the main theorem holds, this is a substantial result: it gives the sharp polynomial growth exponent for effective resistances in the critical one-dimensional long-range percolation model, a quantity of direct relevance to random walks. The paper's architecture—defining Λ(n), proving sub- and supermultiplicativity, and then comparing several resistance types—is natural and is supported by several nontrivial technical contributions, notably the flow-comparison Lemma 2.2, the detailed submultiplicativity proof, and the second-moment bound for point-to-point resistances. The exposition is generally careful and well organized. However, the proof of the box-to-box lower tail in Proposition 5.8 contains a false independence assertion that is load-bearing for the rest of the paper, and Proposition 6.1 is only sketched. These gaps need to be addressed before the main claims can be considered established.

major comments (2)

[§5.3, Proposition 5.8, after Eq. (5.38)] The proof asserts 'independence of R(...) for all odd k' and uses it to bound the probability that at least two of the annulus resistances R_k are below γ^3Λ(N) by M^2 max_i P(R_i < γ^3Λ(N))^2. This independence is false: for k and k+2, the target annulus of R_k is exactly the source annulus of R_{k+2}, and both resistances are decreasing functions of the common edge set E\ C_{\ge N}. By FKG, P(R_k < x, R_{k+2} < x) ≥ P(R_k < x)P(R_{k+2} < x), so the intersection probability can be of order the marginal rather than its square. Consequently the recursion leading to (5.40) and the lower-tail bound for R([-MN,MN],[-2MN,2MN]^c | A_{MN}) do not follow. Since Proposition 5.8 is used in Proposition 6.3 and in the proof of the main theorem, this gap directly threatens Theorem 1.1.
[§6, Proposition 6.1] Proposition 6.1 is the second key ingredient in the proof of Theorem 1.1, but its proof is only a sketch: after stating the new very-good interval definition, the text says 'we will omit certain details' and then asserts the coarse-graining estimate (6.4), the properties (P1)-(P3), and inequality (4.46) without proof. The verification of these facts in the new setting is not a routine line-by-line copy of Section 4, because the definition of very-good intervals and the input Proposition 6.3 differ from Section 3. Since the supermultiplicativity of Λ(n) is essential for the existence of δ, this omission is load-bearing. The authors should either supply the full coarse-graining proof or state precisely which lemmas from Section 4 transfer unchanged and why.

minor comments (6)

[§2.2, Eq. (2.19)] The notation E[·|bE] conditions on the edge set bE, but bE is not explicitly identified with a σ-algebra; please clarify that the conditional expectation is with respect to the σ-algebra generated by bE.
[§3.3, proof of Proposition 3.2] There is an inconsistency in the constants: the text uses both e^{-3C4 m^{1/\log n}} and e^{-3C3 m^{1/\log n}} in the same proof. Please correct the notation.
[§4.2, parameters in (4.10)-(4.13)] The parameters A, λ, a_k, b_k, b_{k-1,k}, and K_* are numerous and somewhat opaque; a short table or a summary of their roles would make the coarse-graining argument much easier to follow.
[§5.3, Definition 5.9] The notation A^c_{2MN,MN} is used for the annulus [-2MN,2MN]\[-MN,MN], but the superscript 'c' normally denotes complement; please use a distinct symbol or define the notation explicitly.
[§5.3, proof of Proposition 5.8] There are several typographical issues: 'at leat' should be 'at least', and 'hods' in Section 5.1 should be 'holds'. Please proofread the final version.
[§7, proof of Theorem 1.1] The application of Fekete's lemma to the sequences a_k and b_k is correct, but a sentence justifying finiteness of log Λ(2^k) (e.g., by the boundedness of the resistance of a single edge) would improve the exposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the exponent delta is derived by Fekete's lemma from independently proven submultiplicativity and supermultiplicativity bounds, and the cited prior-work inputs do not assume the target theorem.

full rationale

The paper's central derivation is self-contained once Lemma 3.3 and the bound delta<1 are accepted as external inputs. Lambda(n) is defined directly from the model (Eq. 1.4), and Theorem 1.1's exponent delta is obtained in Section 7 as the Fekete limit of log Lambda(2^k)/log(2^k) using Proposition 2.1 (submultiplicativity) and Proposition 6.1 (supermultiplicativity). These propositions are proved through renormalization, coarse-graining, and resistance comparisons (Sections 2-6) that do not invoke the conclusion being proved. The main imported bounds are explicitly disclosed in Remark 1.3: delta>0 follows from the authors' prior work [11] and delta<1 follows from [3]. Lemma 3.3 is a strengthened form of [11, Theorem 1.1], but [11] is a separate, parameter-free lower-bound result whose assumptions do not include the existence of delta or the polynomial-growth conclusion; citing it is therefore legitimate external support, not circularity. The use of [3] for the sublinear bound Lambda(M) <= M/(100 C) in the proof of Proposition 5.8 is likewise an independent input, and the eventual Fekete argument does not reuse Proposition 5.8's conclusion as an assumption. The skeptic's objection about independence of annulus resistances in Proposition 5.8 is a correctness risk, not a circularity reduction: even if that independence failed, the claimed derivation would be invalid for probabilistic reasons, but it would not be equivalent to its inputs by construction. No equation is used to define Lambda and then reused as its own prediction, and no fitted parameter is relabeled as a derived exponent. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the model definition, on imported results from [11] and [3], and on standard probabilistic machinery. There are no empirical free parameters: all constants in the proof (α, M, etc.) are existential choices made to satisfy inequalities and do not affect the final statement. No new physical entities are introduced.

assumptions (4)

domain assumption The edge probabilities are p_{i,j}=1-exp(-β ∫∫ |u-v|^{-2} du dv) for |i-j|>1 and p=1 for |i-j|=1, with edges independent.
This defines the β-LRP model studied; eq. (1.1).
domain assumption Lemma 3.3: a slightly stronger version of [11, Theorem 1.1] gives high-probability polynomial lower bounds with constants c*, C*, δ* depending only on β.
Imported from prior work by the same authors; propels the lower bound δ>0 and Prop 4.1. The paper does not re-prove it.
domain assumption δ<1 follows from [3] (Bäumler), presumably bounding resistance from above polynomially with exponent <1.
Stated in Remark 1.3(1); the current paper does not locate or re-prove the bound.
standard math Standard tools: FKG inequality, BK inequality (van den Berg-Kesten), Chernoff bounds, Fekete's lemma, and the variational characterization of effective resistance [7, Prop 2.3].
Used throughout; these are established probabilistic and combinatorial results.

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Pith. "Pith review of The polynomial growth of effective resistances in one-dimensional critical long-range percolation." pith.science (2026). https://pith.science/paper/MLBGQ55F

@misc{pith2026250421378,
  author       = {Pith},
  title        = {Pith review of: The polynomial growth of effective resistances in one-dimensional critical long-range percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLBGQ55F}},
  note         = {Machine review of arXiv:2504.21378}
}

abstract

We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}{\rm d} u{\rm d} v\}$ for $|i-j|>1$ for some fixed $\beta>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 0 to $[-n,n]^c$ and from the interval $[-n,n]$ to $[-2n,2n]^c$ (conditioned on no edge joining $[-n,n]$ and $[-2n,2n]^c$) both grow like $n^{\delta(\beta)}$ for some $\delta(\beta)\in (0,1)$.

Figures

Figures reproduced from arXiv: 2504.21378 by the authors.

**Figure 1.** The illustration for Lemma 2.2. The flow g represents the unit electric flow from x to y, satisfying gw1w, gw2w > 0. When we increase the conductance cw2w (the red rectangle) on the edge ⟨w2, w⟩, the amount of flow g through the edge ⟨w1, w⟩, i.e. gw1w, will decrease. 2.2. Proof of Proposition 2.1. In this section, we aim to employ the renormalization technique and Lemma 2.2 to complete the proof of Proposition 2.1… view at source ↗

**Figure 2.** The illustration for the construction of flow f. The flow g represents the unit electric flow from ϖ(ia) to ϖ(jb) in the graph G. The left figure illustrates step (1), where the blue curve represents the edge ⟨xi→j , yi→j ⟩. The right figure corresponds to step (2), where the blue point represents the set Ii,in, while the dark green points correspond to the set Ii,out. The red curves indicate the flow f from Ii,in t… view at source ↗

**Figure 3.** The illustration for definition of RIi in (3.6). The blue curves represent the long edges in EIi×[(i−1)m,(i+2)m) c , while the green curves represent the edges in EIi×(Ii−1∪Ii+1) . When a unit flow θ enters the interval Ii and then flows out, there are two possible scenarios: either some subflows of θ enter through the blue long edges (as θ1 and θ2) or exit through them (as θ3), or some subflows of θ enter and exit… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: The illustration for Definition 4.11. The red vertices and curves in the figure represent an ak-animal. The black vertices ϖ(i1) and ϖ(i2) are both connected to L. The blue lines denote the annulus Bbk−1,k (ϖ(il)) \ Bbk−1 (ϖ(il)), while the orange lines represent the a…

**Figure 5.** Figure 5: The illustration for Definition 4.12. The red curves represent the two animals close to each other. The red dashed line represents the dashed red edge connecting them that we add. (2) there exist another animal L ′ ∈ Ck and a vertex ϖ(j) ∈ Vn \ L ′ such that ϖ(j) ∼ L ′…

**Figure 6.** Figure 6: The red curves represent two ak-animals, L and L ′ . In the top figure, the green curve represents a long edge that we can add to both L and L ′ according to the ak-LEAF operation. In the bottom figure, we illustrate the process of coloring the green long edge and the …

**Figure 7.** Figure 7: The illustration for the “depth-first” exploration process. In the figure, the process is generated by animal neighbor {Ji} 7 i=1, which are marked within the circles. the corresponding exploration sequence is represented by (J ′ i ) 13 i=1. For this exploration, we h…

**Figure 8.** Figure 8: The illustration for the proof of the claim about the independence. The red curves represent some red animal L ′ ∈ Le al (l < k − 1) that can be upgraded to some red ak−1-animal L. The black points are the vertices connected to L ′ directly. The blue and green lines r…

**Figure 9.** Figure 9: The illustration for the definition of separation points. The blue dashed curves in the graph represent the absence of long edges directly connecting i and {0, 1, · · · , i−2} ∪ {i+ 2, · · · , m −1}, while the red dashed curve represents the absence of long edge direct…

**Figure 10.** Figure 10: The illustration for the flow passing through the separation intervals. The blue lines represent the separation intervals. When the flow travels from 0 to mn−1, it must first traverse the left side of the separation interval [iln,(il +1)n), specifically the interval […

**Figure 11.** Figure 11: The illustration for (5.37). The red curves represent the subflow g, which does not pass through any long edges in C≥N (shown as dashed curves). Therefore, when the subflow g starting from [−MN, MN], it must enter each annulus of width N, denoted as AMN+kN,MN+(k−1)N f…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Spectral dimensions for one-dimensional critical long-range percolation
math.PR 2025-05 conditional novelty 7.0 of 10

For critical long-range percolation on the integers, the quenched and annealed spectral dimensions of the random walk both exist and equal 2/(1 + δ), with δ the effective-resistance exponent from a companion paper.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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J. Bäumler. Distance in 1 ∥x−y∥2d percolation models for all dimensions.Commun. Math. Phys. , 404:1495– 1570, 2023

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work page

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