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Spectral dimensions for one-dimensional critical long-range percolation

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Pith's one-line read The simple random walk on critical one-dimensional long-range percolation has both quenched and annealed spectral dimensions equal to 2/(1+δ).

desk verdict Settles the open d=1, s=2 critical case for the spectral dimension; the answer is almost certainly right, but the written proof misquotes its key tail estimate in Lemma 3.6 and leans on the companion resistance paper. read the letter →

arxiv 2505.15037 v2 pith:Q5DJX2BS submitted 2025-05-21 math.PR

Zherui Fan , Lu-Jing Huang This is my paper · ORCID
classification math.PR MSC 60K3582B2782B43
keywords long-rangepercolationrandomwalkspectraldimensionheatkerneleffectiveresistancecriticalquenchedandannealedone-dimensional
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 establishes the exact spectral dimension of the simple random walk on the one-dimensional critical long-range percolation model, where edges between distant sites are present with probability about β|i-j|^{-2} and nearest neighbors are always present. The main result (Theorem 1.1) is that the diagonal heat kernel at time 2n decays as $n^{-1/(1+\delta)}$ up to logarithmic factors, almost surely, and that its expectation decays exactly as $n^{-1/(1+\delta)}$ up to constants. Consequently, both the quenched and annealed spectral dimensions of the walk are $2/(1+\delta)$, where $\delta\in(0,1)$ is an exponent governing the growth of the expected maximal effective resistance in intervals of length n. This resolves an open problem left by earlier work on spectral dimensions of long-range percolation clusters, which had not covered the critical one-dimensional case.

What carries the argument

The argument rests on a general criterion (Proposition 2.3, imported from the references) that converts high-probability volume growth and resistance growth into heat-kernel decay and spectral dimensions for strongly recurrent walks. The needed inputs are verified for the β-LRP model with volume function $\phi(r)=r^{1/\delta}$ and resistance function $\varphi(r)=r$, in the metric given by the effective resistance itself. The decisive new ingredient is a high-moment bound (Proposition 3.1) controlling the diameter of the resistance metric on an interval: the r-th moment of the maximal resistance from 0 to any point of [0,n) is at most a constant times $\Lambda(n)^r$, where $\Lambda(n)$ is the maximal expected resistance. This bound is proved by decomposing the interval into dyadic blocks, using an induction on block size to control the moments of restricted resistances, and then feeding in the polynomial growth $\Lambda(n) \asymp n^\delta$. The volume tail bounds (Lemmas 3.4 and 3.6) and the resistance-to-boundary tail bound (Lemma 3.7) then assemble into the required high-probability estimates.

What would settle it

For a fixed β, simulate the β-LRP on an interval of length N with periodic boundary conditions, estimate Λ(n) for n = 10, 20, ..., N/2 by Monte Carlo averaging of the maximum effective resistance, and fit δ from the log-log slope. Then sample E[p_{2n}(0,0)] for n up to, say, $N^{{1/(1+δ)}}$/10 and check whether $n^{{1/(1+δ)}}$ E[p_{2n}] converges to a positive finite constant. If the product drifts to 0 or ∞, or if the fitted exponents disagree, the theorem's heat-kernel exponent is contradicted.

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Extended reading notes

Core claim

The central claim is that for every β>0, the β-LRP random walk satisfies, almost surely for all large n, $c_1 n^{-1/(1+\delta)}(\log n)^{-\gamma_1} \le p_{2n}(x,x) \le c_2 n^{-1/(1+\delta)}(\log n)^{\gamma_1}$, and for every n the annealed quantity obeys $c_3 n^{-1/(1+\delta)} \le E[p_{2n}(x,x)] \le c_4 n^{-1/(1+\delta)}$. The exponent δ is imported from the companion result that the maximal expected effective resistance in an interval of length n grows polynomially as $n^\delta$. The proof verifies the hypotheses of a general theorem for strongly recurrent random walks by showing that, in the effective-resistance metric, the volume of the ball of radius r grows like $r^{1/\delta}$ and the resistance from the origin to the complement of that ball grows like r. Passing through that theorem gives both the quenched heat-kernel bounds and the spectral-dimension value $2D/(D+\alpha)=2/(1+\delta)$, and a separate argument gives the matching annealed bounds.

Load-bearing premise

The proof takes as given, rather than proving, that the expected maximal effective resistance Λ(n) grows polynomially as n^δ for some δ in (0,1); if that companion result failed, the volume and resistance estimates would not match the required growth rates.

Editorial extensions

If this is right

  • Both the quenched and annealed spectral dimensions of the walk equal $2/(1+\delta)$, so the diagonal heat kernel decays as $n^{-1/(1+\delta)}$ under both laws, up to logarithmic factors in the quenched case.
  • The walk's typical displacement after n steps and the number of distinct sites it visits both grow like $n^{1/(1+\delta)+o(1)}$, as the paper notes follows from the same estimates.
  • The critical one-dimensional case (s=2), which earlier work had left open, now has a definite spectral dimension, completing the picture for this family of models.
  • For every β>0 the value of the spectral dimension is determined by the resistance exponent δ, and the result holds uniformly in that parameter.

Reading between the lines

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

  • Simulations could test the exponent pairing by estimating δ from the growth of the maximal expected resistance and checking whether the annealed heat kernel times n^{1/(1+δ)} stays bounded away from 0 and ∞.
  • The same volume and resistance verification would likely yield off-diagonal heat-kernel bounds in the resistance metric, though the paper restricts itself to the diagonal case.
  • Since δ may vary with β, the spectral dimension 2/(1+δ) may also vary; the paper does not study this dependence, but its result makes that question well-posed.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper studies the simple random walk on critical one-dimensional long-range percolation, where long edges between i and j are present with probability approximately β|i-j|^{-2}. The main result, Theorem 1.1, gives matching quenched and annealed bounds for the diagonal heat kernel p_{2n}(x,x), up to logarithmic factors in the quenched case and exactly polynomial in the annealed case, and identifies both spectral dimensions as 2/(1+δ), where δ is the polynomial growth exponent of the maximum expected effective resistance in a finite interval, imported from the companion paper [10]. The proof strategy is to apply the Kumagai--Misumi framework: set the volume scale as φ(r)=r^{1/δ}, the resistance scale as φ(r)=r, verify Assumptions (A1)--(A3) via estimates on volume tails and resistance tails, and then invoke Propositions 2.2 and 2.3. The paper also proves a high-moment bound for resistance diameters (Proposition 3.1) and derives the required volume lower and upper tail estimates (Lemmas 3.4, 3.6, and 3.7).

Significance. If the result is correct, it resolves an open question from Can--Croydon--Kumagai [7] and provides the first precise spectral-dimension computation for the critical one-dimensional LRP model, including logarithmic corrections in the quenched heat-kernel bounds. The proof route is natural and economical: it reduces the spectral dimension to the resistance-growth exponent δ from the companion paper, with no free parameters in the final formulas. The paper is not self-contained, however; it relies on [10, Theorem 1.1] and [10, Corollary 1.2] for both the value of δ and tail estimates, and several of the quotations and proof steps in Section 3 contain technical gaps that need repair. The central idea is sound and likely correct, but the manuscript in its present form does not fully establish the claimed theorem.

major comments (5)
  1. [Section 3.1, Lemma 3.2] The lemma as stated asserts only that δ=δ(β)>0, but every later application requires δ∈(0,1): Lemma 3.4 explicitly says δ∈(0,1), and Proposition 2.3(2) requires D=1/δ≥1. The paper neither proves the upper bound δ<1 nor quotes a version of [10, Theorem 1.1] that includes it. Since D≥1 is a condition for the spectral-dimension formula 2D/(D+α), the full statement of the imported resistance-growth theorem must be given and verified.
  2. [Section 3.1, Lemma 3.3, Eq. (3.3)] The induction in Lemma 3.3 does not close as written. Equation (3.3) bounds E[(max_i R_i^{2^{k-1}})^2] by 16m^2 max_{x,y}(E[R^{2^{k-1}}])^2, but Jensen's inequality gives the reverse direction, and the quantity that needs control is E[max_i R_i^{2^k}]. The passage from the recurrence (3.6) to the claimed bound (3.7) is also delegated to an argument in [10, Page 56] without the necessary verification. Since Proposition 3.1 and Lemma 3.4 depend on this lemma, a complete proof of the high-moment bound is required.
  3. [Section 3.2, Lemma 3.4] The Markov bound in the proof of Lemma 3.4 has an exponent mismatch. With p=2/δ, Markov's inequality gives P(max R ≥ r) ≤ C r^{-2/δ} Λ(...)^{2/δ}, not r^{-4/δ} Λ(...)^{4/δ} as displayed. The claimed q1=4 follows only if one chooses p=4/δ (or a larger integer), not with the stated p=2/δ. This is a local error, but it affects the lower-tail volume estimate needed for Assumption (A1).
  4. [Section 3.2, Lemma 3.6, Eq. (3.17)] Lemma 3.6 needs an upper bound for the lower-tail event {R(0,[-n,n]^c)<r}, but equation (3.17) states P(R(0,[-n,n]^c)≥r) ≤ (r n^{-δ})^{q0}. With n≈λ r^{1/δ}, the asserted upper tail has right-hand side of order λ^{-q0δ}, while the event R≥r is typical by Lemma 3.2 (since Λ(n)≍n^δ≍λ^δ r). Thus (3.17) as printed contradicts Lemma 3.2. If [10, Corollary 1.2] is a lower-tail estimate, the inequality should be reversed; otherwise Lemma 3.6's volume upper tail is unsupported. The exact statement of [10, Corollary 1.2] should be included.
  5. [Section 3.2, Lemmas 3.6 and 3.7] The paper uses [10, Corollary 1.2] in two different directions: in Lemma 3.6 as an upper-tail bound on R(0,[-n,n]^c) and in Lemma 3.7 as a lower-tail bound. Because the corollary is not stated in the present paper and its direction is misquoted in Lemma 3.6, the reader cannot verify which tail estimates are actually available. Please state the full corollary explicitly and check that both applications are consistent with it.
minor comments (5)
  1. [Section 3.1, Lemma 3.2] The statement of Lemma 3.2 should include the range δ∈(0,1) and clarify that this range is part of [10, Theorem 1.1], not an additional assumption introduced here.
  2. [Section 3.1, proof of Proposition 3.1] In the proof, the text first chooses r>20/δ and then says 'for each r>10/δ'; the two thresholds should be made consistent.
  3. [Section 3.1, Lemma 3.3] The constant C_{2,2^k} appears without definition in the displayed estimate following Eq. (3.3); also the exponent on the constant appears to be written inconsistently with the preceding line.
  4. [Section 3.2, Lemma 3.6, Eq. (3.16)] The Chernoff bound for #E_n should explicitly verify that the threshold λr^{1/δ}/2 exceeds e times the mean; the displayed chain of inequalities is otherwise not fully justified for all r≥1 and λ≥1.
  5. [Section 3.2, Lemma 3.4] The reduction to integers by taking floors of 1/δ and λ^{-1}r^{1/δ} should be accompanied by a short justification that the constants in the final bounds remain uniform.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral-dimension derivation is conditional on the companion resistance-growth theorem but does not reduce to its own inputs.

full rationale

The paper's main result is obtained by inserting the volume and resistance exponents (D = 1/δ, α = 1) into the Kumagai–Misumi framework (Propositions 2.2–2.3). The exponent δ is not defined in terms of the spectral dimension; it is the growth exponent of effective resistance, imported from the same-authors preprint [10, Theorem 1.1], quoted as Lemma 3.2. The target quantity d_s = 2/(1+δ) is a separately measurable heat-kernel decay exponent, and the proof fits no parameter to heat-kernel data. Hence the main claim is not equivalent by construction to its input. The self-citation to [10] is load-bearing, but it is independent mathematical evidence under the stated rules: it is a parameter-free statement about resistance growth, not about heat kernels, and is externally checkable. A correctness concern, not a circularity, is that (3.17) quotes [10, Corollary 1.2] with the upper-tail inequality P(R(0,[-n,n]^c) ≥ r) ≤ (r n^{-δ})^{q0}, whereas (3.15) requires a lower-tail bound P(R(0,[-n,n]^c) < r); if the intended corollary is indeed a lower-tail estimate this is a fixable sign typo, but if not, Lemma 3.6's verification of Assumption (A1) is unsupported. This does not change the circularity verdict.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. Its central claim rests on three external inputs: the resistance exponent theorem and tail estimates from the same-authors preprint [10], and the general random-walk framework of [14]. The model parameter β is a fixed input, not fitted. The proof does not use data or code.

assumptions (3)
  • domain assumption The effective-resistance exponent δ(β)∈(0,1) exists and Λ(n,β) ≍ n^δ, together with the tail estimates of [10, Corollary 1.2].
    Imported from the same-authors companion paper [10] and quoted as Lemma 3.2. The entire setup ϕ(r)=r^{1/δ} and φ(r)=r depends on this.
  • standard math The Kumagai-Misumi framework [14, Propositions 2.2 and 2.3, Remark 2.4] correctly yields the heat kernel and spectral dimension bounds under hypotheses (A1), (A2), (A3), (2.6), and (2.7).
    The proof of Theorem 1.1 is an application of this published framework; it is not re-derived.
  • domain assumption The second-moment resistance bound of [10, Proposition 5.1] and the recurrence arguments around [10, (5.15)-(5.16)] are correct.
    Lemma 3.3's induction for high moments depends on these companion results, which are only cited, not restated.

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Pith. "Pith review of Spectral dimensions for one-dimensional critical long-range percolation." pith.science (2026). https://pith.science/paper/Q5DJX2BS

@misc{pith2026250515037,
  author       = {Pith},
  title        = {Pith review of: Spectral dimensions for one-dimensional critical long-range percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5DJX2BS}},
  note         = {Machine review of arXiv:2505.15037}
}
abstract

Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $\beta>0$ and with probability 1 for $|i-j|=1$. We prove that both the quenched and annealed spectral dimensions of the associated simple random walk are $2/(1+\delta)$, where $\delta\in (0,1)$ is the exponent of the effective resistance in the LRP model, as derived in [10, Theorem 1.1]. Our work addresses an open question from [7, Section 5].

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