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REVIEW 3 major objections 2 minor 75 references

Asymptotic expansion of the critical point for oriented percolation in high dimensions

T0 review · 3 major / 2 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims the critical value of nearest-neighbor oriented percolation on $\mathbb Z^d$ has an asymptotic expansion in powers of $d^{-1}$, proved by the lace expansion.

desk verdict The submitted full text is the wrong manuscript (a physics.optics paper), so the lace-expansion proof of the 1/d critical-point expansion is entirely absent and the math claim cannot be checked. read the letter →

arxiv 2508.12299 v1 pith:XMZIG5JZ submitted 2025-08-17 math.PR

Noe Kawamoto This is my paper · ORCID
classification math.PR MSC 60K3582B43
keywords orientedpercolationcriticalpointasymptoticexpansionlacehigh-dimensionallatticesnearest-neighborprobability
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

The paper's claim is that the critical probability for nearest-neighbor oriented percolation on $\mathbb Z^d$, the threshold at which an infinite directed open cluster first appears, admits a rigorous asymptotic expansion in powers of $d^{-1}$ as the dimension grows. The announced proof route is the lace expansion, a diagrammatic method that in high dimensions controls the connectivity function well enough to extract the threshold order by order. A precise critical-point expansion matters because it turns a non-perturbative lattice problem into a systematic computation and gives quantitative benchmarks for large-dimensional connectivity. The supplied full text is a different manuscript on optical skyrmions, so the derivation announced in the abstract is not present in the provided material; the summary above follows the paper's stated claim.

What carries the argument

The load-bearing object is the lace expansion, a diagrammatic expansion for the two-point function of a random spatial process: the probability that occupation spreads from one site to another is written as a sum over 'laces,' configurations of mutually intersecting paths, and in high dimensions only sufficiently sparse diagrams survive at each order. In this paper the expansion is meant to convert the non-perturbative connectivity problem for oriented percolation into a power series in $1/d$, from which the critical point can be extracted order by order. The specific lace structures and their bounds are not visible in the supplied text.

What would settle it

A concrete falsifier would be to carry the lace expansion one order beyond the paper's stopping point and exhibit a diagram whose contribution is not bounded by the claimed power of $d^{-1}$; absent such a diagram, one can also compute $p_c$ in moderately high dimensions by high-precision simulation and check whether the observed remainder matches the first omitted term of the series.

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Extended reading notes

Core claim

The paper's central claim, stated in its abstract, is that the critical point of nearest-neighbor oriented percolation on $\mathbb Z^d$ admits an asymptotic expansion in powers of $d^{-1}$ as $d\to\infty$. In this setting, each directed bond between neighboring lattice sites is open with probability $p$, and the critical value $p_c(\mathbb Z^d)$ is the threshold below which every connected cluster dies out almost surely and above which an infinite directed cluster exists. The announcement says the proof relies heavily on the lace expansion, meaning the argument is expected to show, diagram by diagram, that the threshold differs from its leading approximation by a series of corrections suppressed by inverse powers of the dimension. No coefficients are quoted in the abstract, and the supplied full text is a different manuscript, so this paragraph records the claim as made rather than a derivation that can be inspected.

Load-bearing premise

The load-bearing premise is that the lace expansion converges in the high-dimensional nearest-neighbor setting and that its truncation to the required order in $1/d$ is valid; the supplied text does not include the proof of that premise.

Editorial extensions

If this is right

  • If the expansion holds, $p_c(\mathbb Z^d)$ can in principle be computed to any fixed order in $1/d$ by extending the lace expansion, so the threshold becomes a systematic series rather than a single limiting value.
  • The series makes quantitative the sense in which high-dimensional oriented percolation is close to its branching-process limit, with the gap between the two expressed as computable corrections.
  • The rigorous expansion supplies a calibration target for numerical simulations of oriented percolation in moderately high dimensions, where finite-size extrapolations can be checked against the asymptotic series.

Reading between the lines

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

  • A natural extension, not claimed by the paper, is that the same order-by-order lace-expansion scheme would yield analogous $d^{-1}$ expansions for related high-dimensional stochastic models, such as the contact process, whose critical point is computed by similar diagrammatic methods.
  • Readers could test the asymptotic claim numerically by estimating $p_c(\mathbb Z^d)$ for $d=5$ through $d=12$ and checking that the remainder after the leading terms shrinks at the rate predicted by the first omitted power of $d^{-1}$; agreement would support the expansion, while a stable nonvanishing remainder would put pressure on it.
  • Because the supplied material contains none of the derivation, the expansion's coefficients should be treated for now as announced rather than established; this caution is an editorial observation about the available text, not a finding about the mathematics.
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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

3 major / 2 minor

Summary. The abstract of arXiv:2508.12299 announces an asymptotic expansion, in powers of d^{-1} as d tends to infinity, of the critical point for nearest-neighbor oriented percolation on Z^d, with the proof said to rely heavily on the lace expansion. The submitted full text, however, is the entirely different manuscript arXiv:2508.12305v1, 'Seeing through randomness with topological light' by Peters et al., which concerns experiments on optical skyrmions and information transfer through random media. No oriented percolation model, no lattice Z^d, no critical point, and no lace expansion appear anywhere in the body. Consequently the submitted manuscript contains no derivation of the advertised result.

Significance. If the claimed theorem were proved, it would be a worthwhile contribution: rigorous high-dimensional asymptotic expansions for the critical point of nearest-neighbor oriented percolation are of genuine interest, and a lace-expansion derivation would likely be influential. However, as submitted, the paper contains no theorem statement, no proof, and no verifiable mathematical content beyond the abstract sentence. There are no machine-checked proofs, reproducible code, or derivations to credit. The potential significance of the claimed result cannot offset the complete absence of its demonstration.

major comments (3)
  1. [Abstract and Full text] The full text is arXiv:2508.12305v1, an unrelated physics.optics paper on optical skyrmions. It contains no oriented percolation, no lattice Z^d, no critical point, and no lace expansion. The single-sentence abstract is the only mathematical content, so the central claim of the paper is entirely unsubstantiated by the submitted material.
  2. [Abstract] No theorem is stated: there is no formula for the critical point, no explicit coefficients of the 1/d expansion, and no error term or dimension range. Even if the intended body were present, the abstract alone could not support the advertised asymptotic expansion, and there is no way to check the convergence or truncation of the lace expansion that the proof allegedly relies on.
  3. [Full text, Equations (1)-(6)] The equations in the body describe Stokes parameters and skyrmion wrapping numbers; they have no connection to the abstract's claim. This confirms that the text submitted for review is not the manuscript described in the abstract, so the proof is absent rather than merely difficult to verify.
minor comments (2)
  1. [Full text, header] The header of the full text carries arXiv:2508.12305v1; the authors should supply the correct source file for arXiv:2508.12299 before resubmission.
  2. [General] Once the correct manuscript is provided, the abstract should state the theorem explicitly, including the coefficients and error order; until then, the report cannot comment on the mathematical exposition, notation, or figures.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detectable: the supplied text contains no part of the claimed derivation, and the abstract asserts no step that reduces to its own inputs.

full rationale

The abstract of arXiv:2508.12299 states that the critical point of nearest-neighbor oriented percolation on Z^d is studied through an asymptotic expansion in powers of d^{-1}, with the proof relying on the lace expansion. The body supplied with the review, however, is the unrelated physics.optics manuscript arXiv:2508.12305, 'Seeing through randomness with topological light.' That text contains no oriented percolation model, no critical point, no d^{-1} expansion, and no lace expansion argument. A circularity finding requires quoting the paper and exhibiting a specific reduction in which an output is equivalent to an input by construction, or in which a fitted parameter is renamed as a prediction. No such step is present in the supplied material: the abstract's claim is simply unverifiable here because the proof is absent. Missing support is not circularity, and no self-citation, fitted-input-as-prediction, or definitional equivalence can be identified. Accordingly, the honest finding is no significant circularity, with score 0.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The ledger is necessarily sparse for the stated paper because the supplied body is a different manuscript; the only usable input is the one-sentence abstract. The lace expansion itself is the main unstated assumption.

assumptions (1)
  • domain assumption The lace expansion is applicable and converges in the high-dimensional nearest-neighbor oriented percolation regime needed for the claimed expansion.
    The abstract says the proof relies heavily on the lace expansion, but no statement of the convergence conditions appears in the available material.

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Cite this review

Pith. "Pith review of Asymptotic expansion of the critical point for oriented percolation in high dimensions." pith.science (2026). https://pith.science/paper/XMZIG5JZ

@misc{pith2026250812299,
  author       = {Pith},
  title        = {Pith review of: Asymptotic expansion of the critical point for oriented percolation in high dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XMZIG5JZ}},
  note         = {Machine review of arXiv:2508.12299}
}
abstract

We study an asymptotic expansion of the critical point for the nearest-neighbor oriented percolation on $\mathbb Z^d$ in powers of $d^{-1}$ as $d\rightarrow \infty$. The proof relies heavily on the lace expansion.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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