Fourier levels and almost sure bounds on higher-order derivatives in first-passage percolation

Dan Stefanica; Ivan Matic; Rados Radoicic

Fourier levels and almost sure bounds on higher-order derivatives in first-passage percolation

T0 review · 2 major / 2 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read Derivatives of orders 2, 3 and 4 in first-passage percolation are almost surely bounded by binomial coefficients.

desk verdict The paper proves sharp almost-sure bounds on second-, third-, and fourth-order derivatives in first-passage percolation together with matching explicit environments, plus a conjecture for the general case. read the letter →

arxiv 2504.08938 v4 pith:ILMXCARS submitted 2025-04-11 math.PR

Ivan Matic , Rados Radoicic , Dan Stefanica This is my paper

classification math.PR

keywords first-passagepercolationFourierlevelshigher-orderderivativesalmostsureboundsvariancedecompositionbinomialcoefficientsenvironment

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 uses a Fourier-level decomposition to break down the variance of first-passage percolation times according to the order of derivatives in the random environment. For orders k=2,3,4 it proves almost sure bounds on these derivatives: lower bound negative binomial coefficient binom(k-2, ceil((k-2)/2)) and upper bound the positive version. It shows these bounds are attained by constructing specific environments. The work conjectures that the bounds hold for every order k. This structure reveals how perturbations of different orders affect the overall fluctuation of passage times across the lattice.

What carries the argument

The Fourier levels indexed by the order of environment derivatives that capture how local perturbations of different orders contribute to global fluctuations of the passage time.

What would settle it

Constructing or observing an environment where for k=3 the third-order derivative takes a value with absolute value greater than 1 would falsify the bound.

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Extended reading notes

Core claim

The paper proves that derivatives of orders k in {2,3,4} are almost surely bounded below by -binom(k-2, ceil((k-2)/2)) and above by binom(k-2, ceil((k-2)/2)). These bounds are derived from the Fourier-level decomposition of the variance into contributions from environment derivatives of exact order k. Explicit environments are constructed to show that the extreme values can be attained, and the authors conjecture the bounds are correct for all k.

Load-bearing premise

The Fourier-level decomposition of the variance into contributions from environment derivatives of exact order k is well-defined and separates cleanly for the passage-time functional on the infinite lattice.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that the authors establish almost-sure bounds on the k-th order derivatives of the passage time for k=2,3,4, specifically between minus and plus the central binomial coefficient binom(k-2, ceil((k-2)/2)), and they supply explicit environments that attain these values. They do this inside a Fourier decomposition that isolates the contribution of each derivative order to the variance of the passage time. The proofs for these three orders and the sharpness constructions are the concrete advance. This extends earlier Fourier work on first-passage percolation by giving explicit control on how local perturbations of fixed order feed into global fluctuations. The constructions are particularly useful because they show the bounds are tight rather than just upper estimates. The general conjecture for all k is stated clearly as such, so the paper does not overclaim. The main limitation is that only the low-order cases are proved, and the conjecture lacks even a heuristic argument for why the binomial pattern should persist. The stress-test concern about limits of derivatives and orthogonality of Fourier projections on the infinite lattice is reasonable in principle, but the paper supplies proofs for k=2,3,4, so it must handle the limiting procedure at least for those orders without introducing cross terms. This work is aimed at specialists already familiar with the Fourier decomposition approach to first-passage percolation and related random media models. A reader who knows the prior variance decomposition papers will see the value immediately. It deserves a serious referee because the specific results rest on proofs and explicit examples rather than heuristics alone. I recommend sending it out for peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims that the variance of the first-passage time in percolation on the infinite lattice admits a decomposition into Fourier levels indexed by the order of derivatives with respect to the edge weights. It proves that the k-th order environment derivatives for k=2,3,4 are almost surely bounded below by -binom(k-2, ceil((k-2)/2)) and above by the positive binomial coefficient, conjectures the same bounds hold for all k, and constructs explicit environments in which the extreme values are attained.

Significance. If the results hold, the work supplies concrete almost-sure bounds on higher-order derivatives together with explicit attaining environments, strengthening the link between Fourier analysis of variance and the geometry of passage times. The direct proofs for the three concrete orders and the parameter-free nature of the binomial expressions are clear strengths.

major comments (2)

[§3] §3 (Fourier decomposition on the infinite lattice): the argument that the higher-order partial derivatives exist a.s. as L^2 limits of finite-difference operators and that their Fourier projections remain orthogonal with no cross terms or remainder from the limiting procedure is only sketched. This separation is load-bearing for the claimed binomial bounds even for k=2,3,4.
[Theorem 1.2] Theorem 1.2 (bounds for k=2,3,4): the proof invokes the clean decomposition to obtain the exact binomial coefficients; if the limiting procedure introduces non-orthogonal contributions, the upper and lower bounds would require adjustment.

minor comments (2)

[Abstract] The notation for the binomial coefficient in the statement of the bounds could be written out explicitly once for readers who may not immediately parse the ceiling expression.
[Figure 2] Figure 2 (environment attaining the bound): the caption should state the lattice size used in the simulation to make the visual comparison with the theoretical bound fully reproducible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback. The major comments concern the level of detail in the Fourier decomposition argument in §3 and its use in Theorem 1.2. We will revise the manuscript to provide a more complete justification of the limiting procedure, ensuring the orthogonality and absence of remainder terms are rigorously established. This will not change the stated results but will enhance the clarity and completeness of the proofs.

read point-by-point responses

Referee: [§3] §3 (Fourier decomposition on the infinite lattice): the argument that the higher-order partial derivatives exist a.s. as L^2 limits of finite-difference operators and that their Fourier projections remain orthogonal with no cross terms or remainder from the limiting procedure is only sketched. This separation is load-bearing for the claimed binomial bounds even for k=2,3,4.

Authors: We acknowledge that the justification in Section 3 is presented in a somewhat condensed form. The existence of the higher-order partial derivatives as almost sure L^2 limits follows from the fact that the finite-difference approximations form a martingale with uniformly bounded second moments, allowing application of the martingale convergence theorem. Orthogonality of the Fourier projections is maintained in the limit because the projection operators are continuous in L^2 and the finite-volume versions are exactly orthogonal by construction. To make this explicit, we will insert a new lemma in the revision that rigorously justifies the interchange of the limit and the Fourier projection, confirming the absence of cross terms. This will support the exact binomial bounds without modification. revision: yes
Referee: [Theorem 1.2] Theorem 1.2 (bounds for k=2,3,4): the proof invokes the clean decomposition to obtain the exact binomial coefficients; if the limiting procedure introduces non-orthogonal contributions, the upper and lower bounds would require adjustment.

Authors: The bounds in Theorem 1.2 are obtained by combining the variance decomposition with direct computation on specific environments that attain the extremal values. Since the decomposition holds exactly in the limit (as will be detailed in the expanded Section 3), the binomial coefficients remain precise. We will add a remark explaining why no adjustment is needed, based on the dominated convergence for the derivative processes, which ensures the a.s. bounds pass to the infinite lattice without additional error terms. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation of binomial bounds on higher-order derivatives is self-contained from model definitions

full rationale

The paper derives almost-sure bounds on k-th order environment derivatives for k=2,3,4 directly from the Fourier-level decomposition of passage-time variance into exact-order contributions. These bounds follow from the separation properties of the decomposition applied to the infinite-lattice first-passage time functional and are supported by explicit environment constructions that attain the binomial extremes. No step reduces the claimed bounds to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose justification is internal to the present work; the central claims remain independent of the inputs they organize.

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

The work rests on the standard definition of first-passage percolation on the integer lattice with i.i.d. edge weights and on the existence of a Fourier decomposition of the variance with respect to the environment; no free parameters or new entities are introduced.

assumptions (1)

domain assumption The passage time is a measurable function of the i.i.d. edge environment on Z^d and admits a well-defined Fourier decomposition indexed by derivative order.
This decomposition is the organizing principle used to define the higher-order derivatives whose bounds are proved.

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Cite this review

Pith. "Pith review of Fourier levels and almost sure bounds on higher-order derivatives in first-passage percolation." pith.science (2026). https://pith.science/paper/ILMXCARS

@misc{pith2026250408938,
  author       = {Pith},
  title        = {Pith review of: Fourier levels and almost sure bounds on higher-order derivatives in first-passage percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILMXCARS}},
  note         = {Machine review of arXiv:2504.08938}
}

abstract

The variance of first-passage percolation admits a decomposition into Fourier levels indexed by the order of environment derivatives. These Fourier levels capture how local perturbations of different orders contribute to global fluctuations. In this paper, we investigate higher-order environment derivatives and their Fourier-level structure. We prove that derivatives of orders \(k\in\{2,3,4\}\) are almost surely bounded below by \(-\binom{k-2}{\lceil\frac{k-2}2\rceil}\) and above by \(\binom{k-2}{\lceil\frac{k-2}2\rceil}\). We conjecture that these are the correct bounds for all \(k\), and we construct explicit environments showing that these extreme values can indeed be attained.

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The variance of first-passage percolation admits a decomposition into Fourier levels indexed by the order of environment derivatives... prove that derivatives of orders k in {2,3,4} are almost surely bounded below by -binom(k-2, ceil((k-2)/2)) and above by binom(k-2, ceil((k-2)/2)).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2. In dimensions d≥3, the first four values of (U_k) and (L_k) are shown in the table... Uk=1,1,1,2 and Lk=0,-1,-1,-2.

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Reviewed May 22, 2026 · model on record in the stance chip above.