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arxiv: 2604.21364 · v1 · submitted 2026-04-23 · 🧮 math.PR

Shadow and percolation III: chemical distance in continuous landscapes with correlations

David Vernotte (IF) This is my paper

Pith reviewed 2026-05-09 20:56 UTC · model grok-4.3

classification 🧮 math.PR
keywords chemical distanceexcursion setsGaussian fieldspercolationsupercritical regimecontinuous fieldsrandom landscapescorrelations
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The pith

For almost every supercritical level, chemical distances in the excursion set equal Euclidean distances up to constants.

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

The paper proves that for a slope field alpha coming from a smooth centered Gaussian field f, the excursion sets at almost all supercritical levels l have the property that connected points can be joined by paths whose length is comparable to their straight-line Euclidean distance. This holds with high probability over the random field. A reader cares because it shows that percolation clusters in these continuous correlated settings are robustly connected without forcing long detours, much like in simpler discrete models. The proof adapts classical percolation ideas to the continuous case with its specific difficulties.

Core claim

We show that for almost every supercritical level l in the Lebesgue sense, with high probability the chemical distance between two points connected in the excursion set at level l is comparable to the usual Euclidean distance between those two points. This holds for the slope field alpha associated to a smooth planar centered Gaussian field f, despite alpha being continuous but not differentiable everywhere and having long range correlations whose law is not fully known.

What carries the argument

The excursion set of the slope field alpha at a fixed supercritical level l, consisting of points where alpha is at most l. This set's connected components are used to define the chemical distance as the length of the shortest path inside the set.

If this is right

  • The connected components behave like thick regions without macroscopic obstacles.
  • The comparability is uniform for almost all levels, indicating stability across the supercritical phase.
  • Techniques from discrete Bernoulli percolation can be extended to continuous fields with correlations.
  • High probability statements apply to typical realizations of the Gaussian field.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Simulations of Gaussian fields could verify the distance comparability for chosen levels.
  • This geometric control might help analyze the scaling limits of the interfaces in these models.
  • Similar results could be sought for other continuous random fields beyond Gaussians.

Load-bearing premise

The slope field alpha is obtained from a smooth planar centered Gaussian field f whose probability law is still not completely understood, and the result is proven only for Lebesgue-almost every supercritical level.

What would settle it

Finding, for some supercritical level l, a positive probability event where two connected points in the excursion set have a chemical distance much larger than their Euclidean distance, for example due to a narrow winding path in the set.

Figures

Figures reproduced from arXiv: 2604.21364 by David Vernotte (IF).

Figure 1
Figure 1. Figure 1: Illustration of the proof of Proposition 2.5. On the left the two [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
read the original abstract

We study some geometric properties of the excursion set of a slope field alpha associated to a smooth, planar, centered, Gaussian field f. That, is we consider the set of all points such that the value of alpha is at most l where l is a real parameter called the level. We restrict our attention to the levels l that are supercritical. We show that for almost such l, in the sense of the Lebesgue measure, then with high probability the chemical distance between two points connected in the excursion set at level l is comparable to the usual Euclidean distance between those two points. This result is in the spirit of the Antal Pisztora theorem for Bernoulli percolation. However, many new difficulties arise such as the fact that alpha is a continuous field (not differentiable everywhere) with long range correlations and whose law is still not well understood.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for the excursion set of the slope field α (derived from a smooth planar centered Gaussian field f) at supercritical levels l, for Lebesgue-almost every such l, the chemical distance between points connected in the excursion set is comparable to Euclidean distance with high probability. This extends the Antal-Pisztora theorem from Bernoulli percolation to this continuous, non-differentiable setting with long-range correlations, with the almost-everywhere restriction used to navigate the incompletely characterized law of α.

Significance. If the result holds, it is a meaningful advance in percolation theory for random continuous landscapes, providing a chemical-distance analogue that could inform models with correlated Gaussian fields. The paper is credited for explicitly addressing the technical hurdles of continuity, non-differentiability, and long-range dependence while delivering a falsifiable comparability statement restricted to a set of full Lebesgue measure.

minor comments (3)
  1. [§2] The definition of the slope field α and its derivation from f should be stated with an explicit equation (e.g., in §2) to make the long-range correlation structure transparent before the main theorem.
  2. [Theorem 1.1 (or equivalent)] Clarify the precise notion of 'high probability' (e.g., the measure on the underlying probability space and the dependence on the level l) in the statement of the main result, as the continuous setting may require additional uniformity arguments.
  3. [§1] Add a short comparison paragraph in the introduction contrasting the independence assumptions in the classical Antal-Pisztora proof with the correlation-handling techniques used here.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main result, and recommendation for minor revision. We are pleased that the work is viewed as a meaningful advance extending the Antal-Pisztora theorem to continuous correlated landscapes. Below we respond to the referee's summary.

read point-by-point responses

  1. Referee: REFEREE SUMMARY: The manuscript proves that for the excursion set of the slope field α (derived from a smooth planar centered Gaussian field f) at supercritical levels l, for Lebesgue-almost every such l, the chemical distance between points connected in the excursion set is comparable to Euclidean distance with high probability. This extends the Antal-Pisztora theorem from Bernoulli percolation to this continuous, non-differentiable setting with long-range correlations, with the almost-everywhere restriction used to navigate the incompletely characterized law of α.

    Authors: We thank the referee for this precise summary. The almost-everywhere restriction on the level l is indeed essential, as the law of the slope field α remains incompletely characterized; this is discussed in the introduction and prevents a stronger statement for all supercritical levels. The proof establishes the stated comparability with high probability under the given conditions. revision: no

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper presents a rigorous probabilistic result establishing comparability between chemical and Euclidean distances in the excursion set of a slope field alpha (derived from a smooth planar centered Gaussian field f) for Lebesgue-almost every supercritical level l, with high probability. This is framed as an extension of the Antal-Pisztora theorem to a continuous, correlated, non-differentiable setting. The restrictions to almost-every l and high-probability statements are standard technical devices in percolation theory to handle the field's properties and incompletely characterized law; they do not reduce the claim to a tautology or fitted input. No self-definitional steps, predictions that are statistically forced by construction, load-bearing self-citations, or ansatzes smuggled via prior work are present in the provided text. The derivation is self-contained against external benchmarks such as classical percolation results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available, so ledger is limited to explicitly stated setup assumptions.

axioms (2)
  • domain assumption f is a smooth, planar, centered Gaussian field
    Stated as the basic object whose slope field alpha is studied.
  • domain assumption Levels l are supercritical
    Result restricted to supercritical regime where connectivity occurs.

pith-pipeline@v0.9.0 · 5433 in / 1291 out tokens · 35530 ms · 2026-05-09T20:56:04.213459+00:00 · methodology

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Reference graph

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