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arxiv: 2411.10333 · v3 · submitted 2024-11-15 · 🧮 math.PR

Subcritical annulus crossing in spatial random graphs

Emmanuel Jacob , Benedikt Jahnel , Lukas L\"uchtrath This is my paper

Pith reviewed 2026-05-23 17:40 UTC · model grok-4.3

classification 🧮 math.PR MSC 60K35
keywords continuum percolationannulus crossingspatial random graphslong edgescritical intensityweight-dependent random connection modelsubcritical regimemultiscale argument
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The pith

Annuli in spatial random graphs are crossed by a single long edge or not at all below a critical intensity.

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

The paper proves that in continuum percolation models obeying sparseness, translation invariance, and spatial decorrelation, the critical intensity for crossing annuli, denoted hat lambda_c, is positive precisely when long edges occur with positive probability. This equivalence is sharp when the model has a modicum of independence. As a result, for small intensities, annuli either remain uncrossed or are crossed directly by one long edge rather than by a sequence of shorter connections. The multiscale argument used in the proof also shows that the decay of the crossing probability matches the decay of long-edge probabilities. The result is applied to the weight-dependent random connection model by giving checkable conditions based on a decay coefficient zeta.

Core claim

For general continuum percolation models with the stated properties, annuli are either not crossed for small intensities or crossed by a single edge. The critical annulus-crossing intensity hat lambda_c is determined by the occurrence of long edges, and this relation is sharp under a modicum of independence. The proof provides a direct link between the decay rates of annulus-crossing probabilities and long-edge probabilities through a multiscale renormalization.

What carries the argument

The critical annulus-crossing intensity hat lambda_c, defined as the infimum of intensities where annuli can be crossed and shown to coincide with the onset of long edges.

Load-bearing premise

The models obey sparseness, translation invariance, and spatial decorrelation, and for the sharpness part, they possess a modicum of independence.

What would settle it

An observation of an annulus being crossed by a chain of multiple short edges at an intensity where the probability of any long edge is zero would contradict the claim that crossings require long edges.

Figures

Figures reproduced from arXiv: 2411.10333 by Benedikt Jahnel, Emmanuel Jacob, Lukas L\"uchtrath.

Figure 1
Figure 1. Figure 1: Phase diagram in γ and α for the interpolation model constructed with a long-range profile with δ > 2 in (a) and δ ∈ (1, 2) in (b). The ζ < 0 phase in (a) is shaded in grey while the ζ > 0 phases are not shaded. The values of ζ in the corresponding parameter regimes are shown. The solid line in (a) marks the phase transition ζ = 0. Dashed lines represent no change of behaviour. two characteristics of typic… view at source ↗
Figure 2
Figure 2. Figure 2: Examples for the soft Boolean model, Figure [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Phase diagram for γ and δ for the soft Boolean model with local interference. Represented from left to right the phase transition for ζ < 0 for ξ = 0, 0.3, 0.6, 0.9. 3 Further examples We briefly discuss additional examples and generalisations of the previous section. We do not examine these models in as much detail as the WDRCMs but illustrate, by example, how the derived results can be applied to models … view at source ↗
Figure 4
Figure 4. Figure 4: Sketch of (13) in d = 2. A path starting inside B(10r) and leaving B(20r), where no edge longer than r is used. The red vertex on the path is close to the center of the red ball of the covering of ∂B(10r) and it is the starting point for the event G (red balls). Further, the covering of the annulus B(10.5r) \ B(9.5r) is indicated. The same applies to the blue vertex on the path, which lies close to the cen… view at source ↗
read the original abstract

We consider general continuum percolation models obeying sparseness, translation invariance, and spatial decorrelation. In particular, this includes models constructed on general point sets other than the standard Poisson point process or the Bernoulli-percolated lattice. Moreover, in our setting the existence of an edge may depend not only on the two end vertices but also on a surrounding vertex set and models are included that are not monotone in some of their parameters. We study the critical annulus-crossing intensity $\widehat{\lambda}_{c}$, which is smaller or equal to the classical critical percolation intensity $\lambda_{c}$ and derive a condition for $\widehat{\lambda}_{c}>0$ by relating the crossing of annuli to the occurrence of long edges. This condition is sharp for models that have a modicum of independence. In a nutshell, our result states that annuli are either not crossed for small intensities or crossed by a single edge. Our proof rests on a multiscale argument that further allows us to directly describe the decay of the annulus-crossing probability with the decay of long edges probabilities. We apply our result to a number of examples from the literature. Most importantly, we extensively discuss the weight-dependent random connection model in a generalised version, for which we derive sufficient conditions for the presence or absence of long edges that are typically easy to check. These conditions are built on a decay coefficient $\zeta$ that has recently seen some attention due to its importance for various proofs of global graph properties.

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 / 2 minor

Summary. The paper considers general continuum percolation models obeying sparseness, translation invariance, and spatial decorrelation (including non-Poisson point sets and non-monotone models). It defines the critical annulus-crossing intensitywidehat{lambda}_c <= lambda_c and derives a condition for widehat{lambda}_c > 0 by relating annulus-crossing events to long-edge probabilities via a multiscale argument; the condition is sharp for models possessing a modicum of independence. The main conclusion is that annuli are either not crossed at small intensities or crossed by a single edge, with the argument also describing the decay rate of crossing probabilities. The result is applied to the weight-dependent random connection model, yielding checkable conditions on a decay coefficient zeta.

Significance. If the result holds, it supplies a general, hypothesis-checkable criterion for subcritical annulus crossing that applies beyond standard Poisson or lattice models and handles non-monotonicity. The multiscale argument that directly ties crossing decay to long-edge decay, together with the explicit sufficient conditions on zeta for the weight-dependent model, strengthens the toolkit for analyzing global properties of spatial random graphs.

minor comments (2)
  1. [Abstract] Abstract: the claim of a 'derivation and sharpness' is stated without any displayed equations, error bounds, or verification that the multiscale argument closes; while this is acceptable in an abstract, the introduction or §2 should supply at least one key relation (e.g., the inequality linking crossing probability to long-edge probability) so readers can assess the argument before the full proof.
  2. [Abstract] Abstract, first paragraph: 'modicum of independence' is used to delimit sharpness but is not defined or exemplified at the point of first use; a brief parenthetical or forward reference to the precise assumption (e.g., a weak FKG or independence condition on edge indicators) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of our results, and recommendation of minor revision. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under explicit hypotheses

full rationale

The paper states a conditional theorem relating annulus-crossing events to long-edge probabilities via a multiscale argument, under explicitly listed assumptions (sparseness, translation invariance, spatial decorrelation) that are independent of the target result. No equations reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the sharpness claim is restricted to models with additional independence and the applications to examples (including weight-dependent random connection models) rely on checkable decay conditions rather than internal renormalization. The central claim therefore does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of point processes and on the existence of a decay coefficient for long edges; no new entities are introduced and no parameters are fitted inside the derivation.

axioms (2)
  • domain assumption Models obey sparseness, translation invariance, and spatial decorrelation
    Invoked in first sentence of abstract as the setting for the general continuum percolation models.
  • domain assumption Existence of an edge may depend on surrounding vertex set and models need not be monotone
    Stated explicitly to enlarge the class beyond classical Poisson or lattice models.

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