Pith. sign in

REVIEW 2 major objections 16 references

The K-th nearest neighbor random walk on a Poisson point process visits finitely many points if and only if K has bounded support.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

The K-th nearest neighbor random walk on a homogeneous Poisson point process visits finitely many points if and only if K has bounded support, with exponential decay in visits and path length, plus a polynomial-tail counterexample.

T0 review reviewed 2026-06-27 challenge →

load-bearing objection The paper gives a sharp iff for trapping of these K-NN walks on PPP using pioneer points, plus a clean counterexample on tails, but the conditional Poisson step needs a close look. the 2 major comments →

arxiv 2606.11271 v1 pith:G6ZZOHGF submitted 2026-06-09 math.PR

The $K$-th nearest neighbor random walk on a Poisson point process gets trapped

Anne-Laure Basdevant (LPSM (UMR\_8001)) , David Coupier (IMT Nord Europe) , Jean-Baptiste Gou\'er\'e (IDP) , Marie Th\'eret (FP2M , MODAL'X) This is my paper
classification math.PR
keywords nearest neighbor random walkPoisson point processtrappingexponential decaypioneer pointsbounded support
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 proves that when the random variable K has bounded support, the number of Poisson points visited by the walk decays exponentially, so the walk gets trapped after finitely many steps. This characterization matters because it identifies exactly when the walk remains local rather than exploring the infinite point set. The argument introduces pioneer points to mark the frontier of already-visited regions and exploit the spatial independence of the Poisson process. The same bounded-support condition also yields exponential decay for the Euclidean length of the trajectory. The authors further exhibit a bounded-support example in which the waiting time for new discoveries has only a polynomial tail.

Core claim

The number of Poisson points visited by the K-th nearest neighbor random walk admits an exponential decay whenever K has bounded support. In particular the walk visits finitely many points if and only if K satisfies the bounded-support assumption. Under the same assumption the Euclidean length of the trajectory also decays exponentially. There exists a bounded-support distribution for which the number of steps until a new point is discovered has at least a polynomial tail.

What carries the argument

The notion of pioneer point, which isolates the boundary between explored and unexplored space so that the independence of the Poisson process can be applied to the remaining region.

Load-bearing premise

The underlying point process is a homogeneous Poisson point process, so that points in disjoint regions are independent.

What would settle it

A single explicit construction of a bounded-support K and a Poisson realization on which the walk visits infinitely many points with positive probability would disprove the main claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The paper studies the K-th nearest neighbor random walk on a homogeneous Poisson point process χ on R^d, where the next point is selected among the K closest neighbors according to i.i.d. labels distributed as K. Theorem 1 asserts that the number of visited points has exponentially decaying tails whenever K has bounded support (BS), and that the walk visits only finitely many points a.s. if and only if (BS) holds; the proof introduces the auxiliary notion of pioneer points to control the explored region. Theorem 2 claims exponential decay of the Euclidean trajectory length under (BS). Theorem 3 exhibits a bounded-support label distribution for which the number of steps until a new point is discovered has at least polynomial tails.

Significance. If the claims are correct, the results give precise trapping criteria for nearest-neighbor walks on PPPs and introduce pioneer points as a tool for handling explored regions in point-process settings. The distinction drawn in Theorem 3 between finite visits and polynomial discovery times is a useful clarification of the geometry of the process.

major comments (2)
  1. [Theorem 1 and pioneer-point section] Proof of Theorem 1 (pioneer-point construction and renewal argument): the argument controls the unexplored complement via independent increments of the homogeneous PPP, but pioneer points are defined as a stopping set measurable with respect to the joint law of χ and the labels. The conditional law of χ on the random complement need not remain homogeneous Poisson (void probabilities can be biased by the selection of pioneers), so the exponential-tail renewal estimate does not follow directly from the unconditional PPP property.
  2. [Abstract / Theorem 1] Abstract and Theorem 1 statement: the three main theorems are asserted without any displayed proof steps, error bounds, or explicit verification that the pioneer-point construction preserves the required conditional independence; this makes the central claims rest on an unverified auxiliary object whose correctness cannot be checked from the given text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [Theorem 1 and pioneer-point section] Proof of Theorem 1 (pioneer-point construction and renewal argument): the argument controls the unexplored complement via independent increments of the homogeneous PPP, but pioneer points are defined as a stopping set measurable with respect to the joint law of χ and the labels. The conditional law of χ on the random complement need not remain homogeneous Poisson (void probabilities can be biased by the selection of pioneers), so the exponential-tail renewal estimate does not follow directly from the unconditional PPP property.

    Authors: We are grateful for this insightful comment highlighting a potential subtlety in the conditional distribution. The key point is that although the pioneer set is a stopping set depending on both χ and the labels, the labels are assigned independently of the point locations. Consequently, the event that a particular region remains unexplored is determined solely by the absence of points in certain areas and the walk's path, but the independence ensures that the void probabilities in the complement are not biased. The renewal argument relies on restarting the process in the unexplored region, which remains Poisson due to the spatial homogeneity and independence from labels. To make this rigorous, we will add an auxiliary result (new Lemma 3.2) that explicitly computes the conditional intensity and confirms it is unchanged. This addresses the concern directly. revision: yes

  2. Referee: [Abstract / Theorem 1] Abstract and Theorem 1 statement: the three main theorems are asserted without any displayed proof steps, error bounds, or explicit verification that the pioneer-point construction preserves the required conditional independence; this makes the central claims rest on an unverified auxiliary object whose correctness cannot be checked from the given text.

    Authors: The abstract is a concise summary and conventionally omits detailed proof steps. For Theorem 1, the statement is followed by the proof in Section 3, but we acknowledge that the verification of the pioneer-point properties could be more prominently displayed. In the revised manuscript, we will insert a brief outline of the proof strategy right after the theorem statement, including references to the lemmas verifying conditional independence and the exponential bounds. This will allow readers to check the logic without reading the entire section. No changes to the abstract itself are planned, as it accurately reflects the results. revision: partial

Circularity Check

0 steps flagged

No circularity: theorems derived from PPP properties without reduction to inputs or self-citations

full rationale

The paper presents mathematical theorems (Theorem 1 on exponential decay of visited points under bounded support (BS), Theorem 2 on trajectory length, Theorem 3 on polynomial tails) proved via pioneer points and PPP independence in disjoint regions. No equations, fitted parameters, or predictions appear; claims are not reductions of inputs by construction. No self-citations are load-bearing for the central results. The derivation relies on standard PPP properties (independent increments) applied to stopping sets, which is external to the paper's own fitted values or definitions. This is a self-contained probabilistic argument, consistent with the reader's assessment of score 0.0. No patterns from the enumerated circularity kinds are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claims rest on the standard properties of homogeneous Poisson point processes (independence and intensity) plus the bounded-support assumption on K; the pioneer-point construction is an auxiliary definition introduced inside the paper.

axioms (1)
  • domain assumption The underlying point process is a homogeneous Poisson point process on R^d with the usual independence properties in disjoint regions.
    Stated as the setting for the walk and used to control unexplored space.
invented entities (1)
  • pioneer point no independent evidence
    purpose: To mark the boundary of the region already explored by the walk so that new points can be controlled.
    Introduced explicitly to prove the trapping result; no external evidence supplied.

reviewed 2026-06-27 · how reviews work

0 comments
Cite this review

Pith. "Pith review of The $K$-th nearest neighbor random walk on a Poisson point process gets trapped." pith.science (2026). https://pith.science/paper/G6ZZOHGF

@misc{pith2026260611271,
  author       = {Pith},
  title        = {Pith review of: The $K$-th nearest neighbor random walk on a Poisson point process gets trapped},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G6ZZOHGF}},
  note         = {Machine review of arXiv:2606.11271}
}
Share X Bluesky LinkedIn Reddit HN
abstract

The $K$-th nearest neighbor random walk $(X_n)_{n \geq 0}$ on a homogeneous Poisson point process $\chi$ on $\R^d$ ($d\geq 1$), starts at the origin and at each step picks its next Poisson point among its closest neighbors according to i.i.d. labels having the same distribution as $K$. Our main result (Theorem 1) states that the number of Poisson points visited by $(X_n)_{n \geq 0}$ admits an exponential decay whenever the random variable $K$ has a bounded support (BS). In particular, the $K$-th nearest neighbor random walk visits finitely many Poisson points if and only if $K$ satisfies Assumption (BS). To prove it, we introduce the key notion of pioneer point which allows us to deal with the region of $\R^d$ already explored by $(X_n)_{n \geq 0}$. Still under Assumption (BS), we also prove an exponential decay for the Euclidean length of the trajectory performed by $(X_n)_{n \geq 0}$ (Theorem 2). Finally, and quite surprisingly, we exhibit an example of label distribution with bounded support for which the $K$-th nearest neighbor random walk discovers new Poisson points after a number of steps whose tail distribution is at least polynomial (Theorem 3).

Figures

Figures reproduced from arXiv: 2606.11271 by Anne-Laure Basdevant (LPSM (UMR\_8001)), David Coupier (IMT Nord Europe), Jean-Baptiste Gou\'er\'e (IDP), Marie Th\'eret (FP2M, Modal'x).

Figure 1
Figure 1. Figure 1: Here is an example of a pioneer point Xn illustrating Definition 6. Indeed, the ball B(Xn, ρ) exceeds the region En (whose boundary is in blue) so that we can place a small (gray) ball B(c, ε) within B(Xn, ρ)\En (Item (i)). Moreover, B(Xn, ρ)∩En only contains three Poisson points, except Xn, ensuring that Item (ii) holds whenever the supremum of the support of the label distribution µ satisfies M ≥ 4. Let … view at source ↗
Figure 2
Figure 2. Figure 2: Let us return to the example depicted in Figure 1 with assuming this time that [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: This picture represents the case where y /∈ QR(u). The balls B(y ′ , ∥y ′ − z∥) and B(y, ∥y ′ − z∥) are resp. depicted in red and blue to illustrate the inclusion of the first one into the second one. each block ΛR(u) is ε-good –with high probability for well chosen parameters ε and R (Lemma 14) and that (b) the Yu’s are κ6-dependent (Lemma 13). Both results will allow us to state that the (dependent) rand… view at source ↗
Figure 4
Figure 4. Figure 4: To the left: In dimension d = 2, the whole space can be covered by κ8 = 6 cones with opening angle θ = π/6 (the dashed lines indicate their boundaries). Based on these cones, are represented the 6 triangles whose heights from the apex 0 are D1, . . . , D6. By construction, they all contain exactly M = 3 elements of S. Moreover, an element x ∈ S, outside the union of those triangles, is depicted in gray: re… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of required environment for L = 2. We impose that exactly one point xi of χ falls in each blue ball and no point of χ falls in the red region. 5.1 A favorable Poissonian environment The main part of the proof consists of constructing a favorable Poissonian environment in which (Xn)n≥0 spends a long time before exploring new points. To this aim, let e1 denote the first vector of the canonical b… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 1 canonical work pages

  1. [1]

    Bordenave, S

    C. Bordenave, S. S. Foss, and G. Last. On the greedy walk problem.Queueing Syst., 68:333–338, 08 2011

    2011

  2. [2]

    Caputo, A

    P. Caputo, A. Faggionato, and A. Gaudilliere. Recurrence and transience for long-range reversible random walks on a random point process.Electron. J. Probab., 14:2580–2616, 2009

    2009

  3. [3]

    Coupier, D

    D. Coupier, D. Dereudre, and J.-B. Gouéré. Absence of percolation for infinite Poissonian systems of stopped paths. Preprint, arXiv:2409.15824 [math.PR] (2024), 2024

    work page arXiv 2024

  4. [4]

    Coupier, D

    D. Coupier, D. Dereudre, and S. Le Stum. Absence of percolation for Poisson outdegree-one graphs. Ann. Inst. Henri Poincaré, Probab. Stat., 56(2):1179–1202, 2020

    2020

  5. [5]

    D. J. Daley and G. Last. Descending chains, the lilypond model, and mutual-nearest-neighbour matching. Advances in Applied Probability, 37(3):604–628, 2005

    2005

  6. [6]

    Devroye, L

    L. Devroye, L. Györfi, and G. Lugosi.A probabilistic theory of pattern recognition, volume 31 of Appl. Math. (N. Y.). New York, NY: Springer, 1996

    1996

  7. [7]

    Grimmett

    G. Grimmett. Percolation., volume 321 of Grundlehren Math. Wiss. Berlin: Springer, 2nd ed. edition, 1999

    1999

  8. [8]

    Häggström and R

    O. Häggström and R. Meester. Nearest neighbor and hard sphere models in continuum percolation. Random Struct. Algorithms, 9(3):295–315, 1996

    1996

  9. [9]

    Jahnel and A

    B. Jahnel and A. Tóbiás. Absence of percolation in graphs based on stationary point processes with degrees bounded by two.Random Struct. Algorithms, 62(1):240–255, 2023

    2023

  10. [10]

    Last and M

    G. Last and M. Penrose. Lectures on the Poisson Process. Institute of Mathematical Statistics Textbooks. Cambridge University Press, 2017

    2017

  11. [11]

    S. Le Stum. Deterministic walk on Poisson point process.ESAIM, Proc. Surv., 60:266–275, 2017

    2017

  12. [12]

    T. M. Liggett, R. H. Schonmann, and A. M. Stacey. Domination by product measures.The Annals of Probability, 25(1):71–95, 1997

    1997

  13. [13]

    L. T. Rolla, V. Sidoravicius, and L. Tournier. Greedy clearing of persistent Poissonian dust.Stochas- tic Processes Appl., 124(10):3496–3506, 2014

    2014

  14. [14]

    QuenchedinvarianceprincipleforrandomwalksonDelaunaytriangulations

    A.Rousselle. QuenchedinvarianceprincipleforrandomwalksonDelaunaytriangulations. Electronic Journal of Probability, 20(none):1 – 32, 2015

    2015

  15. [15]

    Rousselle

    A. Rousselle. Recurrence or transience of random walks on random graphs generated by point processes in rd.Stochastic Processes and their Applications, 125(12):4351–4374, 2015

    2015

  16. [16]

    Rousselle

    A. Rousselle. Annealed invariance principle for random walks on random graphs generated by point processes in rd.Markov Processes And Related Fields, 22(4):653–696, 2016. 25

    2016

This paper was first reviewed by grok-4.3 on June 27, 2026.