pith. sign in

arxiv: 2403.14295 · v3 · submitted 2024-03-21 · 🧮 math.PR

Large and moderate deviations in Poisson navigations

Partha Pratim Ghosh , Benedikt Jahnel , Sanjoy Kumar Jhawar This is my paper

Pith reviewed 2026-05-24 03:21 UTC · model grok-4.3

classification 🧮 math.PR MSC 60F1060G5560K05
keywords Poisson point processlarge deviationsmoderate deviationsrandom navigationrenewal processvertical displacementdirected graphsgeometric networks
0
0 comments X

The pith

The probability of large vertical displacements in cone-based Poisson navigations decays at a precise exponential rate.

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

The paper studies directed paths formed by successive nearest-neighbor connections within a fixed cone on a homogeneous Poisson point process in the plane. It derives both large-deviation and moderate-deviation principles that supply exact exponential decay rates for the probability that the total vertical displacement exceeds a given threshold. These rates are obtained by decoupling the horizontal progress from the vertical increments through renewal theory. A reader would care because the model captures the geometry of random networks in which paths must advance rightward while staying close to a target line. The non-Markovian dependence between steps is controlled so that standard large-deviation tools apply after the renewal reduction.

Core claim

In the non-Markovian routing scheme on a homogeneous Poisson point process, where each point is joined to its nearest point to the right lying inside a fixed cone, the vertical displacement of the resulting path satisfies a large-deviation principle and a moderate-deviation principle; both give explicit exponential rates for the probability of atypically large vertical excursions.

What carries the argument

Renewal-process representation of the horizontal displacements under the cone nearest-neighbor rule, which reduces the vertical increments to a sequence whose dependencies can be bounded uniformly.

If this is right

  • Tail bounds on the vertical displacement become available at all deviation scales between the law of large numbers and the large-deviation regime.
  • The same renewal reduction yields moderate-deviation results interpolating between central-limit and large-deviation behavior.
  • The exponential rates apply directly to the analysis of path stretch and connectivity in planar random geometric graphs.
  • The method extends verbatim to any routing rule whose horizontal increments admit a renewal structure with controlled vertical noise.

Where Pith is reading between the lines

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

  • The same exponential rates could be used to calibrate rare-event simulation algorithms for wireless routing protocols that penalize vertical drift.
  • Analogous deviation principles may hold when the underlying point process is replaced by a stationary ergodic point process with finite intensity.
  • The renewal reduction technique might transfer to higher-dimensional navigations or to models with random cone angles.
  • The explicit rate functions could be compared numerically with those arising in related models such as directed last-passage percolation.

Load-bearing premise

The cone nearest-neighbor rule makes the horizontal steps sufficiently independent that renewal theory can be applied to separate horizontal progress from vertical contributions.

What would settle it

A Monte Carlo simulation on a large finite window of a homogeneous Poisson point process that produces log-probabilities for vertical displacement exceeding level n that deviate from the claimed rate function by more than a fixed additive constant for large n would falsify the result.

Figures

Figures reproduced from arXiv: 2403.14295 by Benedikt Jahnel, Partha Pratim Ghosh, Sanjoy Kumar Jhawar.

Figure 1
Figure 1. Figure 1: Simulated sample path of V¯ for θ = arctan(5). Lemma 1. Consider iid copies {U˜ i}i≥1 of the progress variable U1 ∈ R 2 in the directed-θ￾navigation. Then, {n −1 Pn i=1 U˜ i}n≥1 satisfies the large-deviation principle with rate n and rate function Iλ,θ(u) := sup{hγ, ui − Jλ,θ(γ): γ ∈ R 2 }, where, for all γ ∈ R 2 , Jλ,θ(γ) := log  λ Z ∞ 0 drr exp(−λθr2 ) Z θ −θ dϕexp(γ1r cos ϕ + γ2r sin ϕ)  < ∞. Further,… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration for the proof of Lemma 5. which means Ln(π/2 − θ) ∩ B(Vm, |Vn − Vm|) = ∅. This proves that, for any 0 ≤ m ≤ n − 1, Cπ/2−θ (Vn) ∩ B(Vm, |Vn − Vm|) = ∅, thereby completing the proof. Proof of Lemma 3. We define a sequence of sets {Dn}n≥1 as Dn := Cθ(Vn−1) ∩ B(Vn−1, Rn) ∩ Hc n−1 . Essentially, Dn is the previously unexplored set where we have explored at the n-th step to find Vn. Clearly, {Dn}n≥1… view at source ↗
read the original abstract

We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizontal displacement as well as renewal-process arguments.

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 derives large- and moderate-deviation results for the vertical displacement of paths generated by a non-Markovian directed navigation on a homogeneous Poisson point process in the plane. Each step joins a point to its nearest Poisson point lying to the right inside a fixed cone; the proofs control inter-step dependencies and horizontal displacement randomness via renewal-process arguments to obtain precise exponential decay rates.

Significance. If the claims hold, the work supplies explicit large-deviation rate functions for vertical excursions in a geometrically constrained random geometric graph, a setting relevant to wireless network models and continuum percolation. The explicit use of renewal theory to handle the acknowledged non-Markovian character, together with the parameter-free character of the derivation, constitutes a technical contribution that could be reused in related routing problems.

minor comments (3)
  1. The abstract asserts that 'precise exponential rates of decay' are established but does not name the rate function or the precise scaling regime (large vs. moderate deviations); adding one sentence with the form of the rate would improve readability.
  2. The routing rule is described as 'nearest Poisson point to the right within a cone'; a short paragraph early in the introduction stating the precise cone aperture and the tie-breaking rule (if any) would remove ambiguity for readers unfamiliar with the model.
  3. The renewal-process construction is invoked to control horizontal displacements; a brief remark on the length of the renewal intervals or the moment conditions required for the large-deviation upper bound would help readers assess the scope of the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on standard renewal theory and Poisson point process properties to control step dependencies and horizontal displacements in a cone-based nearest-neighbor routing scheme, yielding exponential decay rates for vertical large deviations. These are independent external mathematical tools with no reduction of claims to self-defined quantities, fitted parameters from the same data, or load-bearing self-citations. The non-Markovian character is handled explicitly via renewal constructions without circular redefinition. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on standard properties of homogeneous Poisson point processes and applicability of renewal theory to the horizontal component; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • standard math Homogeneous Poisson point process in the plane has the usual independence and intensity properties
    Underlying spatial distribution for the navigation points.
  • domain assumption Horizontal displacements permit a renewal-process representation that decouples vertical deviation probabilities
    Invoked to control dependencies across steps.

pith-pipeline@v0.9.0 · 5613 in / 1002 out tokens · 34393 ms · 2026-05-24T03:21:34.652697+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    M. A. Arcones. Large Deviations of Empirical Processes. High Dimensional Probability III, Birkh¨ auser Basel, 205–223, 2003

    work page 2003

  2. [2]

    Asmussen

    S. Asmussen. Applied Probability and Queues: Stochasti c Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 51, Springer, New York. , 2003

    work page 2003

  3. [3]

    Baccelli and C

    F. Baccelli and C. Bordenave. The radial spanning tree of a Poisson point process. Ann. Appl. Probab. , 17:305–359, 2007

    work page 2007

  4. [4]

    Bonichon and J.-F

    N. Bonichon and J.-F. Marckert. Asymptotics of geometri cal navigation on a random set of points in the plane. Adv. Appl. Probab. , 43(4):899–942, 2011

    work page 2011

  5. [5]

    Bordenave

    C. Bordenave. Navigation on a Poisson point process. Ann. Appl. Probab. , 18(2):708–746, 2008

    work page 2008

  6. [6]

    Coupier, K

    D. Coupier, K. Saha, A. Sarkar and V. C. Tran. The 2d-direc ted spanning forest converges to the Brownian web. Ann. Probab., 49(1):435–484, 2021

    work page 2021

  7. [7]

    Dembo and O

    A. Dembo and O. Zeitouni. Large Deviations Techniques an d Applications. Stochastic Modelling and Applied Probability 38, Springer-Verlag Berlin Heidelberg , 1998

    work page 1998

  8. [8]

    Eichelsbacher and M

    P. Eichelsbacher and M. L¨ owe. Moderate deviations for i id random variables. ESAIM: Probab. Stat. , 7:209– 218, 2003

    work page 2003

  9. [9]

    Hirsch, B

    C. Hirsch, B. Jahnel, P. Keeler and R. I. A. Patterson. Tra ffic flow densities in large transport networks. Adv. App. Probab. , 49(4):1091–1115, 2017

    work page 2017

  10. [10]

    den Hollander

    F. den Hollander. Large Deviations. American Mathematical Soc. , 2000. 21

    work page 2000

  11. [11]

    C. D. Howard and C. M. Newman. Euclidean models of first-p assage percolation. Prob. Theory Relat. Fields , 108:153–170, 1997

    work page 1997

  12. [12]

    Jahnel and W

    B. Jahnel and W. K¨ onig. Probabilistic Methods for Tele communications. Birkh¨ auser Compact Textbooks in Mathematics, 2020

    work page 2020

  13. [13]

    M. Ledoux. On moderate deviations of sums of iid vector r andom variables. Ann. Inst. H. Poincar´ e Probab. Stat., 28:267–280, 1992

    work page 1992

  14. [14]

    R. Roy, K. Saha and A. Sarkar. Scaling limit of a drainage network model on perturbed lattice. arXiv:2302.09489, 2023

    work page arXiv 2023

  15. [15]

    R. Roy, K. Saha, A. Sarkar and V. C. Tran. Random directed forest and the Brownian web. Ann. Inst. H. Poincar´ e Probab. Stat., 52(3):1106-1143, 2016

    work page 2016

  16. [16]

    S. R. S. Varadhan. Large Deviations and Applications. SIAM, Philadelphia, 1984. (Partha Pratim Ghosh) Technische Universit ¨at Braunschweig, Universit ¨atsplatz 2, 38106 Braun- schweig, Germany Email address : p.pratim.10.93@gmail.com; partha.ghosh@tu-braunschwe ig.de (Benedikt Jahnel) Technische Universit ¨at Braunschweig, Universit ¨atsplatz 2, 38106 Br...

    work page 1984