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arxiv: 2605.23718 · v1 · pith:CQ4GLHQTnew · submitted 2026-05-22 · 🧮 math.PR

Geodesics and Wandering Exponents in Brochette First-Passage Percolation

Maxime Marivain (X) This is my paper

Pith reviewed 2026-05-25 03:31 UTC · model grok-4.3

classification 🧮 math.PR
keywords first-passage percolationgeodesicswandering exponentsBrochette modellong-range dependencetransversal deviationpassage-time distribution
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The pith

Geodesics exist in Brochette percolation and their wandering deviation H_n follows an explicit order set by the passage-time distribution near its minimum.

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 a first-passage percolation model where all edges on the same horizontal or vertical line share the same random passage time, geodesics from the origin to the point n e_1 exist under mild conditions on the passage-time law. It then shows that the largest sideways deviation H_n of such a geodesic scales in a way that depends on how the distribution behaves close to its smallest possible value. This dependence allows the derivation of precise wandering exponents even though the model has long-range correlations along lines. Readers interested in random growth or interface motion may care because the result shows that certain forms of dependence do not destroy the possibility of exact scaling laws for path fluctuations.

Core claim

In the Brochette first-passage percolation model, where edges on the same axis-parallel line share a common random passage time, geodesics exist under mild assumptions, and the maximal transversal deviation H_n of the geodesic from the origin to n e_1 has an order of magnitude determined by the behavior of the passage-time distribution near its infimum; these facts produce explicit wandering exponents for this dependent model.

What carries the argument

The maximal transversal deviation H_n, whose order is controlled by the lower tail of the passage-time distribution.

If this is right

  • Existence of geodesics from the origin to n e_1 under mild assumptions.
  • The order of magnitude of H_n is determined by the behavior of the passage-time distribution near its infimum.
  • Explicit wandering exponents are obtained in this long-range dependent setting.

Where Pith is reading between the lines

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

  • The tail conditions on passage times may determine fluctuations in other models with line-wise dependence.
  • Simulations with different distributions near the infimum could test the predicted scaling of H_n.
  • The method might extend to higher dimensions or altered dependence structures.

Load-bearing premise

The passage-time distribution must satisfy specific regularity or tail conditions near its infimum to fix the order of H_n.

What would settle it

Finding a passage-time distribution satisfying the mild assumptions but where H_n fails to follow the predicted order would falsify the result.

Figures

Figures reproduced from arXiv: 2605.23718 by Maxime Marivain (X).

Figure 1
Figure 1. Figure 1: The initial path Γ 1 n is the concatenation of γ1,n (in green), the black path and γ3,n (in blue). The path Γ 2 n is the concatenation of γ1,n (in green), γ2,n (in red), and γ3,n (in blue). Thus we have P(Hn > εn) ≥ P(T(Γ2 n ) − T(Γ1 n ) ≤ 0) ≥ P(T(Γ2 n ) − T(Γ1 n ) ≤ 0 and σn(ε) < ρn(ε)) ≥ P n 3 (σn(ε) − ρn(ε)) + (ε + √ ε)n(µn + νn) ≤ 0  = P  n 1 β (ρn(ε) − σn(ε)) ≥ 3(ε + √ ε)n 1 β (µn + νn)  . Now, u… view at source ↗
Figure 2
Figure 2. Figure 2: The construction of Γn going out of J−⌊Kn⌋, ⌊Kn⌋K 2 ). Therefore {H(Γn) > Kn} ⊂ {T(Γn) ≥ Knςn(K)}. Thus we have P(H(Γn) > Kn) ≤ P(T(Γn) ≥ Knςn(K)) ≤ P(T(Γn) ≥ Knςn(K) and ςn(K) ≥ K − β+1 2β n − 1 β ) + P(ςn(K) < K− β+1 2β n − 1 β ) ≤ P(T(Γn) ≥ K β−1 2β n 1− 1 β ) | {z } =:Un + P((Kn) 1 β ςn(K) < K− β−1 2β ) | {z } =:Vn . Now, since the passage times are bounded and using Theorem 1.7 in [Mar26], there exist… view at source ↗
Figure 3
Figure 3. Figure 3: The path Γ 2 n is the concatenation of γ1,n (in blue), γ2,n (in purple) and γ3,n (in green). bound we construct a path Γ 2 n such that H(Γ2 n ) > εn β β+d−1 and we show that this path is asymptotically better than Γ 1 n . (See [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The path Γ 2 n is the concatenation of γ1,n (in blue), γ2,n (in purple) and γ3,n (in green). Moreover, we have T (0, x) = min  T˜ (0, x), Tˇ (0, x)  ≥ min  T˜ (0, x), Kn β β+d−1 a  . Thus, if En occurs we have sup x∈Dn, K 2 (0) T (0, x) ≥ min  2 3 Kn β β+d−1 a, Kn β β+d−1 a  = 2 3 Kn β β+d−1 a. Therefore P [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

We study geodesics in the Brochette first-passage percolation model, where edges on the same axis-parallel line share a common random passage time, inducing long-range dependence. We focus on the maximal transversal deviation H n of geodesics from the origin to ne 1 . We prove existence of geodesics under mild assumptions and establish the order of magnitude of H n depending on the behavior of the passage-time distribution near its infimum. These results yield explicit wandering exponents in this dependent setting.

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 / 0 minor

Summary. The paper studies geodesics in the Brochette first-passage percolation model, where edges on the same axis-parallel line share a common random passage time, inducing long-range dependence. It proves existence of geodesics under mild assumptions and establishes the order of magnitude of the maximal transversal deviation H_n of geodesics from the origin to n e_1, depending on the behavior of the passage-time distribution near its infimum. These results yield explicit wandering exponents in this dependent setting.

Significance. If the results hold, this work is significant for extending first-passage percolation theory to a long-range dependent setting, where explicit wandering exponents are typically unavailable. The mild assumptions and explicit dependence on the passage-time distribution near its infimum allow for precise control, providing a concrete advance over standard bounds in the literature. The mathematical treatment of existence and order-of-magnitude statements under dependence is a clear strength.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for the positive assessment of its significance in extending FPP results to a long-range dependent setting. The recommendation is listed as uncertain, but the report contains no specific major comments to address. We remain available to provide any additional clarifications or details that would resolve the uncertainty.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract states results as theorems proving geodesic existence under mild assumptions and deriving the order of H_n (hence wandering exponents) from the passage-time distribution's behavior near its infimum. No equations, fitted parameters presented as predictions, or self-citations are visible in the provided text that would reduce any load-bearing step to an input by construction. The claims are framed as independent mathematical derivations in a dependent setting, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work appears to rest on standard probability axioms plus model-specific regularity conditions near the infimum that are not enumerated here.

pith-pipeline@v0.9.0 · 5599 in / 1090 out tokens · 36539 ms · 2026-05-25T03:31:33.833170+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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