Brochette first-passage percolation

Maxime Marivain (CMAP)

arxiv: 2501.11398 · v1 · submitted 2025-01-20 · 🧮 math.PR

Brochette first-passage percolation

Maxime Marivain (CMAP) This is my paper

Pith reviewed 2026-05-23 05:22 UTC · model grok-4.3

classification 🧮 math.PR

keywords first-passage percolationBrochette modeltime constantshape theorempoint-to-point convergenceL1 diamondinfinite passage times

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The pith

In the Brochette first-passage percolation model, equal passage times along lines produce point-to-point convergence to a time constant whose vanishing allows varied behaviors, with the limiting shape always the L1 diamond.

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

The paper defines a first-passage percolation process on the plane in which every edge on a given straight line receives the exact same random passage time. It proves that the minimal travel time from the origin to a distant point converges to a deterministic time constant after scaling by distance, and it classifies the possible values and behaviors of this constant when it reaches zero. A shape theorem then shows that the set of points reachable within time t grows asymptotically into the L1 diamond, and the same convergence holds when passage times may be infinite.

Core claim

We establish a point-to-point convergence theorem, identifying the time constant. In particular, we explore the case where the time constant vanishes and demonstrate the existence of a wide range of possible behaviours. Next, we prove a shape theorem, showing that the limiting shape is the L1 diamond. Finally, we extend the analysis by proving a point-to-point convergence theorem in the setting where passage times are allowed to be infinite.

What carries the argument

The Brochette model, in which passage times are identical for every edge lying on the same line, imposing a rigid linear correlation structure that drives both the convergence and the L1-diamond shape.

If this is right

The time constant can be computed or bounded for many distributions, including cases that produce zero speed.
When the time constant is zero the model still admits a range of sublinear growth regimes.
The L1 diamond is the only possible limiting shape under the line-wise constancy assumption.
Point-to-point convergence continues to hold after allowing infinite passage times on some lines.

Where Pith is reading between the lines

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

The line-wise constancy may be viewed as a discrete analogue of a random Riemannian metric that is constant along rays.
The L1 diamond suggests that effective travel speed is governed by the maximum projection onto the coordinate axes.
Similar line-wise models on other lattices could be analyzed by the same renewal-type arguments.

Load-bearing premise

The model requires that passage times are identical for all edges lying on the same line.

What would settle it

A single explicit distribution of line-wise passage times for which the scaled reachable set fails to converge to the L1 diamond in the Hausdorff metric would falsify the shape theorem.

read the original abstract

We investigate a novel first-passage percolation model, referred to as the Brochette first-passage percolation model, where the passage times associated with edges lying on the same line are equal. First, we establish a point-to-point convergence theorem, identifying the time constant. In particular, we explore the case where the time constant vanishes and demonstrate the existence of a wide range of possible behaviours. Next, we prove a shape theorem, showing that the limiting shape is the $L^1$ diamond. Finally, we extend the analysis by proving a point-to-point convergence theorem in the setting where passage times are allowed to be infinite.

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.

Desk Editor's Note private letter to a colleague

This paper defines a Brochette FPP model with equal passage times along lines and proves point-to-point convergence, vanishing-time behaviors, and an L1 diamond shape theorem for it.

read the letter

The main takeaway is a new first-passage percolation variant where every edge on the same horizontal or vertical line gets the identical passage time. They prove a point-to-point convergence result that pins down the time constant, examine the vanishing case and list several possible behaviors, establish a shape theorem with the L1 diamond as the limit, and extend the convergence statement to allow infinite times. That correlation structure is the actual object of study, not an extra assumption layered on top of a standard model. The L1 shape follows directly once times are constant per line, since crossing a full row or column costs the same fixed amount no matter the number of edges traversed. The vanishing-time analysis adds concrete regimes that standard i.i.d. models do not always display so explicitly. The abstract stays light on the underlying distribution family and moment conditions, which usually need to be stated for almost-sure convergence arguments in this area; if the full proofs supply those and handle the built-in dependence without extra fitting, the claims are on solid ground. No circularity appears in the stated results. This is aimed at people already working on correlated or exactly solvable FPP models. A reader who wants to see what changes when line-wise constancy is imposed will get usable statements to compare against other variants. It is worth sending to a referee who can check the technical details of the proofs.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces the Brochette first-passage percolation model on Z^2 in which all edges on the same horizontal or vertical line share an identical passage time. It claims to prove a point-to-point convergence theorem identifying the time constant (including a detailed analysis of the vanishing case and the range of possible behaviors), a shape theorem establishing that the limiting shape is the L^1 diamond, and an extension of the point-to-point convergence result to the setting in which passage times may be infinite.

Significance. If the stated theorems are correct, the work supplies explicit convergence and shape results in a strongly correlated FPP model whose line-wise dependence structure permits closed-form identification of the time constant and limiting shape; this is a meaningful addition to the literature on dependent percolation models. The treatment of the vanishing-time-constant regime and the infinite-passage-time extension further clarifies boundary behavior that is typically intractable in the i.i.d. setting.

minor comments (2)

The abstract states that the time constant is identified but does not indicate the moment or tail assumptions placed on the common line values; the main text should make these hypotheses explicit at the outset of each theorem statement.
Notation for the common passage time along each line (horizontal versus vertical) should be introduced once and used consistently; the current abstract leaves the distinction between the two families of lines implicit.

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 provided in the report, so we have no specific points to address. We will incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity; results follow from explicit model definition

full rationale

The paper introduces a model whose defining feature (identical passage times along each line) is stated as the modeling premise, not derived. All claimed theorems (point-to-point convergence, time-constant analysis, L1-diamond shape theorem, and infinite-time extension) are proved directly from this premise and standard first-passage percolation techniques. No parameter fitting, self-citation chains, or ansatzes are invoked as load-bearing steps; the correlation structure is the object of study rather than an unverified input. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or new postulated entities; the central modeling choice is the line-equality constraint itself.

pith-pipeline@v0.9.0 · 5617 in / 1107 out tokens · 68176 ms · 2026-05-23T05:22:36.773070+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.4 and Theorem 1.5: passage times identical on each integer line δ ∈ Δ, convergence T(0,nx)/n → a||x||_1
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.9 and Alexander-duality-free shape proof via envelopes and variance bounds

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.