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Classification of the limit shape for 1+1-dimensional FPP

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In a planar random-metric model with deterministic vertical weights, a flat vertical edge of the limit shape exists exactly when the horizontal weight law has an atom at the lower endpoint of its support.

desk verdict A sharp flat-edge classification in a new FPP variant, with an original proof that currently has a fixable factor-of-two gap in the key case. read the letter →

arxiv 2411.13030 v2 pith:MCNDVOQX submitted 2024-11-20 math.PR

Malte Hassler This is my paper · ORCID
classification math.PR MSC 60K3560F1082B43
keywords firstpassagepercolationlimitshapeflatedgetimeconstantdirectedlargedeviationsgeodesicssite
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a planar first-passage-percolation model in which every vertical edge costs 1 and horizontal edges carry independent random costs drawn from a law $G$. The main result is a sharp classification: the vertical direction on the limiting metric ball (the limit shape) is part of a flat straight edge precisely when $G$ has an atom at the infimum $t_0$ of its support. When that atom is present, the time constant $\Lambda(v)$ is exactly $t_0+v$ for all sufficiently large slopes $v$; when it is absent, $\Lambda(v)>t_0+v$ for every $v$, so the boundary has no vertical flat segment and the limit shape is not a polygon. This matters because limit-shape geometry is largely open in classical FPP, and this simplified model yields a complete answer decided by a single checkable property of the distribution.

What carries the argument

The argument's central object is the time constant $\Lambda(v)=\lim_{n\to\infty} n^{-1}T((0,0),(n,\lceil vn\rceil))$ and the comparison line $t_0+v$, which is always a lower bound and, by Proposition 14, an asymptote. The dichotomy is carried by three interlocking mechanisms. First, the exact limit shape of the directed SJ-model, the model where paths may only step east and north, is used to show that if $G$ has an atom at $t_0$ then $\Lambda(v)=t_0+v$ for all sufficiently large $v$, and to build a two-point Bernoulli distribution that lies below $G$ in stochastic order. Second, geodesics are coarse-grained over large rectangles whose trapezoidal crossing events are approximately independent Bernoulli variables, coupling the microscopic model to a site-percolation model; right- and left-tail large-deviation bounds, including Lemma 13 for site percolation, make low-passage-time macroscopic paths exponentially rare, which forces $\Lambda(v)>t_0+v$ when the atom is absent. Third, for the derivative bounds, a random shear map replaces the continuous shear used in continuous polymer models; the discrete identity $\Delta V(z)=|z+1|-|z|$ turns a tilted passage time into a count of up-, right-, and down-turns along the geodesic.

What would settle it

Take an atom-free $G$ with $t_0=0$, for instance Lebesgue measure on $[0,1]$, and estimate the time constant $\Lambda(v)$ for a fixed $v>0$ from finite-$n$ passage times. If $\lim_{n\to\infty} T((0,0),(n,\lceil vn\rceil))/n$ equals $v$ rather than being strictly larger, then the claimed strict inequality fails and the Main Theorem's 'only if' direction would be refuted.

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Extended reading notes

Core claim

The paper's central claim is the dichotomy stated in the Main Theorem: for $G\neq\delta_0$, with $t_0$ the left endpoint of the support of $G$, the point $(0,1)$ lies on a flat edge of the limit shape if and only if $G(\{t_0\})>0$. Equivalently, the time constant satisfies $\Lambda(v)\ge t_0+v$ for all $v\ge 0$, and equality holds for an interval of slopes exactly when the lower endpoint carries positive mass; otherwise $\Lambda(v)>t_0+v$ for every finite $v$. A second result, Theorem 3, gives almost-sure bounds on the upper and lower derivatives of $\Lambda$ in terms of the asymptotic density of up-, right-, and down-turns along geodesics, obtained by a random shear construction. The paper also proves a full shape theorem for the model without moment assumptions.

Load-bearing premise

The argument's load-bearing premise is that, in the coarse-grained site-percolation comparison, paths with unusually small passage time are exponentially rare once the per-edge probability of a cheap crossing is small enough. If that large-deviation estimate fails at the scale used in the proof, the strict inequality $\Lambda(v)>t_0+v$ for atom-free $G$ is not established.

Editorial extensions

If this is right

  • If $G(\{t_0\})>0$, the flat vertical edge is present and $\Lambda(v)=t_0+v$ for every slope $v\ge (1-G(t_0))/G(t_0)$, so the atom mass controls the extent of the edge.
  • If $G(\{t_0\})=0$, the limit shape has no flat edge through $(0,1)$ and is not a polygon; the time constant stays strictly above the asymptote $t_0+v$ at every finite slope.
  • For any non-deterministic $G$, $\partial_+\Lambda(0)<1$, so the limit shape is never the $\ell^1$ diamond; randomness alone rounds off the corner.
  • A positive linear density of down-turns along geodesics would imply $\partial_+\Lambda(v)<1$ and hence rule out the flat vertical edge, reducing the shape question to a counting problem for geodesic turns.
  • The large-deviation bounds hold without moment assumptions because high horizontal weights can be bypassed by short vertical detours, making the shape theorem available for every distribution $G$ on $[0,\infty)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same atom-at-the-infimum criterion is a natural conjecture for classical FPP limit shapes, but transferring it would require replacing the deterministic vertical weights and the exact directed model used here by a different coarse-graining.
  • A simulation-friendly diagnostic suggested by Theorem 3 is to estimate the long-run frequency of down-turns along geodesics; if the frequency is bounded below by a positive constant, the derivative bound gives a quantitative route to ruling out the flat vertical edge.
  • The explicit threshold $(1-G(t_0))/G(t_0)$ predicts how the atom mass tunes the flat edge: a very small atom confines the flat edge to slopes near the vertical, while a large atom pushes the shape toward the $\ell^1$ diamond.
  • The random shear construction is a discrete replacement for continuous shear maps and could apply to other integer-valued variational problems, such as discrete directed polymers, whenever one needs directional derivatives of a variational quantity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies a modified planar first-passage percolation model in which all vertical edges have deterministic weight 1 and horizontal edges have i.i.d. weights with a general distribution G on [0,∞). The main result is a sharp classification: the point (0,1) lies on a flat edge of the limit shape if and only if G has an atom at t0, the infimum of the support of G. The proof combines subadditivity and large-deviation estimates for the original model, explicit formulas for the directed SJ-model, a new random-shearing construction, and a reduction to a macroscopic site-percolation model. The paper also proves existence of the time constant and limit shape, gives derivative bounds relating geodesic turn counts to derivatives of the time constant, and establishes several auxiliary continuity and large-deviation results.

Significance. If the main theorem is correct, it provides one of the few exact, non-perturbative classifications of flat edges in a planar FPP-type model, and it does so without moment assumptions on the weight distribution. The random-shearing technique and the macroscopic site-percolation coupling are original and potentially useful beyond this paper. The central claim is falsifiable and parameter-free, and the paper is generally well organized. However, the proof of Proposition 15, which is the load-bearing step for the main theorem, contains a concrete constant error in the final case analysis, and an auxiliary counting lemma is false as stated. These issues are local and appear fixable, but they must be corrected before the proof can be accepted.

major comments (3)
  1. [Section 3.4, cases (S5a)/(S5b) and Eq. (17)] The case (S5b) as stated does not yield the deficit claimed in Eq. (17). If the macroscopic path φ(γ) has D ≤ δn/(64K(v+δ/2)) downward moves, while the number Z of zero-weight macroscopic sites satisfies Z ≥ δn/(32K(v+δ/2)) from (S3b) and (S4b), then the site count gives T(φ(γ)) ≤ N + V + 1 + 2D − Z ≤ N + V + 1, with no negative term. The negative term −δ/(64(v+δ/2)) n/K in Eq. (17) would require the stronger bound D ≤ δn/(128K(v+δ/2)). Since (S5b) is the case that completes the proof that a distribution without an atom at t0 forces Λ(v) > t0 + v, this factor-of-two error is load-bearing and must be fixed by adjusting the threshold in (S5b) and the subsequent constants.
  2. [Section 3.4, application of Lemma 13 after Eq. (17)] The application of Lemma 13 after Eq. (17) is invalid as written because the parameter ε is chosen too large. Lemma 13 is invoked with ε = δ/(27(v+δ/2)), while the deficit in Eq. (17) is δ/(64(v+δ/2)); since δ/27 > δ/64, the event that a path has passage time at most N(1+v/(2(v+δ)) − δ/(64(v+δ/2))) is not contained in the event that it has passage time at most N(1+v/(2(v+δ)) − δ/(27(v+δ/2))). After the correction in the previous comment, the deficit becomes δ/(128(v+δ/2)), so ε must be chosen no larger than that value. The constants in Lemma 13, Proposition 20, and the choice of K must be made consistent with this smaller deficit.
  3. [Section 4.2, Lemma 23] Lemma 23 is false as stated. For M = k = 2, the printed formula returns 9, but the number of 2-tuples of integers whose absolute values sum to 2 is 8. The correct count is S(M,k) = Σ_{ρ=1}^{min(k,M)} C(k,ρ) C(M−1,ρ−1) 2^ρ. The error comes from counting positive ρ-tuples with an extra positive leftover via C(M−1,ρ), which misses the case where the nonzero coordinates sum exactly to M and introduces spurious cases. Since Corollary 24 relies on this lemma, the proof of Corollary 24, and consequently the proofs of Lemma 11 and Lemma 1 that use it, need a corrected count or an alternative direct bound on the number of semi-directed paths.
minor comments (4)
  1. [Section 3.4, cases (S2a) and (S4a)] The displayed lower bounds in cases (S2a) and (S4a) appear to be missing the factor δ: the extra passage time should be δ/(4K(v+δ/2)) n and δ min(2,Kε/4)/(32K(v+δ/2)) n, respectively.
  2. [Section 3.3, Corollary 17] The definition of f*_p contains a typo: f*_p(n) should be ⌈vn⌉, not (n,⌈vn⌉).
  3. [Section 4.1, Lemma 21] The proof of Lemma 21 uses T both for the passage time and for the stopping time; renaming the stopping time, say to τ, would avoid confusion.
  4. [Section 2.1, Proposition 4] The proof of uniqueness for continuous G does not explicitly account for geodesics with different numbers of vertical edges; the equality of passage times then involves an integer offset in addition to the horizontal-weight sums, and this case should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the only flagged issue is a constant mismatch, which concerns correctness rather than circularity.

full rationale

The paper's derivation chain is self-contained relative to its stated external inputs. The Main Theorem is reduced, via the geometry of the limit-shape boundary (Definition 1 and convexity), to the dichotomy: either the time constant eventually equals the asymptote v + t0 (atom case, Proposition 14) or it stays strictly above v + t0 (no-atom case, Proposition 15). Proposition 14 uses Seppäläinen's exact directed shape result (Theorem 6) and a continuity-under-truncation argument (Lemma 11); Proposition 15 constructs a two-point Bernoulli comparison model Ber(G,v), defines the gap ε from that directed model, and then uses independent large-deviation estimates (Theorems 4 and 5) and the site-percolation bound (Lemma 13). None of these inputs is fitted to the conclusion: ε is derived from the comparison model, not from the unknown Λ(v), and the site-percolation result is external to the paper rather than a self-citation. The reviewer's flagged factor-of-two mismatch between (S5b) and the deficit in (17) is a concrete correctness issue in the printed constants, but it is not a circular step: if the bound fails, the proof of Proposition 15 is incomplete, but it does not reduce the theorem to its own assumptions or to a self-referential definition. The references contain no load-bearing self-citations by the author, and no known result is merely renamed as a new one. Therefore the paper should receive score 0 for circularity, with the constant mismatch left to a correctness assessment rather than a circularity finding.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's proof uses no fitted parameters. The central claim rests on standard FPP theorems (Seppäläinen, Cox, Kesten, Liggett et al.) and on the paper's own large deviation and concentration estimates, which are derived within the paper. The random shear maps and macroscopic site percolation are mathematical proof devices, not proposed physical entities.

assumptions (6)
  • standard math Subadditive ergodic theorem of Kingman/Liggett
    Used in Section 4.1 to prove existence of the time constant (Theorem 1).
  • standard math Seppäläinen's exact limit shape for directed SJ-model with two-point distributions (Theorem 6, [10])
    Crucial for Proposition 14 and the Bernoulli comparison in Proposition 18.
  • standard math Cox's continuity of the time constant for Bernoulli FPP and Kesten's large deviation results for FPP ([3], [8])
    Used in Lemma 10 and Lemma 13 for continuity and large deviations of the site percolation model.
  • standard math Liggett-Schonmann-Stacey domination by product measures ([9])
    Used in the proof of Lemma 13 to compare dependent site percolation with a product measure.
  • standard math Concentration inequalities (Boucheron-Lugosi-Massart entropy tensorization) [16]
    Used in the proof of Theorem 8 to establish the concentration bound for passage times.
  • domain assumption Model definition: vertical edges have deterministic weight 1, horizontal edges are i.i.d. with distribution G on [0,∞)
    This defines the model studied in the paper (Section 1.1).

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Cite this review

Pith. "Pith review of Classification of the limit shape for 1+1-dimensional FPP." pith.science (2026). https://pith.science/paper/MCNDVOQX

@misc{pith2026241113030,
  author       = {Pith},
  title        = {Pith review of: Classification of the limit shape for 1+1-dimensional FPP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MCNDVOQX}},
  note         = {Machine review of arXiv:2411.13030}
}
read the original abstract

We introduce a simplified model of planar first passage percolation where weights along vertical edges are deterministic. We show that the limit shape has a flat edge in the vertical direction if and only if the random distribution of the horizontal edges has an atom at the infimum of its support. Furthermore, we present bounds on the upper and lower derivative of the time constant.

Figures

Figures reproduced from arXiv: 2411.13030 by the authors.

Figure 1
Figure 1. A semi-directed path p from the origin to (n, ⌈vn⌉) with n = 7 and v = 0.7. Pioneer points are shown in green and fp = (0, 2, 2, 1, −1, −1, 4, 6, 5). 2.3. An equivalent model. We now rewrite our model in a way that resembles the polymers studied in [19]. For each semi-directed, non-intersecting path p from the origin to (n, ⌈vn⌉), define γ = γ(p) = (fp(0), ..., fp(n + 1)). See [PITH_FULL_IMAGE:figures/full_fig_p006… view at source ↗
Figure 2
Figure 2. A semi-directed path starting from the origin together with the associated macroscopic vertices from AK (blue) and A′ K (red). Shaded regions indicate a trapezoidal crossing. We will now make a case distinction on γ when γ is the geodesic and show that in each case T(γ) = T((0, 0),(n, ⌈vn⌉)) ≥ (t0 + v + ε0)n for some ε0 > 0 surely or with exponentially high probability in n. By Theorem 4, this yields Proposition 15.… view at source ↗

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Reference graph

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22 extracted references · 22 canonical work pages

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