{"id":"1db3c272-5c5c-4809-91fe-03793622a405","arxiv_id":"2411.13030","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a planar FPP model with deterministic vertical weights, the limit shape has a flat vertical edge if and only if the horizontal edge weight distribution charges its infimum.","lead":"A simplified first passage percolation model with random horizontal edge weights and deterministic vertical weights has a flat limit-shape edge exactly when the horizontal weight distribution has an atom at its smallest value. The paper also proves new bounds linking the time constant's derivatives to the geometry of geodesics.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The S5 threshold in the proof of Proposition 15 is off by a factor of 2: as written, (S5b) yields no deficit in (17), so Lemma 13 cannot be applied at that step.","rationale":"The reader's verdict identifies the same region of the proof as fragile, but frames the issue as the validity of Lemma 13 and the coarse-grained coupling. My stress-test finds the more immediate obstruction one step earlier: the counting in the S5 case, as printed, does not produce the deficit claimed in (17). The large deviation bound Lemma 13 itself is plausibly correct for small p, but the manuscript applies it to an inequality that is not currently established. I also note that Lemma 23 is false as stated, since for M=1, k=2 it gives 5 tuples instead of 4; however, that auxiliary counting lemma is not the main theorem's central load-bearing step, and a corrected stars-and-bars count would likely repair it. The main theorem is probably correct and the proof structure is coherent, but the manuscript must fix the factor-of-two inconsistency before Proposition 15 is rigorous. This supports the reader's CONDITIONAL verdict rather than a rejection.","tokens_in":21194,"tokens_out":21635,"duration_ms":258808,"concrete_test":"Re-derive the S5 case with the corrected threshold: replace (S5b) by D ≤ δn/(128K(v+δ/2)) and recompute the site count as N + V + 2D + 1 with Z ≥ δn/(32K(v+δ/2)). If the resulting bound is T(φ(γ)) ≤ N[1 + v/(2(v+δ)) − δ/(64(v+δ/2))], then apply Lemma 13 with ε = δ/(64(v+δ/2)) rather than δ/(27(v+δ/2)), and verify from Proposition 20 that P(Q_K) < p_ε for all sufficiently large K. If this re-derivation closes the gap, the central theorem stands and the issue is a typographical factor error; if not, Proposition 15 lacks a valid case analysis.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main theorem rests on Proposition 15, whose proof ends with the S5 case in Section 3.4. Let N = n/K, V = vn/(2K(v+δ)), and let D be the number of downward moves of the macroscopic path φ(γ). A semi-directed macroscopic path with horizontal displacement N and vertical displacement V visits N + V + 2D + 1 sites, because each downward move requires an extra upward move to preserve the net displacement. From (S4b), the number Z of zero-weight sites on φ(γ) satisfies Z ≥ δn/(32K(v+δ/2)). The text assumes D ≤ δn/(64K(v+δ/2)) in (S5b). Then 2D − Z ≤ δn/(32K(v+δ/2)) − δn/(32K(v+δ/2)) = 0, so the passage time of φ(γ) is at most N + V + 1, i.e. N(1 + v/(2(v+δ))), with no negative term. This contradicts the claimed bound (17), which contains −δ/(64(v+δ/2))N. That negative term would follow only if D ≤ δn/(128K(v+δ/2)); in that case 2D − Z ≤ δn/(64K(v+δ/2)) − δn/(32K(v+δ/2)) = −δn/(64K(v+δ/2)). Thus the printed constants in (S5b) and (17) are inconsistent. In addition, Lemma 13 is applied with ε = δ/(27(v+δ/2)), whereas the deficit in (17) is δ/(64(v+δ/2)); one must use ε no larger than the deficit. This is a concrete, fixable factor-of-two error, but it is load-bearing because S5b is the case that completes the proof that a distribution without an atom at t0 forces Λ(v) > t0 + v.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21477,"tokens_out":16939,"duration_ms":167124,"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":[{"comment":"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.","section":"Section 3.4, cases (S5a)/(S5b) and Eq. (17)"},{"comment":"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.","section":"Section 3.4, application of Lemma 13 after Eq. (17)"},{"comment":"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.","section":"Section 4.2, Lemma 23"}],"minor_comments":[{"comment":"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.","section":"Section 3.4, cases (S2a) and (S4a)"},{"comment":"The definition of f*_p contains a typo: f*_p(n) should be ⌈vn⌉, not (n,⌈vn⌉).","section":"Section 3.3, Corollary 17"},{"comment":"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.","section":"Section 4.1, Lemma 21"},{"comment":"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.","section":"Section 2.1, Proposition 4"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is attractive and the proof strategy is inventive. The factor-of-two error in the S5 case and the mismatch in the Lemma 13 parameter are localized, but they affect the concluding step of Proposition 15, so the revision should be carefully checked for similar constant bookkeeping errors in the macroscopic coupling. I do not see grounds for rejection, provided the constants are corrected and Lemma 23 is repaired or replaced."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's the short version: this paper is a real contribution, and the main theorem is likely true, but the proof as written has a load-bearing gap that is concrete and fixable. I agree with the conditional verdict.\n\nWhat's new: the model (undirected FPP with deterministic vertical weights) appears not to have been studied before; the Main Theorem gives a sharp iff: the vertical flat edge exists exactly when the horizontal weight distribution has an atom at the infimum of its support. That is a clean, exact classification in a planar FPP variant, which is rare. The random-shearing argument in Section 2 is a fresh idea, and the reduction to macroscopic site percolation in Section 3 is inventive. Theorem 3's derivative bounds are a useful byproduct, even though they are not needed for the main proof.\n\nThe soft spot is Section 3.4. The stress-test note is correct: in case (S5b), the printed threshold D ≤ δn/(64K(v+δ/2)) gives 2D − Z ≤ 0, so the passage time of the macroscopic path has no negative deficit and (17) does not follow. You would need D ≤ δn/(128K(v+δ/2)) to get the stated deficit. This is a factor-of-two error, but it is in the case that completes the no-atom direction, so Proposition 15 is not justified as written. I believe it is fixable—the proof structure is coherent—but it has to be fixed before the main theorem is accepted.\n\nThere are smaller issues. Lemma 23 is false as stated ('sum to M' should be 'at most M'); the proof and Corollary 24 both use the corrected version, so this is a typo-level fix. The S2a bound in the same section also looks off by a factor of δ (the text has 1/(4K(v+δ/2)) where it should have δ/(4K(v+δ/2))), and the application of Lemma 13 uses ε = δ/(27(v+δ/2)) when the deficit in (17) is only δ/(64(v+δ/2)), so the constants need to be reconciled. I do not share the reader's concern about Lemma 21—the exponential tail estimate seems valid—but it is worth a second look.\n\nBottom line: send it to a serious referee. The central claim is significant, the techniques are original, and the gaps are of the sort that referee reports exist to catch. No circularity, no data issues—it is a pure math paper that needs a repair pass, not a rewrite. I would welcome it at a reading group once the constants are cleaned up; as it stands, a reading group would spend most of its time on bookkeeping.\n\nBest,","headline":"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.","tokens_in":22128,"tokens_out":11847,"would_cite":true,"duration_ms":102449,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F10","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["first passage percolation","limit shape","flat edge","time constant","directed percolation","large deviations","geodesics","site percolation"],"falsifier":"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.","tokens_in":20862,"feed_emoji":"📐","tokens_out":13145,"duration_ms":115853,"temperature":0.7,"pith_summary":"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.","feed_headline":"Flat vertical edge appears iff weights have an atom at their minimum","feed_subtitle":"One atom in the horizontal weights decides whether the limit shape has a flat vertical edge; without it, the shape is not a polygon.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $[0,\\infty)$."],"supporting_citations":[{"why":"Supplies the exact limiting shape for the directed SJ-model, which the proof uses to locate the asymptote $t_0+v$ and to derive the flat-edge threshold when $G$ has an atom at $t_0$.","marker":"[10]"},{"why":"Gives continuity of the time constant for Bernoulli edge weights, used in Lemma 10 and in the site-percolation comparison.","marker":"[3]"},{"why":"Provides the continuity of the time constant under truncation of large weights, used in Lemma 11 and Proposition 14.","marker":"[5]"},{"why":"Supplies the right-tail large-deviation estimate for passage times, used as Theorem 4 to control geodesics that are too long or too costly.","marker":"[7]"},{"why":"Provides the standard FPP large-deviation results invoked in Lemma 13 for the site-percolation model.","marker":"[8]"},{"why":"Establishes the stochastic domination by product measures that couples the dependent edge weights of the site-percolation model to Bernoulli FPP.","marker":"[9]"},{"why":"Provides the entropy tensorization and symmetrized log-Sobolev inequality behind the concentration bound, from which the left-tail large deviations in Theorem 5 follow.","marker":"[16]"}],"fun_headline_variants":["Flat edge in FPP iff weights have atom at min","Atom in weights dictates flat limit shape edge","FPP limit shape flatness decided by single atom","One atom controls flat edge of FPP shape","Limit shape flat iff horizontal weights have atom"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Flat edge in FPP iff weights have atom at min","Atom in weights dictates flat limit shape edge","FPP limit shape flatness decided by single atom","One atom controls flat edge of FPP shape","Limit shape flat iff horizontal weights have atom"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1411,"prompt_tokens":782,"completion_tokens":629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":398,"completion_tokens_details":{"reasoning_tokens":557}},"tokens_in":398,"tokens_out":629,"duration_ms":6425,"temperature":1.0,"reasoning_tokens":557,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:00:51.974871+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"(1998)Exact limiting shape for a simplified model of first-passage percolation on the plane","cited_arxiv_id":null,"evidence_quote":"Supplies the exact limiting shape for the directed SJ-model, which the proof uses to locate the asymptote $t_0+v$ and to derive the flat-edge threshold when $G$ has an atom at $t_0$."},{"cited_title":"(1980) The time constant of first-passage percolation on the square lattice","cited_arxiv_id":null,"evidence_quote":"Gives continuity of the time constant for Bernoulli edge weights, used in Lemma 10 and in the site-percolation comparison."},{"cited_title":"and Kesten, H","cited_arxiv_id":null,"evidence_quote":"Provides the continuity of the time constant under truncation of large weights, used in Lemma 11 and Proposition 14."},{"cited_title":"and Kesten, H","cited_arxiv_id":null,"evidence_quote":"Supplies the right-tail large-deviation estimate for passage times, used as Theorem 4 to control geodesics that are too long or too costly."},{"cited_title":"(1986) Aspects of first passage percolation","cited_arxiv_id":null,"evidence_quote":"Provides the standard FPP large-deviation results invoked in Lemma 13 for the site-percolation model."},{"cited_title":"and Stacey, A.M","cited_arxiv_id":null,"evidence_quote":"Establishes the stochastic domination by product measures that couples the dependent edge weights of the site-percolation model to Bernoulli FPP."},{"cited_title":"(2013) Concentration inequalities","cited_arxiv_id":null,"evidence_quote":"Provides the entropy tensorization and symmetrized log-Sobolev inequality behind the concentration bound, from which the left-tail large deviations in Theorem 5 follow."}],"review_version":1}