{"id":"7af7b03f-1b56-43a2-91c2-db1445e42d49","arxiv_id":"2412.03320","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The rescaled first-passage percolation metric on the unit box satisfies a large deviation principle at speed n, with a rate function given by an integral against Hausdorff measure of a local cost.","lead":"This paper proves that in first-passage percolation, the random metric restricted to a box obeys a large deviation principle at speed n, giving the exponential rate of rare events where all distances are unusually small. The rate function is described in three equivalent forms, one of which is an integral of a local cost over the box.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The identification J^(∞)=J in the proof of Theorem 1.10 has an unjustified step: (5.40) does not imply the liminf bound is at least min_{K_ε} J.","rationale":"The reader's weakest-assumption analysis focused on the exponential moment condition (Moment), which is indeed necessary and explicitly discussed in Section 1.3; I do not see a flaw there. The more serious issue I find is in the proof of Theorem 1.10, where the step from (5.40) to the lower bound by min_{K_ε} J^(∞) is not justified by the preceding inequalities. This is a genuine gap in a key identification, but it appears repairable by a standard compactness-and-monotonicity argument, and the central theorem is likely true. The reader's verdict of CONDITIONAL is therefore appropriate, though for a different reason than the cross-reference typos in Lemma 5.4. Since my concern does not change the overall verdict, I mark verdict_should_be as UNCHANGED while noting the proof needs repair.","tokens_in":39188,"tokens_out":51345,"duration_ms":482623,"concrete_test":"Rewrite the converse inequality in the proof of Theorem 1.10 without the K_ε device: use the LDP upper bound on the closed set F_ε to get liminf_n −(1/n) log P(LD^-_n(D,ε)) ≥ inf_{F_ε} J^(∞), then prove directly that inf_{F_ε} J^(∞) → J^(∞)(D) as ε→0 by taking a minimizing sequence, using compactness of the sublevel {J^(∞) ≤ J^(∞)(D)+δ}, Proposition 4.4, and lower semicontinuity. If this direct argument succeeds, the concern is a localized proof gap; if inf_{F_ε} J^(∞) can remain strictly below J^(∞)(D), then the rate-function identification in Theorem 1.10 lacks a valid proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.10 (Section 5.3), after defining F_ε and K_ε = F_ε ∩ D^L_μ, the text claims from (5.40) that liminf_n −(1/n) log P(ˆT_n ∈ F_ε) ≥ min_{D'∈K_ε} J^(∞)(D'). The LDP upper bound for the closed set F_ε gives only liminf_n −(1/n) log P(ˆT_n ∈ F_ε) ≥ inf_{D'∈F_ε} J^(∞)(D'), and since K_ε ⊂ F_ε, one has inf_{F_ε} J^(∞) ≤ min_{K_ε} J^(∞), not the reverse. Inequality (5.40) only says that outside D^L_μ the rate is > J^-(D); it does not rule out points in F_ε \\ K_ε with J^(∞) strictly between J^-(D) and min_{K_ε} J^(∞), which would make inf_{F_ε} J^(∞) < min_{K_ε} J^(∞). The subsequent compactness argument therefore does not establish the intended conclusion J^-(D) ≥ J^(∞)(D). This step is load-bearing because the equality J = J^- on D_μ is the final identification of the LDP rate function. The gap appears patchable by a direct argument using compactness of the sublevel {J^(∞) ≤ J^(∞)(D) + δ} together with Proposition 4.4 and lower semicontinuity, but as written the proof is incomplete at this point.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies lower-tail large deviations for the rescaled first-passage percolation metric \\widehat T_n on the unit box. The main results are: (Theorem 1.1) existence of an elementary rate function J_pp for point-to-point lower-tail events, with convexity, monotonicity and a zero set governed by the time constant; (Theorem 1.7) under (SubC) and (StrongShape), a rate function J^- for events of the form {\\widehat T_n \\le D}, expressed through highway networks, as a supremum over disjoint Lipschitz paths, and as an integral against 1-dimensional Hausdorff measure of a local cost; and (Theorem 1.10) under (SubC) and (Moment), a full large deviation principle at speed n for (\\widehat T_n) with good rate function J equal to J^- on D_\\mu and infinite outside. The proofs use subadditivity and FKG for the elementary rate function, hub arguments and highway networks for the variational formulas, and exponential approximation by truncated passage times for the full LDP.","tokens_in":39554,"tokens_out":3778,"duration_ms":37447,"significance":"If the proof is completed, this is a substantial contribution: it gives the first metric-level large deviation principle at speed n for the lower tail of FPP, with explicit and mutually consistent formulas for the rate function, and it complements the author's earlier speed-n^d LDP for upper-tail deviations. The paper also correctly identifies the finite-exponential-moment assumption as essentially sharp, since exp(-t) tails already break the speed-n LDP. The treatment is detailed and self-contained, with a careful topological framework for pseudometrics, highway networks, and exponential approximation. The main results are clearly stated and the connection between the three expressions for J^- is nontrivial and well motivated.","major_comments":[{"comment":"The inference from (5.40) to 'lim ... ≥ min_{K_ε} J^(∞)' is not justified. Since F_ε = K_ε ∪ (F_ε \\ D^L_μ), inequality (5.40) only says that the infimum of J^(∞) over the complement of D^L_μ is strictly larger than J^-(D); it does not say that it is at least min_{K_ε} J^(∞). Points of F_ε outside D^L_μ could have J^(∞) strictly between J^-(D) and min_{K_ε} J^(∞), making inf_{F_ε} J^(∞) smaller than min_{K_ε} J^(∞). This step is load-bearing: it is the only argument giving the converse inequality J^-(D) ≥ J^(∞)(D), which identifies the LDP rate function as J^- on D_μ. The gap appears repairable by a direct compactness argument on sublevel sets of J^(∞) together with Proposition 4.4 and lower semicontinuity, but as written the proof is incomplete at this point.","section":"§5.3, proof of Theorem 1.10, Eq. (5.40)"}],"minor_comments":[{"comment":"The sentence 'let L > 0 be the number provided by Lemma 5.6' should refer to Lemma 5.2, not Lemma 5.6; Lemma 5.2 is the long-geodesic estimate used here, whereas Lemma 5.6 states the LDP for ~T^(b).","section":"§5.2, proof of Lemma 5.4"},{"comment":"There are several typographical errors ('satsifying', 'intger', 'Lispchitz', repeated missing spaces) that do not affect the mathematics but should be corrected in a revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"I concur with the stress-test concern: the gap in Section 5.3 is real but localized. The paper is otherwise strong, and the issue appears repairable within the manuscript's scope, so major revision rather than rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. It proves the speed-n LDP for the lower tail of the FPP metric—the natural complement to the author's earlier speed-n^d upper-tail result—and the three expressions for J^- are genuinely useful. The proof is mostly in good shape, but I found a real gap in the final identification step of Theorem 1.10 that the author needs to fix.\n\nWhat's new: Theorem 1.10 is the headline: under (SubC) and (Moment), the rescaled metric satisfies an LDP at speed n with good rate J. The point-point rate Jpp is extended to all directions. The three formulas for J^- — highway-network sum, sup over disjoint Lipschitz paths, and the Hausdorff-measure integral — are new, and the area-formula argument for the third is elegant. The construction of a highway network for any D in D_mu and the hub argument under (StrongShape) are solid. The paper is careful with moment assumptions; Section 1.3's discussion of why speed n fails for exp(-t) tails is honest and correct.\n\nThe soft spot: in Section 5.3, after defining F_epsilon and K_epsilon, the author claims that (5.40) lets him replace inf over F_epsilon with min over K_epsilon. It doesn't. (5.40) only says that outside D^L_mu the rate is > J^-(D), which does not rule out points in F_epsilon \\ K_epsilon with rate strictly between J^-(D) and min_{K_epsilon}. Since this inequality is the bridge to J^(infinity) = J, the proof is incomplete at that point. I think it is patchable (use compactness of the sublevel and Proposition 4.4), but the patch is not in the text. There is also the cross-reference typo in Lemma 5.4 (citing Lemma 5.6 instead of 5.2) and a garbled sentence nearby, but those are minor.\n\nRecommendation: send it to a serious referee. The main theorems are important and likely correct; a referee can push for the fix. I would not cite the LDP as-is until the gap is closed, but the rate-function formulas and the highway machinery should be citable.","headline":"This paper proves the natural speed-n lower-tail LDP for the FPP metric and is worth serious refereeing, but the final identification step in Theorem 1.10 has a genuine gap that needs patching.","tokens_in":40073,"tokens_out":4261,"would_cite":false,"duration_ms":36304,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the rescaled random metric in first-passage percolation satisfies a large deviation principle at speed $n$, with a rate function that vanishes exactly on the time-constant norm and is given by an integral of a local…","keywords":["first-passage percolation","large deviation principle","random metric","rate function","time constant","highway network","Hausdorff measure","lower tail"],"falsifier":"In $d=2$ with bounded weights (for instance $\\nu$ uniform on $[0,1]$), approximate the right-hand side of (1.31) numerically for the target $D=\\tfrac12\\mu$ by discretizing a highway network, then simulate $\\hat T_n$ on boxes up to $n=2000$ and estimate the slope of $\\log P(\\hat T_n \\le D+\\varepsilon)$; if the measured slope does not approach the computed integral as $n$ grows, the LDP rate is wrong.","tokens_in":39018,"feed_emoji":"📏","tokens_out":9820,"duration_ms":80664,"temperature":0.7,"pith_summary":"This paper establishes a large deviation principle for the random metric of first-passage percolation on $\\mathbb{Z}^d$. The object is $\\hat T_n(x,y)=\\frac1n T_{[0,n]^d}(\\lfloor nx\\rfloor,\\lfloor ny\\rfloor)$, the rescaled shortest passage time inside a box. The main theorem says that, when edge weights have all exponential moments, the probability that $\\hat T_n$ is close to a target pseudometric $D$ decays like $\\exp(-nJ(D))$, with $J(D)>0$ exactly when $D$ lies below the time-constant norm $\\mu$, and $J(D)=\\infty$ otherwise. Because $J$ is written as an integral of a local cost over the box, it turns a global random-geometry problem into a variational calculus problem. The result is a large deviation principle at speed $n$ for the metric-level object and completes, together with a known speed-$n^d$ LDP, the picture of large-deviation speeds for this model.","feed_headline":"Metric large deviations occur at speed n in first-passage percolation","feed_subtitle":"Below the time-constant norm, deviations from the random metric decay like exp(-nJ), with J an integral of local costs.","key_machinery":"The argument rests on three objects. The elementary rate function $J_{\\mathrm{pp}}(x,\\zeta)=\\lim_{n\\to\\infty}-\\frac1n\\log P(T(0,nx)\\le n\\zeta)$ prices point-to-point lower-tail deviations and is the 'integrand' of the more complex rate functions. A highway network for a pseudometric $D$ is a countable family of injective, pairwise disjoint $D$-geodesics whose closure reconstructs $D$ as a limit of metrics built by 'activating' these geodesics; it plays the role of a skeleton supporting the deviation event. The path gradient $(P\\text{-grad }D)_z(u)$ measures the infinitesimal $D$-length of a path leaving $z$ with velocity $u$, and Lemma 4.9 says that, almost everywhere, it equals the norm $\\mu$ except along highways in the tangent direction. FKG subadditivity, Jensen's inequality, the area formula, and a hub argument from the shape theorem connect these pieces.","core_discovery":"Under (SubC) and (Moment), Theorem 1.10 states that the random pseudometrics $(\\hat T_n)_{n\\ge1}$ satisfy the large deviation principle at speed $n$ with the good rate function $J(D)=J^-(D)$ for $D\\in\\mathcal D_\\mu$ and $J(D)=\\infty$ otherwise. Theorem 1.7 gives $J^-(D)$ three equivalent expressions (under the weaker assumption (StrongShape)): as the sum, along a 'highway network' of pairwise disjoint $D$-geodesics, of integrals of the elementary point-to-point rate function $J_{\\mathrm{pp}}$ against the metric derivative; as a supremum of the same integrals over all finite or countable families of disjoint 1-Lipschitz paths; and, most intrinsically, as the integral over $X$ of $\\max_{u\\in S^2} J_{\\mathrm{pp}}(u,(P\\text{-grad }D)_z(u))$ against the $1$-dimensional Hausdorff measure. All of these are finite only if $J^-(D)<\\infty$, and the rate function satisfies strict monotonicity: if $D_1\\le D_2$ and $J^-(D_2)<\\infty$, then $J^-(D_1)>J^-(D_2)$.","pith_inferences":["One could test whether the same highway-and-integral structure transfers to directed first-passage percolation or to chemical distance in supercritical percolation, where the local-cost integrand would change but the variational form over disjoint paths might persist; the paper does not explore this.","The strict monotonicity (1.32) suggests that $J$ orders the space $\\mathcal D_\\mu$ by cost, so one could use $J$ as a quantitative 'distance to $\\mu$' among pseudometrics, which the paper does not claim.","A concrete check: for small $d$ and bounded weights, approximate $J^-(D)$ by discretizing a highway network and computing the one-dimensional integral, then compare with Monte Carlo estimates of the exponential decay of $P(\\hat T_n\\le D+\\varepsilon)$; agreement would support, and disagreement would refute, the formula."],"forward_implications":["The probability of a lower-tail deviation $\\{\\hat T_n \\le D+\\varepsilon\\}$ decays as $\\exp(-(J^-(D)+o(1))n)$, so abnormally fast travel is exponentially costly with an explicit cost.","The rate function is strictly decreasing under pointwise order of metrics, so smaller target metrics cost strictly more whenever the larger one has finite cost.","Localization: because $J^-(D)$ is an integral of a local cost against Hausdorff measure, the global cost is additive along spacetime highways, which makes it possible to compare different targets term by term.","Moment threshold: all exponential moments are needed; with only $\\exp(-t)$ tails the speed-$n$ LDP cannot hold, so Theorem 1.10 cannot be extended to light-tailed but not exponentially integrable weights.","Together with a previously known LDP at speed $n^d$ for bounded weights, the two results imply no intermediate speed yields an LDP with a finite positive rate."],"supporting_citations":[{"why":"Supplies Kesten's lower-tail estimates (1.5) and the upper-tail superexponential bound under (Moment); Theorem 1.1's J_pp and Lemma 5.1 build on these.","marker":"[17]"},{"why":"Provides the shape theorem and the time constant mu, plus standard FPP estimates such as (2.4) and the hub-type lemma used in Appendix A.","marker":"[2]"},{"why":"Previous LDP at speed n^d for bounded passage times; the highway method of this paper is a refinement of it and is used for the three-speed comparison.","marker":"[22]"},{"why":"Supplies the general large-deviation toolkit: exponential equivalence, weak LDP, and passage from weak LDP plus exponential tightness to the LDP in Theorem 1.10.","marker":"[11]"},{"why":"The greedy-lattice-animal estimate that controls the error when truncating unbounded passage times, giving the exponentially good approximation in Lemma 5.9.","marker":"[12]"},{"why":"Cox-Durrett's proof of the shape theorem is adapted to prove the hub lemma (Lemma 4.5) under (StrongShape).","marker":"[8]"},{"why":"Continuity of the time constant under truncation, used in Lemma 3.3 to extend the positivity of J_pp to unbounded weights.","marker":"[14]"},{"why":"The area formula used to convert the highway-sum expression (1.29) into the Hausdorff-integral expression (1.31).","marker":"[18]"}],"fun_headline_variants":["First-passage metric obeys large deviations at speed n","Random metric in FPP: speed-n large deviation principle","New LDP for first-passage random metric at speed n","Rate function integral gives lower-tail metric deviations","Speed-n large deviations for first-passage percolation metrics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the edge-weight distribution has a finite exponential moment of every order; if the tail decays only like $\\exp(-t)$, the paper itself shows that no speed-$n$ LDP can hold, so dropping this assumption destroys Theorem 1.10.","fun_headline_variants_meta":{"raw":{"variants":["First-passage metric obeys large deviations at speed n","Random metric in FPP: speed-n large deviation principle","New LDP for first-passage random metric at speed n","Rate function integral gives lower-tail metric deviations","Speed-n large deviations for first-passage percolation metrics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000735,"raw_usage":{"total_tokens":3305,"prompt_tokens":987,"completion_tokens":2318,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":2238}},"tokens_in":603,"tokens_out":2318,"duration_ms":17077,"temperature":1.0,"reasoning_tokens":2238,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:32:26.574677+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In $d=2$ with bounded weights (for instance $\\nu$ uniform on $[0,1]$), approximate the right-hand side of (1.31) numerically for the target $D=\\tfrac12\\mu$ by discretizing a highway network, then simulate $\\hat T_n$ on boxes up to $n=2000$ and estimate the slope of $\\log P(\\hat T_n \\le D+\\varepsilon)$; if the measured slope does not approach the computed integral as $n$ grows, the LDP rate is wrong.","supporting_citations":[{"cited_title":"Ecole d’Ete de Probabilites de Saint Flour XIV, 1984","cited_arxiv_id":null,"evidence_quote":"Supplies Kesten's lower-tail estimates (1.5) and the upper-tail superexponential bound under (Moment); Theorem 1.1's J_pp and Lemma 5.1 build on these."},{"cited_title":"Auffinger, M","cited_arxiv_id":null,"evidence_quote":"Provides the shape theorem and the time constant mu, plus standard FPP estimates such as (2.4) and the hub-type lemma used in Appendix A."},{"cited_title":"Large deviation principle at speed $n^d$ for the random metric in first-passage percolation","cited_arxiv_id":"2404.09589","evidence_quote":"Previous LDP at speed n^d for bounded passage times; the highway method of this paper is a refinement of it and is used for the three-speed comparison."},{"cited_title":"Procaccia, and Marie Théret","cited_arxiv_id":null,"evidence_quote":"Continuity of the time constant under truncation, used in Lemma 3.3 to extend the positivity of J_pp to unbounded weights."},{"cited_title":"Krantz and H.R","cited_arxiv_id":null,"evidence_quote":"The area formula used to convert the highway-sum expression (1.29) into the Hausdorff-integral expression (1.31)."}],"review_version":1}