{"id":"555f30b4-7f2a-4f93-aca6-49044a6aba66","arxiv_id":"2511.06166","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Planar FPP non-random fluctuations diverge at least as (log n)^{1/2−κ} for any κ>0, under absolute continuity plus either a non-polygonal limit-shape condition or near-deterministic weights.","lead":"This paper proves that in planar first-passage percolation, the non-random part of passage-time fluctuations—the gap between expected travel time and the large-scale time constant—must grow at least like (log n)^{1/2−ε} for any ε>0, under either of two explicit conditions on the edge-weight distribution. This improves the previous log-log lower bound and is the first such divergence result that works in arbitrary directions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Arbitrary-direction version of the main theorem rests on an unproved extension of Lemma 2; compactness of the limit shape does not transfer the midpoint-avoidance estimate.","rationale":"The paper's core mechanism — applying Lemma 1 to a high-probability event A, then comparing geodesics in the modified environment — is sound for x=ne1: the shell construction, the ∥τ∥^2 bound, and the annulus sum give the claimed c log(n)^{1/2−κ}. I checked Claim 1 and found a minor notational gap: the line '≥ T(0,n)(\\tildeω)' should read '≥ T^K(0,n)(\\tildeω)' because the chosen path γ(0,n)(ω) lies in K by A2; with that replacement the claim follows. So the horizontal case is not the weak point. The real soft spot is the extension to arbitrary directions. Lemma 2, the external midpoint-avoidance input, is quoted only for γ(−n,n), and the §3 assertion that the proof is 'similar' with uniform constants by compactness is not a proof. Compactness of the limit shape controls deterministic quantities like μ, but not a probabilistic vertex-hitting bound for geodesics; uniformness of constants there is an extra assumption. If [8] does not contain a direction-uniform version, the main theorem's strongest advertised novelty — arbitrary directions — is unsupported. This matches the Reader's weakest_assumption, so I recommend leaving the CONDITIONAL verdict unchanged pending verification of [8].","tokens_in":5458,"tokens_out":28582,"duration_ms":241322,"concrete_test":"Check Dembin–Elboim–Peled [8], Theorem 1.2 and its proof: does it yield, for every x∈Z^2 with |x|=n and every v∈Λ(n^{1/33}), the uniform bound P(v ∈ γ(−x,x)) ≤ C n^{−1/16} (or equivalently P(Λ(n^{1/33}) ∩ γ(−x,x)=∅)→1 with rate uniform in the direction of x), or is it stated only for x=(n,0)? If only the horizontal case is stated, verify whether the argument in [8] is explicitly direction-agnostic before accepting the arbitrary-direction assertion; if it is not, the proof of the main theorem should be restricted to coordinate-axis directions or the missing uniform estimate should be proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem claims a lower bound for all x∈Z^2. The proof in §3 is written for x=ne1 and then asserts 'the proof is similar for arbitrary directions' with constants uniform 'essentially due to compactness of the limit shape.' The load-bearing probabilistic input, Lemma 2 (from [8]), is stated only for γ(−n,n), i.e. horizontal endpoints, and its proof uses a vertex-hitting bound of order n^{−1/16} for that specific pair. Nothing in the preprint shows that [8] provides the same avoidance event P(Λ(|x|^{1/33}) ∩ γ(−x,x)=∅)→1 for non-horizontal x with constants uniform in direction. Compactness of the limit shape alone does not transfer a quantitative geodesic-avoidance estimate across directions: the constants could degenerate near directions where the limit shape is flat, and compactness does not force continuity of such constants. If Lemma 2 is genuinely unavailable for arbitrary directions, the theorem's headline claim of divergence in arbitrary directions is unproven, though the horizontal/vertical case would survive. This is an external-input gap that cannot be checked from the preprint alone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a lower bound of order (log |x|)^{1/2-κ} for the non-random fluctuation E[T(0,x)] - μ(x) in planar first passage percolation, for absolutely continuous weight distributions satisfying either (A1) (finite exponential moments and a limit shape that is not a polygon with at most 40 sides) or (A2) (weights supported in [1,1+ε_40]). The strategy follows Nakajima's inequality (2), combines it with the BKS midpoint result of Dembin-Elboim-Peled (Lemma 2) and a Mermin-Wagner type perturbation (Lemma 1). The proof constructs a positive-probability event on which modifying weights inside Λ(n^{1/33}) leaves the geodesic from -x to x unchanged but increases the passage time from 0 to x by c log(n)^{1/2-κ}; subadditivity converts this into the desired expectation lower bound. The central §3 derivation is written for x = n e_1 and asserts that arbitrary directions follow by compactness of the limit shape. The paper also discusses possible strengthenings and limitations in §1.3.","tokens_in":5714,"tokens_out":12442,"duration_ms":106990,"significance":"If the proof is completed, this is a notable result: it improves Nakajima's log log n lower bound to a power of log n, and it would be the first divergence of the non-random fluctuations in all directions. The argument is elegant and concise, and it relies on established external theorems rather than fitted parameters. The Mermin-Wagner lemma is used cleanly, and the summation estimate in Claim 1 is transparent. However, the arbitrary-direction claim rests on a form of the midpoint theorem that is not stated or proved in the manuscript, and the high-probability events A2 and A3 are only sketched. These are fixable, but until fixed the theorem's headline 'arbitrary directions' statement is not fully supported.","major_comments":[{"comment":"The main theorem is for all x∈Z^2, but Lemma 2 is stated only for γ(-n,n), and the proof of §3 is written only for x = n e_1. The sentence 'the proof is similar for arbitrary directions... essentially due to compactness of the limit shape' does not transfer the midpoint-avoidance estimate: compactness of the deterministic limit shape gives no quantitative control on the probability that the geodesic between -x and x avoids Λ(|x|^{1/33}) uniformly in the direction of x. If [8] does contain such a directional statement, it must be stated and cited; otherwise the theorem should be restricted to coordinate directions or supplied with a proof.","section":"§1.2 / Lemma 2 / §3"},{"comment":"The events A2 and A3 are asserted to have high probability without proof. A2 ('any geodesic between points in Λ(n) lies in Λ(Cn)') is attributed to a 'standard large deviation result' with no reference or verification, and A3 is said to follow from Lemma 3, but Lemma 3 only treats geodesics from 0 to ∂Λ(n), whereas A3 requires a statement for all pairs x,y∈Λ(n) and all relevant paths (or geodesic subpaths). The reduction to subcritical Bernoulli percolation and Kesten's large-deviation theorem needs to be written out. These events are used in Claims 1 and 2, so this is load-bearing; I expect it can be fixed.","section":"§3, events A2 and A3"},{"comment":"The proof of Claim 1 switches between T and T^K, and between γ and γ^K, without explicitly distinguishing the K-constrained objects. In particular, the path p is taken from the geodesic γ(0,n)(ω), but the claim concerns T^K(0,n)(ω). Since the argument also uses the assumption that γ(0,n)(ω) is contained in K (via A2), this can be clarified. This is a rigor issue rather than a mathematical error, but it should be fixed.","section":"§3, Claim 1 proof"}],"minor_comments":[{"comment":"The statement contains a typographical issue in the measure notation: ν^n(T_τ(A)) ≥ e^{-‖τ‖^2} ν^n(A)^2 should be typeset cleanly so that the exponent is unambiguous.","section":"Lemma 1"},{"comment":"After defining M_n, the sentence 'Since g_σ is increasing, we have ˜ω ≤ ω on Λ(n^{1/33}) and equal weights outside' should specify that this holds edge by edge, and that outside Λ(n^{1/33}) but inside K the weights are equal because τ_e = 0 there.","section":"§3, before Claim 1"},{"comment":"The union bound over Λ(n^{1/33}) uses the exponent n^{2/33 - 1/16}, which tends to 0 slowly; it would help to state explicitly that the n^{-1/16} bound is uniform over vertices in the box. This is a minor clarity point.","section":"Lemma 2 proof"},{"comment":"There are formatting artifacts in the title and abstract ('p assage', 'Fluctua tions').","section":"Abstract / title"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope and I saw no citation or attribution concerns. The central horizontal-axis argument appears sound and the paper is potentially publishable, but the arbitrary-direction theorem depends on an external input that is not stated in the manuscript, and the A2/A3 high-probability claims are under-justified. I would support publication if the author either supplies the directional midpoint estimate or restricts the statement accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe paper does one real thing: it improves the known lower bound on non-random fluctuations E[T(0,x)]−μ(x) from Nakajima's log log n (and constant-order in some regimes) to log^{1/2−κ} for absolutely continuous weights satisfying (A1) or (A2). The proof is not a routine application: the τ-deformation concentrated on annuli inside Λ(n^{1/33}) with the Mermin-Wagner Lemma 1, plus subadditivity, is a genuine new combination, and the central derivation in §3 checks out. I found no internal contradictions, no circularity, no free parameters. Lemma 1's use is appropriate and Claims 1 and 2 follow from the events defined. The paper is honest about its limits: the outlook explicitly says (A2) is hard to relax and higher dimensions give only a constant bound.\n\nThe soft spot is the arbitrary-directions claim. The main theorem asserts the bound for all x∈Z^2, but Lemma 2 (the BKS midpoint input) is recorded only for horizontal endpoints γ(−n,n). The paper then says 'the proof is similar for arbitrary directions' and that constants are uniform 'essentially due to compactness of the limit shape.' That is not enough. Compactness of the limit shape does not force uniform quantitative control on hitting probabilities for geodesics in near-flat directions; the constants could degenerate. If [8] actually proves a uniform midpoint estimate for all directions, the paper should say so and cite the precise statement. If not, the theorem should be narrowed to the horizontal/vertical (or finitely many lattice directions) or the estimate proved. The horizontal case is not in question, and that alone is already an improvement.\n\nA second, minor issue: A2 and A3 are dismissed as standard large-deviation/union-bound arguments. I believe they are standard, but the reader can't verify the constants without doing the work. A short proof or a precise reference would help.\n\nVerdict: it deserves a serious referee. The main idea is sound and the result is new. The referee should ask for the arbitrary-direction gap to be closed or the statement adjusted. For a reading group, it's worth the time—good example of transferring deep external results into a simple subadditivity argument.","headline":"Solid conditional improvement over Nakajima's log-log bound; the horizontal proof works, but the arbitrary-direction claim leans on an unproved extension of the BKS midpoint theorem.","tokens_in":6219,"tokens_out":5483,"would_cite":true,"duration_ms":48813,"reading_group":"yes","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":"The expected passage time in planar first passage percolation exceeds the time constant by at least (log |x|)^{1/2−κ} in every direction for a large class of edge weights.","keywords":["first passage percolation","non-random fluctuations","time constant","Mermin-Wagner type estimates","BKS midpoint problem","subadditivity","planar lattice","geodesic avoidance"],"falsifier":"Find a direction and a distribution satisfying (A1) or (A2) for which the probability that the geodesic from −x to x passes through a fixed small box around the origin does not vanish as |x|→∞; then the key event A1 would fail and the lower bound would not follow from this argument. A direct numerical check of E[T(0,x)] − μ(x) for large n on a finite grid would also reveal whether the claimed log^{1/2−κ} growth actually occurs.","tokens_in":5326,"feed_emoji":"⏱️","tokens_out":6829,"duration_ms":50925,"temperature":0.7,"pith_summary":"This paper proves that the non-random part of the fluctuation of the passage time in planar first passage percolation diverges as the distance grows, for a broad class of absolutely continuous edge-weight distributions. Concretely, the expected passage time E[T(0,x)] is always at least the time constant μ(x) plus c_κ (log |x|)^{1/2−κ}, for any κ>0 and every point x. This improves the previously known log-log lower bound and is the first such divergence result that holds in all directions. The proof uses a simple subadditivity identity that reduces the problem to a gap between two crossing times, and a probabilistic weight-perturbation argument that forces that gap to be large with positive probability.","feed_headline":"Planar FPP fluctuations: at least (log n)^{1/2} in every direction","feed_subtitle":"New lower bound on non-random fluctuation in first passage percolation, the first to work for all directions.","key_machinery":"The central identity is 2(E[T(0,x)]−μ(x)) ≥ E[T(−x,0)+T(0,x)−T(−x,x)], which follows from subadditivity of passage times and reduces the problem to showing that the sum of two one-sided crossing times typically exceeds the two-sided crossing time by a large amount. The proof then uses a Mermin–Wagner-type bijection (Lemma 1) that raises the weights of a positive fraction of edges in a tiny box by a small amount at a controlled probability cost, together with a midpoint-result (Lemma 2) guaranteeing that the two-sided geodesic avoids that box.","core_discovery":"The main theorem states that for any absolutely continuous edge-weight distribution satisfying either (A1) or (A2), there is a positive constant c_κ such that E[T(0,x)] − μ(x) ≥ c_κ (log |x|)^{1/2−κ} for all x∈Z^2 and all κ>0. The proof modifies the edge weights inside a small box around the origin using a measure-preserving bijection biased to increase the weights of a positive fraction of edges. With uniformly positive probability, this modification leaves the geodesic between two opposite points unchanged while increasing the passage time from the origin to one of them by at least c (log n)^{1/2−κ}. Subadditivity then converts this event into a lower bound on the expectation.","pith_inferences":["The proof is largely modular: only the midpoint-avoidance lemma is needed from the BKS-type machinery, so analogous lower bounds should hold for other planar random metric models where a similar midpoint estimate can be established.","The 'arbitrary directions' statement rests on a compactness argument; verifying the midpoint estimate directly for a non-horizontal pair would give a clean, concrete test of the paper's main claim.","A natural next step is to prove that every non-deterministic absolutely continuous distribution has a limit shape that is not a polygon, which would remove the technical condition in (A1) and widen the class of distributions covered.","If combined with recent noise-sensitivity results, the same weight-modification technique may also yield lower bounds on the random fluctuations, suggesting the two fluctuation components grow at a comparable rate."],"forward_implications":["For any absolutely continuous distribution satisfying (A1) or (A2), the gap E[T(0,x)] − μ(x) grows to infinity at least as fast as (log |x|)^{1/2−κ} in every direction.","This is the first result proving divergence of non-random fluctuations for arbitrary directions, not just coordinate axes.","If a local form of the absolute-deviation lower bound holds, the exponent improves to a strict 1/2, and under a uniform curvature assumption it becomes n^{1/128}.","The result applies both to distributions with finite exponential moments whose limit shape is not a low-sided polygon and to near-deterministic weights supported on [1,1+ε]."],"fun_headline_variants":["Fluctuations in FPP: at least sqrt(log n) in every direction","Planar FPP: non-random jitter hits sqrt(log n) floor","First lower bound for all directions in planar FPP","Sqrt(log n) bound: FPP fluctuations can't stay tiny","Breaking log(log n): FPP fluctuations reach sqrt(log n)"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The theorem depends on the lemma from the cited literature that the geodesic between two opposite points avoids a small box around the origin with probability tending to 1; that lemma is proved only for horizontal pairs in the cited work, and the extension to arbitrary directions is asserted by a compactness argument rather than shown.","fun_headline_variants_meta":{"raw":{"variants":["Fluctuations in FPP: at least sqrt(log n) in every direction","Planar FPP: non-random jitter hits sqrt(log n) floor","First lower bound for all directions in planar FPP","Sqrt(log n) bound: FPP fluctuations can't stay tiny","Breaking log(log n): FPP fluctuations reach sqrt(log n)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000149,"raw_usage":{"total_tokens":985,"prompt_tokens":652,"completion_tokens":333,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":396,"completion_tokens_details":{"reasoning_tokens":253}},"tokens_in":396,"tokens_out":333,"duration_ms":3623,"temperature":1.0,"reasoning_tokens":253,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T23:21:15.677270+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a direction and a distribution satisfying (A1) or (A2) for which the probability that the geodesic from −x to x passes through a fixed small box around the origin does not vanish as |x|→∞; then the key event A1 would fail and the lower bound would not follow from this argument. A direct numerical check of E[T(0,x)] − μ(x) for large n on a finite grid would also reveal whether the claimed log^{1/2−κ} growth actually occurs.","supporting_citations":[],"review_version":1}