{"id":"b45728e7-6324-4ff1-8c18-837df5b03be5","arxiv_id":"2505.03211","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For two-valued i.i.d. edge weights on Z^2, the event that the constrained left-right crossing time exceeds its median is noise sensitive up to width about n^{1/2}, improving the prior n^{1/22}, and conditional curvature assumptions extend this to all widths and yield a new variance lower bound.","lead":"This paper studies how much the fastest left-to-right path through a random grid changes when a small fraction of edge speeds are randomly resampled. It proves that for two-valued speeds, the event that the crossing time is above its median is highly sensitive to such noise up to much wider corridors than previously known, and it improves the state of the art for variance lower bounds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.7's displayed inequality is reversed; Lemma 2.8 needs the opposite high-probability event, so the atomic small-ball step is unsupported as printed.","rationale":"The paper's main unconditional theorem depends on Proposition 2.6, which depends on Lemma 2.8, which depends on Proposition 2.7. The printed Proposition 2.7 is internally inconsistent with its proof and with Lemma 2.8: the proof's first line establishes the complement of the displayed inequality. This is not a disagreement with the time-constant argument; the strict inequality mu_+ > mu is standard, and the concentration argument around (2.7)-(2.8) is plausible. The issue is that, as written, the proof chain contains a false premise: Lemma 2.8 treats {|E| >= alpha n} as a high-probability event, while the displayed Proposition 2.7 makes it exponentially unlikely. Because the correction is mechanical and the proof of Proposition 2.7 itself establishes the needed direction, this is a conditional-accept situation rather than a rejection. No other issue found is as load-bearing: Lemma 2.4 has a similar factor-2 slip in the continuous case, and the passage from {tau <= q} to {tau >= q} needs a small atom-mass check, but both are patchable and do not affect the main atomic noise-sensitivity claim.","tokens_in":21768,"tokens_out":33556,"duration_ms":326162,"concrete_test":"Re-derive Proposition 2.7's statement from its proof: replace the displayed '>= alpha n' by '<= alpha n' and confirm the proof's first inequality P(exists geodesic with <= alpha n b-edges) <= P(T_+(n,k) <= T(n,k) + alpha n) then matches the statement actually proved. Next, re-run Lemma 2.8 under this corrected statement: on the high-probability event {|E| >= alpha n}, verify that (2.5) and the decrement bound yield P(T(n,k) - T_{-r}(n,k) <= c r sqrt(n/k)) <= e^{-c r sqrt(n/k)}. If the printed '>=' version is used verbatim, the event in Lemma 2.8 has exponentially small probability and the binomial estimate is applied on a negligible event, so Proposition 2.6 and Theorem 1.2(1) are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The unconditional result Theorem 1.2(1) rests on the atomic small-ball estimate Proposition 2.6, whose proof (Lemma 2.8) requires that, with probability exponentially close to 1, every geodesic for T(n,k) contains at least alpha n b-edges. Proposition 2.7 is printed with the reverse inequality: it asserts P(exists geodesic gamma with |{e in gamma: t_e = b}| >= alpha n) <= C e^{-c n}, i.e. that many b-edges are unlikely. The proof of Proposition 2.7 actually bounds the opposite event: it starts with P(exists geodesic gamma with |{e in gamma: t_e = b}| <= alpha n) and shows this implies T_+(n,k) <= T(n,k) + alpha n, which is exponentially unlikely via the strict time-constant inequality from [4, Theorem 2.12]. Lemma 2.8 then uses the intended statement to ensure |E| >= alpha n before applying the binomial estimate (2.5). As printed, the event {|E| >= alpha n} has probability at most C e^{-c n}, so the conditional decrement bound T(n,k) - T_{-r}(n,k) >= c r sqrt(n/k) cannot be derived; the Mermin-Wagner perturbation argument for atomic weights is unsupported. This is the load-bearing premise for Theorem 1.2(1).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies first-passage percolation on Z^2 with i.i.d. weights taking two values a<b. For the constrained left-right crossing time tau(n,k), whose vertical fluctuations are bounded by k, the authors prove that the event {tau(n,k) >= q_alpha(tau(n,k))} is noise sensitive for k up to n^{1/2} e^{-C sqrt(log n)}, improving the earlier n^{1/22} range of Ahlberg and De la Riva. Conditional on limit-shape hypotheses, they extend this to k <= n^{1-epsilon} and, under a curvature assumption, to k <= n, including the unrestricted crossing T_n. They also prove variance lower bounds, in particular Var(T_n) >= n^{1/4-epsilon} under curvature and an exponential moment assumption, generalizing a result of Damron, Houdre, and Ozdemir from exponential weights to general absolutely continuous or two-point distributions. The proof strategy is to establish small-ball probability estimates in the tail by perturbing edge weights via a Mermin-Wagner type estimate, then convert these into influence bounds and apply the BKS criterion.","tokens_in":22032,"tokens_out":30611,"duration_ms":294261,"significance":"If the results are correct, this is a substantial advance: the noise-sensitivity range is improved from n^{1/22} to essentially n^{1/2}, the variance lower bound is extended to a much broader class of weight distributions, and the Mermin-Wagner perturbation technique is adapted to atomic distributions in a nontrivial way. The proof architecture is coherent and the paper is transparent about which ingredients come from previous work [10,12]. However, the central unconditional claim rests on a small-ball estimate whose proof, as printed, uses a proposition with the wrong inequality direction. That issue is load-bearing and must be fixed before the results can be accepted.","major_comments":[{"comment":"The displayed inequality in Proposition 2.7 is reversed. The proposition states P( exists a geodesic gamma for T(n,k) with |{e in gamma : t_e = b}| >= alpha n ) <= C e^{-cn}, but the proof immediately bounds the opposite event P( exists a geodesic gamma with |{e in gamma : t_e = b}| <= alpha n ), using the strict time-constant inequality mu^+(e1) > mu(e1). Lemma 2.8 and hence the atomic small-ball estimate Proposition 2.6 require the high-probability event |{e in gamma : t_e = b}| >= alpha n. As printed, Proposition 2.7 makes that event exponentially unlikely, so the derivation of Lemma 2.8 and Proposition 2.6 is unsupported. This is the key step behind Theorem 1.2(1) and Proposition 4.1. The fix is almost certainly to replace the '>=' in the statement by '<=', matching the proof and the subsequent use, but the printed version must be corrected and any dependent statements rechecked.","section":"Section 2.3, Proposition 2.7"},{"comment":"The quantitative statement of Theorem 1.3(1) is ambiguous and appears inconsistent with the proof. The text reads as e^{-C sqrt(log n)/k} (n/k) or e^{-C sqrt(log n)} k (n/k), but the optimization in the proof of Theorem 1.3 produces a bound of the form e^{-C sqrt(log(n/k))} (n/k). This is a substantive difference, especially when k is close to n. The theorem should state the bound with an explicit log(n/k) so that the displayed result matches the derivation.","section":"Theorem 1.3(1) and its proof in Section 3"}],"minor_comments":[{"comment":"In the definition of A_n(alpha,k_n), the argument of the quantile is written as q_alpha(tau(k_n,n)); it should be q_alpha(tau(n,k_n)).","section":"Theorem 1.2"},{"comment":"The quantile in the displayed event is written as q_alpha(tau(n,k)) but should depend on k_n, namely q_alpha(tau(n,k_n)).","section":"Proposition 4.1"},{"comment":"Expressions such as 'log n/k' are used in several places where context indicates log(n/k). Please write log(n/k) explicitly to avoid confusion, particularly in Lemmas 2.11 and 4.4 and in the proof of Proposition 4.1.","section":"Notation throughout"},{"comment":"Section 2 begins with the standing assumption 1 <= k <= n/4, but Theorem 1.3(1) states a bound for all k <= n. The proof of Proposition 2.6 appears to require k <= n/4; please either state that restriction in Theorem 1.3(1) or explain how the range n/4 < k <= n is handled.","section":"Section 2 and Theorem 1.3"}],"recommendation":"major_revision","confidential_remarks":"The reversed inequality in Proposition 2.7 appears to be a typo rather than a fundamental flaw, since the proof establishes exactly the statement needed by Lemma 2.8. However, as submitted, the main unconditional theorem is not proved, so the revision is more than cosmetic. The manuscript should also be checked for similar inequality-direction or indexing slips in the theorem statements and displayed events."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main news: this paper gets the Ahlberg–de la Riva threshold for noise sensitivity of constrained left-right crossings in the two-point model from n^{1/22} up to roughly n^{1/2}, using a small-ball-in-the-tail estimate obtained by a Mermin–Wagner weight perturbation. That's a real advance, not a routine extension. The method is new to this problem and also buys a variance lower bound n^{1/4-epsilon} under (UC) for absolutely continuous weights, generalizing the exponential-only result of Damron–Houdré–Özdemir. The proofs are coherent and the unconditional part is the main event.\n\nThe one load-bearing typo: Proposition 2.7 is printed with the inequality reversed. It claims P(there is a geodesic with at least alpha n b-edges) is exponentially small; the proof actually demonstrates the complementary event, that having at most alpha n b-edges is exponentially unlikely, and Lemma 2.8 uses the complementary statement to feed the atomic small-ball estimate. As printed, the small-ball argument for atomic weights does not go through. The fix is straightforward—swap the inequality in the statement—but a referee should require it. I do not see any other issues of that magnitude. The (UC) assumption is unverified for any distribution, so the all-k and variance results are conditional; the paper says so plainly.\n\nBottom line: worth a serious referee. The unconditional improvement alone justifies publication once Proposition 2.7 is corrected. I'd bring it to reading group.","headline":"Improves the n^{1/22} noise-sensitivity threshold to n^{1/2-epsilon} in the two-point FPP model with a genuinely new Mermin–Wagner small-ball method, but Proposition 2.7's reversed inequality needs a correction before the atomic-weight argument is sound.","tokens_in":22581,"tokens_out":3077,"would_cite":true,"duration_ms":28682,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"In two-point first-passage percolation, above-quantile crossing events are noise sensitive for corridor widths up to about $n^{1/2}$, and under limit-shape assumptions for all widths.","keywords":["first-passage percolation","noise sensitivity","left-right crossing","Mermin-Wagner estimate","small ball probability","variance lower bound","two-point weight distribution","limit shape curvature"],"falsifier":"Compute, for two-point weights on $[0,n]\\times[0,2k]$ with $k\\approx n^{1/2}$, the probability that every minimal left-right crossing uses at least $\\alpha n$ heavy (weight-$b$) edges for a fixed $\\alpha>0$: if this probability does not tend to 1 exponentially, the small-ball argument at the $n^{1/2}$ scale fails. A direct check of the claimed conclusion would measure the covariance $E[1_{A_n}(t)1_{A_n}(t^\\epsilon)]-E[1_{A_n}(t)]^2$ for $\\epsilon=0.01$ and $k=n^{1/2}$; if it does not decay to 0 as $n$ grows, the noise-sensitivity statement is false.","tokens_in":21569,"feed_emoji":"🎲","tokens_out":15363,"duration_ms":120216,"temperature":0.7,"pith_summary":"In first-passage percolation on $\\mathbb{Z}^2$ with i.i.d. edge weights taking two values $a<b$, this paper asks whether the event that a constrained left-right crossing time lies above a fixed quantile becomes asymptotically independent from the same event after a tiny random resampling of edge weights. The main unconditional result is that for corridor width $k_n \\leq e^{-C\\sqrt{\\log n}} n^{1/2}$ the above-quantile event is noise sensitive, improving the known range from $k\\leq n^{1/22-\\epsilon}$ to essentially $n^{1/2}$ up to a subexponential factor; with extra limit-shape assumptions the conclusion extends to $k\\leq n^{1-\\epsilon}$ and then to all $k\\leq n$, covering the full-square crossing $T_n$. The same small-ball machinery gives $\\operatorname{Var}(\\tau(n,k)) \\geq e^{-C\\sqrt{\\log n}} n$ for $k\\leq n$ for atomic or absolutely continuous weights, and under curvature plus an exponential moment gives $\\operatorname{Var}(T_n) \\geq c n^{1/4-\\epsilon}$, generalizing a variance bound previously known only for exponential weights.","feed_headline":"Above-median crossing events are noise sensitive","feed_subtitle":"Tiny resampling of edge weights makes an above-median crossing event asymptotically unpredictable from the original","key_machinery":"The load-bearing object is a small-ball probability estimate for $T(n,k)$, the minimal passage time among left-right paths in the rectangle $[0,n]\\times[0,2k]$ whose vertical displacement is at most $k$. The estimate bounds $P(T(n,k)\\in[a,a+1])$ for $a$ in the upper tail by $C e^{C/\\epsilon} \\sqrt{k/n}\\, P(T(n,k)\\leq a+1)^{1-\\epsilon}$ plus an exponentially small error, with an extra $\\log(n/k)$ factor for atomic weights. To prove it, the paper uses a Mermin-Wagner type estimate: a change-of-measure inequality that allows a small deterministic drift $r/\\sqrt{kn}$ to be added to every edge weight in the cylinder, while controlling the probability of a tail event in the original law by probabilities in the drifted laws. Because the drifted crossing time moves by a bounded amount over a grid of drifts, the set of drifts for which $T(n,k)$ can land in a fixed unit interval has small measure, yielding the small-ball bound. This bound is then used twice: it forces enough probability mass away from the quantile to give the variance lower bound, and it controls the probability that an edge is pivotal for the above-quantile event, so the BKS influence criterion, which derives noise sensitivity from vanishing sum of squared edge influences, applies.","core_discovery":"The paper's central claim, Theorem 1.2, is that for the two-point weight distribution on $\\{a,b\\}$, for each $\\alpha\\in(0,1)$, the sequence $A_n(\\alpha,k_n)=\\{\\tau(n,k_n)\\geq q_\\alpha(\\tau(n,k_n))\\}$ is noise sensitive whenever $k_n\\leq e^{-C\\sqrt{\\log n}}n^{1/2}$, with $C$ depending only on $G$ and $\\alpha$. Noise sensitivity means that for every $\\epsilon>0$ the covariance between the indicator of $A_n$ evaluated on the original edge configuration and on a configuration where each edge is independently resampled with probability $\\epsilon$ tends to $0$. The proof achieves this through a small-ball estimate for the restricted crossing time in the tail, obtained by Mermin-Wagner type perturbations of edge weights rather than through the moderate-deviation estimates used by [1]. Under the hypothesis that the limit shape is not a polygon with few sides, the result holds for all $k\\leq n^{1-\\epsilon}$; under an additional positive-curvature hypothesis it holds for all $k\\leq n$, giving in particular the noise sensitivity of the event $\\{T_n\\geq q_\\alpha(T_n)\\}$. The same estimate gives $\\operatorname{Var}(\\tau(n,k))\\geq e^{-C\\sqrt{\\log n}}n$ under (ABS) or (ATO), and under (UC) plus an exponential moment gives $\\operatorname{Var}(T_n)\\geq c n^{1/4-\\epsilon}$.","pith_inferences":["The small-ball perturbation method should transfer to exactly solvable first-passage models with explicit limit shapes, such as rotationally invariant or harmonic models; confirming noise sensitivity of above-median crossings there would test the mechanism outside the two-point distribution.","The paper's own remark that an $O(1)$ bound on the intersection of a geodesic with any vertical line would push the unconditional result to $k\\leq e^{-C\\sqrt{\\log n}} n$ suggests the true threshold may lie far above $n^{1/2}$; a numerical study of total influence as a function of $k$ could indicate whether the $n^{1/2}$ cutoff is real or an artifact of the proof.","The same estimation scheme could be applied to other geometric observables, such as minimal surfaces or tube crossings, where a small-ball estimate in the upper tail would yield analogous noise sensitivity and variance statements; this is a testable extension the paper does not pursue."],"forward_implications":["For two-point weights $\\{a,b\\}$, the above-quantile crossing event $A_n(\\alpha,k_n)$ is noise sensitive whenever $k_n \\leq e^{-C\\sqrt{\\log n}} n^{1/2}$; this is unconditional and improves the prior $n^{1/22}$ range of [1].","If the limit shape is not a polygon with few sides, the same noise sensitivity holds for every $k\\leq n^{1-\\epsilon}$; if additionally the shape has positive curvature in the horizontal direction, it holds for every $k\\leq n$, which includes the event that the full left-right crossing $T_n$ lies above its median.","For every $k\\leq n$, $\\operatorname{Var}(\\tau(n,k)) \\geq \\exp(-C\\sqrt{\\log n})\\, n$ holds for both atomic two-point and absolutely continuous weight distributions; under curvature and an exponential moment, $\\operatorname{Var}(T_n)\\geq c n^{1/4-\\epsilon}$.","The $n^{1/4-\\epsilon}$ variance lower bound for $T_n$ generalizes the earlier exponential-distribution result of [8] to all absolutely continuous weight distributions with an exponential moment.","In dimension $d$ the same proof gives $\\operatorname{Var}(\\tau(n,k)) \\geq \\exp(-C\\sqrt{\\log n})\\, k\\,(n/k^{d-1})$, so the variance mechanism is not special to two dimensions."],"supporting_citations":[{"why":"Defines the restricted crossing problem and proves noise sensitivity for $k\\leq n^{1/22}$; serves as the baseline and motivation that this paper improves.","marker":"[1]"},{"why":"Supplies the BKS influence criterion used to turn vanishing sum of squared influences into noise sensitivity.","marker":"[6, Theorem 1.3]"},{"why":"Provides the strict time-constant inequality $\\mu^+(e_1)>\\mu(e_1)$ that guarantees a positive density of heavy edges on geodesics for Proposition 2.7.","marker":"[4, Theorem 2.12]"},{"why":"Supplies the Mermin-Wagner type change-of-measure estimate that underlies the weight perturbations.","marker":"[10, Lemma 2.12 and Remark 2.15]"},{"why":"Contributes the tail small-ball probability ideas for point-to-point passage times adapted in Proposition 2.2.","marker":"[12]"},{"why":"Proves the variance lower bound for exponential weights that Theorem 1.3 generalizes to absolutely continuous distributions.","marker":"[8]"},{"why":"Supports the strict time-constant comparison behind Proposition 2.7.","marker":"[15]"},{"why":"Provides the strict inequality for the time constant when the environment is made more variable, feeding Proposition 2.7.","marker":"[17]"}],"fun_headline_variants":["Sharper noise sensitivity for minimal crossing in FPP","Variance lower bound of n^{1/4-ε} for crossing time","Mermin-Wagner trick extends FPP noise sensitivity","Two-point weights: noise sensitivity up to n^{1/2}","Variance bound extends to absolutely continuous weights"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing input for the unconditional result is that with probability exponentially close to 1, every geodesic for the restricted crossing $T(n,k)$ contains at least a fixed positive fraction of heavy (weight-$b$) edges; if that fails, the small-ball estimate and the noise-sensitivity conclusion at corridor width near $n^{1/2}$ collapse. The printed Proposition 2.7 states the opposite bound, while the proof and Lemma 2.8 use the direction stated here.","fun_headline_variants_meta":{"raw":{"variants":["Sharper noise sensitivity for minimal crossing in FPP","Variance lower bound of n^{1/4-ε} for crossing time","Mermin-Wagner trick extends FPP noise sensitivity","Two-point weights: noise sensitivity up to n^{1/2}","Variance bound extends to absolutely continuous weights"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001149,"raw_usage":{"total_tokens":4904,"prompt_tokens":1226,"completion_tokens":3678,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":842,"completion_tokens_details":{"reasoning_tokens":3593}},"tokens_in":842,"tokens_out":3678,"duration_ms":25480,"temperature":1.0,"reasoning_tokens":3593,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:58:08.218342+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for two-point weights on $[0,n]\\times[0,2k]$ with $k\\approx n^{1/2}$, the probability that every minimal left-right crossing uses at least $\\alpha n$ heavy (weight-$b$) edges for a fixed $\\alpha>0$: if this probability does not tend to 1 exponentially, the small-ball argument at the $n^{1/2}$ scale fails. A direct check of the claimed conclusion would measure the covariance $E[1_{A_n}(t)1_{A_n}(t^\\epsilon)]-E[1_{A_n}(t)]^2$ for $\\epsilon=0.01$ and $k=n^{1/2}$; if it does not decay to 0 as $n$ grows, the noise-sensitivity statement is false.","supporting_citations":[{"cited_title":"Is 'being above the median' a noise sensitive property?","cited_arxiv_id":"2308.16388","evidence_quote":"Defines the restricted crossing problem and proves noise sensitivity for $k\\leq n^{1/22}$; serves as the baseline and motivation that this paper improves."},{"cited_title":"Small ball probabilities for the passage time in planar first-passage percolation","cited_arxiv_id":"2406.10971","evidence_quote":"Contributes the tail small-ball probability ideas for point-to-point passage times adapted in Proposition 2.2."},{"cited_title":"Fluctuation bounds for first-passage perco- lation on the square, tube, and torus","cited_arxiv_id":null,"evidence_quote":"Proves the variance lower bound for exponential weights that Theorem 1.3 generalizes to absolutely continuous distributions."},{"cited_title":"Strict inequalities for the time constant in first passage percolation","cited_arxiv_id":null,"evidence_quote":"Supports the strict time-constant comparison behind Proposition 2.7."},{"cited_title":"Inequalities for the time constant in first-passage percolation","cited_arxiv_id":null,"evidence_quote":"Provides the strict inequality for the time constant when the environment is made more variable, feeding Proposition 2.7."}],"review_version":1}