REVIEW 2 major objections 4 minor 1 cited by
Noise sensitivity and variance lower bound for minimal left-right crossing of a square in first-passage percolation
T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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 $\operatorname{Var}(T_n)\geq c n^{1/4-\epsilon}$.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Section 2.3, Proposition 2.7] 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.
- [Theorem 1.3(1) and its proof in Section 3] 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.
minor comments (4)
- [Theorem 1.2] 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)).
- [Proposition 4.1] 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)).
- [Notation throughout] 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 2 and Theorem 1.3] 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.
Circularity Check
No significant circularity: the derivation is self-contained, and cited prior-work lemmas are general results that do not assume the target noise-sensitivity claim.
full rationale
The derivation chain is self-contained with respect to the central claims. The small-ball estimates (Propositions 2.2 and 2.6), the variance lower bound (Theorem 1.3), and the influence bound feeding the BKS criterion (Proposition 4.1) are proved from the model assumptions (ATO), (ABS), (EXP), and the geometric assumptions (≥s sides) and (UC). No fitted constant is renamed as a prediction, and the target events A_n(α,k_n) do not appear as inputs to their own proof. The cited tools from the authors' earlier work, namely Lemma 2.1 (a Mermin–Wagner type estimate from [10]) and Proposition 5.5 (a geodesic directional estimate from [10, Proposition 3.1]), are general, parameter-free results whose stated assumptions do not include noise sensitivity or the variance lower bound, so they count as independent support under the review rules; [12] is used only for ideas in the small-ball argument, not as an unverified load-bearing premise. The main external input, the strict time-constant inequality from [4, Theorem 2.12], is a standard result from a published monograph. One apparent local defect should be recorded as a correctness risk rather than a circularity: Proposition 2.7 is printed with the inequality P(∃ geodesic γ with |{e∈γ: t_e=b}| ≥ αn) ≤ Ce^{-cn}, while its proof and Lemma 2.8 require the opposite high-probability statement; if the reversal is not a typographical error, the atomic small-ball step is unsupported as printed, but this is a proof-support gap, not a reduction of the conclusion to its inputs. No self-definitional, fitted-input, uniqueness-importation, or renaming circularity appears.
Assumptions & free parameters
assumptions (8)
- standard math BKS theorem characterizing noise sensitivity via sum of squared influences.
- standard math Mermin-Wagner type estimate, Lemma 2.1, from [10, Lemma 2.12]: existence of monotone bijections g_τ with the hypercontractive inequality (2.4).
- standard math Strict time constant inequality µ^+(e1)>µ(e1) from [4, Theorem 2.12], based on van den Berg-Kesten and Marchand.
- standard math Concentration of passage times, Theorem 5.3 from [10] (Talagrand plus Alexander), valid under (ABS)+(EXP) or (ATO).
- domain assumption Assumption (ATO): weights are in {a,b} with 0<a<b<∞.
- domain assumption Assumption (≥s sides): the limit shape is not a polygon with fewer than s sides.
- domain assumption Assumption (UC): positive curvature µ(e1+h e2)-µ(e1) ≥ c h^2 for small h.
- standard math Proposition 5.5 from [10, Proposition 3.1] on geodesics avoiding wrong directions under (≥s sides).
Cite this review
Pith. "Pith review of Noise sensitivity and variance lower bound for minimal left-right crossing of a square in first-passage percolation." pith.science (2026). https://pith.science/paper/LQLXTNLY
@misc{pith2026250503211,
author = {Pith},
title = {Pith review of: Noise sensitivity and variance lower bound for minimal left-right crossing of a square in first-passage percolation},
year = {2026},
howpublished = {\url{https://pith.science/paper/LQLXTNLY}},
note = {Machine review of arXiv:2505.03211}
}
abstract
We study first-passage percolation on $\mathbb Z ^2$ with independent and identically distributed weights, whose common distribution is uniform on $\{a,b\}$ with $0<a<b<\infty $. Following Ahlberg and De la Riva, we consider the passage time $\tau (n,k)$ of the minimal left-right crossing of the square $[0,n]^2$, whose vertical fluctuations are bounded by $k$. We prove that when $k\le n^{1/2-\epsilon}$, the event that $\tau (n,k)$ is larger than its median is noise sensitive. This improves the main result of Ahlberg and De la Riva which holds when $k\le n^{1/22-\epsilon }$. Under the additional assumption that the limit shape is not a polygon with a small number of sides, we extend the result to all $k\le n^{1-\epsilon }$. This extension follows unconditionally when $a$ and $b$ are sufficiently close. Under a stronger curvature assumption, we extend the result to all $k\le n$. This in particular captures the noise sensitivity of the event that the minimal left-right crossing $T_n=\tau (n,n)$ is larger than its median. Finally, under the curvature assumption, our methods give a lower bound of $n^{1/4-\epsilon }$ for the variance of the passage time $T_n$ of the minimal left-right crossing of the square. We prove the last bound also for absolutely continuous weight distributions, generalizing a result of Damron--Houdr\'e--\"Ozdemir, which holds only for the exponential distribution. Our approach differs from the previous works mentioned above; the key idea is to establish a small ball probability estimate in the tail by perturbing the weights for tail events using a Mermin--Wagner type estimate.
Figures
Forward citations
Cited by 1 Pith paper
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Lower bounds on non-random fluctuations in planar first passage percolation
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.
Reference graph
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