Lower bounds on non-random fluctuations in planar first passage percolation

Malte Hassler

REVIEW 3 major objections 4 minor 9 references

Lower bounds on non-random fluctuations in planar first passage percolation

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2511.06166 v2 pith:EFOSXSUH submitted 2025-11-08 math.PR

Malte Hassler This is my paper · ORCID

classification math.PR MSC 60K3560F1082B43

keywords firstpassagepercolationnon-randomfluctuationstimeconstantMermin-WagnertypeestimatesBKSmidpointproblemsubadditivityplanarlatticegeodesicavoidance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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].

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—

The 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.

The 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.

A 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.

Verdict: 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[§1.2 / Lemma 2 / §3] 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.
[§3, events A2 and A3] 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.
[§3, Claim 1 proof] 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.

minor comments (4)

[Lemma 1] 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.
[§3, before Claim 1] 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.
[Lemma 2 proof] 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.
[Abstract / title] There are formatting artifacts in the title and abstract ('p assage', 'Fluctua tions').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation relies on independent external theorems and contains no fitted-input or self-citation circularity.

full rationale

The paper's argument is self-contained in the relevant sense: it derives the lower bound on E[T(0,x)] - mu(x) from independent external inputs — Lemma 1 (Mermin-Wagner type estimate from Dembin–Elboim–Peled [8]), Lemma 2 (the BKS midpoint problem, also from [8]), Lemma 3 (from Kesten [3]), and the subadditivity identity E[T(-x,x)] >= mu(2x). None of these inputs is equivalent to the target conclusion. The main theorem's hypotheses (A1)/(A2) are assumptions used to invoke the external midpoint theorem, not consequences of the conclusion. There are no fitted parameters presented as predictions: the constants c_kappa and c_3 are obtained by explicit estimates, and the positive-probability event M_n is not calibrated to force the claimed lower bound. There is also no self-citation chain: the author does not cite any of their own prior work, and [8] is an independent external result. The only notable issue is that the main theorem is stated for all directions while the detailed proof is written for x = ne_1, with the manuscript asserting 'the proof is similar for arbitrary directions' and uniformity of constants 'essentially due to compactness of the limit shape.' That is a potential gap in proof coverage, not circularity: it concerns whether the external Lemma 2 or its uniform-direction analogue supports the claim, not whether the paper's conclusion is assumed among its inputs. Under the hard rules, missing or insufficient support is a correctness risk and does not count as a circular step unless the result is shown to reduce to its inputs by construction, which is not the case here.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof does no empirical fitting. All auxiliary constants are existential and chosen to satisfy inequalities; κ is part of the universal quantification. The scientific content rests on external theorems from [3] and [8], listed as axioms, plus the explicit hypotheses (A1)/(A2).

assumptions (5)

domain assumption Absolute continuity of ν gives a.s. uniqueness of geodesics, so γ(a,b) is well-defined.
Used in §1.4 and throughout the events A1/A2/A3.
standard math Time-constant/limit-shape theorem: T(0,xn)/n→μ(x) a.s. and the limit shape is compact and convex.
Invoked in §1 and for the compactness/uniformity claim in §3; from [1,2].
standard math Kesten's large-deviation theorem for supercritical Bernoulli percolation/FPP (Thm 5.2 in [3]).
Used in Lemma 3 and to justify A2/A3.
standard math Lemma 1: Mermin-Wagner-type estimate from [8, Lemma 2.12].
The core weight-deformation tool in §3; accepted as a quoted theorem.
standard math Lemma 2: BKS midpoint theorem from [8, Thm 1.2/1.5] under (A1) or (A2).
Provides the high-probability avoidance event A1; the deepest external input.

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Cite this review

Pith. "Pith review of Lower bounds on non-random fluctuations in planar first passage percolation." pith.science (2026). https://pith.science/paper/EFOSXSUH

@misc{pith2026251106166,
  author       = {Pith},
  title        = {Pith review of: Lower bounds on non-random fluctuations in planar first passage percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFOSXSUH}},
  note         = {Machine review of arXiv:2511.06166}
}

abstract

The fluctuations of the passage time in first passage percolation are of great interest. We show that the non-random fluctuations in planar FPP are at least of order $\sqrt{\log n}$ under some conditions that are known to be met for a large class of absolutely continuous edge weight distributions. This improves the ${\log(\log(n))}$ bound proven by Nakajima and is the first result showing divergence of the fluctuations for arbitrary directions. Our proof is an application of recent work by Dembin, Elboim and Peled on the BKS midpoint problem and the development of Mermin-Wagner type estimates.

Figures

Figures reproduced from arXiv: 2511.06166 by the authors.

**Figure 1.** The geodesics from (−n, 0) to (n, 0) are the same for both environments ω and ˜ω because they do not intersect Λ(n 1/33). But for geodesics going through the origin, the passage time for ω is larger. Since ˜ω ∈ AK 3 , the geodesic γ(n, 0)(ω) must go through edges e in each of the rectangular annuli of width n 1/66 where the edge weight is at least τe larger compared to ˜ω. References [1] Hammersley, J.M. and Welsh,… view at source ↗

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Reference graph

Works this paper leans on

9 extracted references · 1 linked inside Pith

[8]

(2024)Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

Dembin B., Elboim, D and Peled, R. (2024)Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation. Geom. Funct. Anal., Vol. 44, pp. 733–797

2024
[1]

and Welsh, D.J.A

Hammersley, J.M. and Welsh, D.J.A. (1965)First passage percolation, subadditive processes, stochastic net- works and generalized renewal theoryBernoulli–Bayes–Laplace Anniversary Volume

1965
[2]

and Durrett, R

Cox, J.T. and Durrett, R. (1981)Some limit theorems for percolation with necessary and sufficient conditions. Annals of Probability, Vol. 9, pp. 583–603

1981
[3]

(1986)Aspects of first passage percolation

Kesten, H. (1986)Aspects of first passage percolation. ´Ecole d’´ et´ e de probabilit´ es de Saint-Flour, XIV-1984, Lecture Notes in Math., Vol. 1180, pp. 125–264, Springer, Berlin

1986
[4]

Newman, C. M. and Piza, M. (1995)Divergence of shape fluctuations in two dimensions. Annals of Probab., Vol. 23, pp. 977–1005

1995
[5]

and Hanson, J

Auffinger, A., Damron, M. and Hanson, J. (2015)Rate of convergence of the mean for sub-additive ergodic sequences. Advances in Math., Vol. 285, pp. 138-181

2015
[6]

(2020)Lower bounds for fluctuations in first-passage percolation for general distributions

Damron, M., Hanson, J., Houdr´ e, C., Xu, C. (2020)Lower bounds for fluctuations in first-passage percolation for general distributions. Annales de l’Institut Henri Poincar´ e, Vol. 56.2, pp.1336–1357

2020
[7]

(2021)Divergence of non-random fluctuation for Euclidean first-passage percolation

Nakajima, S. (2021)Divergence of non-random fluctuation for Euclidean first-passage percolation. Stochastic Analysis, Random Fields and Integrable Probability, Vol. 87. pp. 363-380

2021

Show all 9 references

[9]

Dembin B. and Elboim, D (2025)Noise sensitivity and variance lower bound for minimal left-right crossing of a square in first-passage percolationpreprint arXiv:2505.03211 Cornell University, Department of Mathematics, Malott Hall, 212 Garden A venue, 14853 Ithaca, USA Email ad...

2025 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [8]

(2024)Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

Dembin B., Elboim, D and Peled, R. (2024)Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation. Geom. Funct. Anal., Vol. 44, pp. 733–797

2024

[2] [1]

and Welsh, D.J.A

Hammersley, J.M. and Welsh, D.J.A. (1965)First passage percolation, subadditive processes, stochastic net- works and generalized renewal theoryBernoulli–Bayes–Laplace Anniversary Volume

1965

[3] [2]

and Durrett, R

Cox, J.T. and Durrett, R. (1981)Some limit theorems for percolation with necessary and sufficient conditions. Annals of Probability, Vol. 9, pp. 583–603

1981

[4] [3]

(1986)Aspects of first passage percolation

Kesten, H. (1986)Aspects of first passage percolation. ´Ecole d’´ et´ e de probabilit´ es de Saint-Flour, XIV-1984, Lecture Notes in Math., Vol. 1180, pp. 125–264, Springer, Berlin

1986

[5] [4]

Newman, C. M. and Piza, M. (1995)Divergence of shape fluctuations in two dimensions. Annals of Probab., Vol. 23, pp. 977–1005

1995

[6] [5]

and Hanson, J

Auffinger, A., Damron, M. and Hanson, J. (2015)Rate of convergence of the mean for sub-additive ergodic sequences. Advances in Math., Vol. 285, pp. 138-181

2015

[7] [6]

(2020)Lower bounds for fluctuations in first-passage percolation for general distributions

Damron, M., Hanson, J., Houdr´ e, C., Xu, C. (2020)Lower bounds for fluctuations in first-passage percolation for general distributions. Annales de l’Institut Henri Poincar´ e, Vol. 56.2, pp.1336–1357

2020

[8] [7]

(2021)Divergence of non-random fluctuation for Euclidean first-passage percolation

Nakajima, S. (2021)Divergence of non-random fluctuation for Euclidean first-passage percolation. Stochastic Analysis, Random Fields and Integrable Probability, Vol. 87. pp. 363-380

2021