Large deviation principle at speed $n$ for the random metric in first-passage percolation

FP2M; Julien Verges (IDP; Modal'x)

REVIEW 1 major objections 2 minor 1 cited by

Large deviation principle at speed $n$ for the random metric in first-passage percolation

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2412.03320 v1 pith:OIGHZXKT submitted 2024-12-04 math.PR

Julien Verges (IDP , FP2M , MODAL'X) This is my paper · ORCID

classification math.PR MSC 60K3560F10

keywords first-passagepercolationlargedeviationprinciplerandommetricratefunctiontimeconstanthighwaynetworkHausdorffmeasurelowertail

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $J^-(D_1)>J^-(D_2)$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

What'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.

The 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.

Recommendation: 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.

Referee Report

1 major / 2 minor

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.

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 (1)

[§5.3, proof of Theorem 1.10, Eq. (5.40)] 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.

minor comments (2)

[§5.2, proof of Lemma 5.4] 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).
[Throughout] There are several typographical errors ('satsifying', 'intger', 'Lispchitz', repeated missing spaces) that do not affect the mathematics but should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rate functions are constructed by subadditivity and then verified by independent bounds; no fitted input is renamed as a prediction.

full rationale

The paper's derivation chain is not circular. The elementary rate function Jpp is defined in Theorem 1.1 by a subadditive/FKG argument applied directly to point-to-point passage times; it is not defined in terms of the metric rate functions J^- or J. The rate J^- is constructed from Jpp, highway networks, and the gradient-by-paths functional, and Theorem 1.7 proves the three expressions rather than assuming them. Theorem 1.10's rate function is identified through truncated processes and exponential approximation (Dembo--Zeitouni theory), with J^(∞)=J obtained from lower/upper bounds; no fitted parameter is renamed as a prediction and no quantity is equal to its input by definition. The citations to the author's prior work [22] concern a highway-network lemma and a standard weak-LDP lemma whose proof is said to be copied verbatim; they are not invoked as an unexplained uniqueness or ansatz, and the central result is otherwise proved in the text. The proof gap flagged around (5.40) is a correctness issue (the liminf bound does not follow from (5.40) alone), not a circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard FPP tools (FKG, BK, subadditivity) and imported external results (shape theorem, Kesten's path-mass estimate, Dembo-Gandolfi-Kesten greedy animal bound, geometric measure theory). No constants are fitted to data; no new physical entities are introduced.

assumptions (7)

standard math FKG inequality and BK inequality for i.i.d. edge weights
Used throughout Sections 3-5 to factor probabilities of simultaneous decreasing/increasing events; see e.g. proof of Theorem 1.1 (Section 3) and Lemma 4.8.
domain assumption Time constant convergence for FPP (shape theorem)
T(0,nx)/n converges to mu(x) in probability for all x, cited as [5] Theorem 4; used to identify J_pp(x,zeta)=0 iff zeta>=mu(x) (Lemma 3.3) and to set the limiting metric.
domain assumption Kesten's Proposition 5.8 on greedy path mass
Provides the exponential decay rate beta for long self-avoiding paths with small total weight; used in Lemma 5.2 to prove long geodesics are super-exponentially unlikely.
domain assumption Dembo-Gandolfi-Kesten Lemma 4.2 on greedy lattice animals
Controls the upper tail of the mass collected by a large animal; used in Lemma 5.9 to show the truncated metrics are exponentially good approximations.
standard math Geometric measure theory: area formula, metric derivative, Vitali covering theorem
Used in Section 2.2 and Lemma 4.9 to convert path integrals to Hausdorff-measure integrals and to extract the local expression of the rate function.
domain assumption Boundary assumptions (SubC), (Moment), (StrongShape)
These define the regime of validity: subcritical bond probability, finite exponential moments, and a d+xi moment of the minimum edge weight; stated in the introduction and Section 1.2.
standard math Large deviation framework (Dembo-Zeitouni)
Provides the definition of LDP, exponential tightness, and the weak-LDP-to-LDP transfer used in Section 5.

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Cite this review

Pith. "Pith review of Large deviation principle at speed $n$ for the random metric in first-passage percolation." pith.science (2026). https://pith.science/paper/OIGHZXKT

@misc{pith2026241203320,
  author       = {Pith},
  title        = {Pith review of: Large deviation principle at speed $n$ for the random metric in first-passage percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIGHZXKT}},
  note         = {Machine review of arXiv:2412.03320}
}

abstract

Consider standard first-passage percolation on $\mathbb Z^d$. We study the lower-tail large deviations of the rescaled random metric $\widehat{\mathbf T}_n$ restricted to a box. If all exponential moments are finite, we prove that $\widehat{\mathbf T}_n$ follows the large deviation principle at speed $n$ with a rate function $J$, in a suitable space of metrics. Moreover, we give three expressions for $J(D)$. The first two involve the metric derivative with respect to $D$ of Lipschitz paths and the lower-tail rate function for the point-point passage time. The third is an integral against the $1$-dimensional Hausdorff measure of a local cost. Under a much weaker moment assumption, we give an estimate for the probability of events of the type $\{\widehat{\mathbf T}_n \le D \}$.

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Ambrosio and P

L. Ambrosio and P. Tilli. Topics on Analysis in Metric Spaces . Oxford Lecture Mathematics and. Oxford University Press, 2004

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U pper tail large deviations in ﬁrst passage percolation

Riddhipratim Basu, Allan Sly, and Shirshendu Ganguly. U pper tail large deviations in ﬁrst passage percolation. Communications on Pure and Applied Mathematics , 74(8):1577– 1640, 2021. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.22010, arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.22010, doi:10.1002/cpa.22010

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Raphaël Cerf and Marie Théret. Weak shape theorem in ﬁrst passage percolation with inﬁnite passage times. Annales de l’Institut Henri Poincaré, Probabilités et Stat istiques, 52(3):1351 – 1381,

work page

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Large deviations in ﬁrst-pa ssage percolation

Yunshyong Chow and Yu Zhang. Large deviations in ﬁrst-pa ssage percolation. The Annals of Applied Probability, 13(4):1601 – 1614, 2003. doi:10.1214/aoap/1069786513

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A variational formula for large deviations in ﬁrst-passage percolation under tail estimates

Clément Cosco and Shuta Nakajima. A variational formula for large deviations in ﬁrst-passage percolation under tail estimates. The Annals of Applied Probability , 33(3):2103 – 2135, 2023. doi:10.1214/22-AAP1861

work page doi:10.1214/22-aap1861 2023

[8] [8]

Theodore Cox and Richard Durrett

J. Theodore Cox and Richard Durrett. Some Limit Theorems for Percolation Processes with Necessary and Suﬃcient Conditions. The Annals of Probability , 9(4):583 – 603, 1981. doi:10.1214/aop/1176994364

arXiv 1981