Higher-order derivatives of first-passage percolation with respect to the environment

Dan Stefanica; Ivan Matic; Rados Radoicic

arxiv: 2504.08935 · v4 · pith:5MGQ6HSSnew · submitted 2025-04-11 · 🧮 math.PR

Higher-order derivatives of first-passage percolation with respect to the environment

Ivan Matic , Rados Radoicic , Dan Stefanica This is my paper

Pith reviewed 2026-05-22 19:48 UTC · model grok-4.3

classification 🧮 math.PR

keywords first-passage percolationhigher-order derivativespassage time variancetwo-point edge weightsenvironment derivatives

0 comments

The pith

The variance of the passage time in first-passage percolation equals an expression built from higher-order derivatives of the passage time with respect to the edge weights.

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

The paper works with first-passage percolation on a lattice where every edge weight is an independent random variable that takes only the two values a or b. It defines higher-order partial derivatives of the random passage time with respect to these individual edge weights. The central result states that the variance of the passage time can be rewritten exactly in terms of these derivatives. The authors then derive algebraic identities, bounds, and structural properties that the derivatives satisfy.

Core claim

When edge weights are i.i.d. and supported on the two-point set {a, b}, the passage time becomes a differentiable function of the environment. Higher-order derivatives of this function exist and satisfy the identity that the variance of the passage time equals a sum of squared or product terms involving these derivatives evaluated at the two possible weight configurations.

What carries the argument

Higher-order partial derivatives of the first-passage time function with respect to the individual edge weights under the two-point distribution.

If this is right

The variance of the passage time admits an explicit representation without summing over all possible paths.
The derivatives obey recursive relations and size bounds that follow from the two-point support.
Higher moments or other statistics of the passage time may admit similar derivative expansions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same derivative construction could be tested on models with more than two weight values by taking limits or approximations.
The approach supplies a calculus-based route to concentration inequalities that might complement geometric arguments used in the literature.

Load-bearing premise

The edge weights must be independent and supported exactly on two values so that the passage time can be differentiated with respect to each weight while remaining a well-defined random variable.

What would settle it

A direct numerical check on a small finite grid with weights in {a, b} that computes both the variance of the passage time and the proposed derivative expression and finds they differ.

read the original abstract

We introduce and study derivatives in first-passage percolation with edge weights given by i.i.d. random variables supported on ${a,b}$. We show that the variance of the passage time can be expressed in terms of these derivatives. We further analyze their structure and establish several fundamental properties and bounds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines higher-order derivatives for FPP on two-point weights and uses them to write an exact expression for the variance of the passage time.

read the letter

The main thing here is a new identity that writes the variance of the passage time directly in terms of higher-order derivatives with respect to the edge weights, but only when those weights sit on the two-point set {a, b}. By moving to discrete support the authors avoid the usual non-differentiability problems that appear with continuous weights, and they then derive some structural properties and bounds for the derivatives themselves. That is the concrete advance. If the algebra holds, it supplies an exact relation rather than another bound or approximation, which is uncommon in this area. The two-point restriction is the obvious limitation. It makes the derivatives well-defined but also confines the result to a narrow class of environments. Most work on first-passage percolation uses continuous or unbounded distributions, so it is not yet clear how much of the technique survives outside this case. The abstract gives no numerical checks or simple examples, which would have helped show whether the expressions are usable in practice. This is a paper for specialists already working on fluctuations or exact computations in percolation models. A reader who needs new identities for variances in random media might find the approach worth following up. It is not broad enough to interest a general probability audience. I would send it to referees. The claim is specific, the method differs from standard ones, and the central identity is worth checking in detail even if the scope stays limited.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces higher-order derivatives of the first-passage time T with respect to the edge weights in first-passage percolation on the lattice, where the weights are i.i.d. random variables supported on the two-point set {a, b}. The central claim is that Var(T) admits an exact expression in terms of these derivatives. The authors further analyze the structure of the derivatives and establish fundamental properties together with bounds.

Significance. If the main identity holds, the work supplies an exact, derivative-based formula for the variance of the passage time in a discrete-weight FPP model. This is potentially useful for fluctuation analysis, as the two-point support removes the non-differentiability obstacles that appear with continuous weights. The derivation proceeds directly from the min-plus structure of T rather than from external approximations.

minor comments (2)

[Abstract] The abstract states that the variance 'can be expressed in terms of these derivatives' but does not indicate the precise order of differentiation or the algebraic form of the identity; a single clarifying sentence would improve readability.
[Introduction] Notation for the higher-order derivatives (e.g., how mixed partials with respect to distinct edges are indexed) should be introduced explicitly in the first section where they appear, to avoid ambiguity when the same symbol is reused for different orders.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. No major comments appear in the report, so we have no specific points requiring point-by-point response.

Circularity Check

0 steps flagged

Derivation self-contained from model definitions

full rationale

The paper defines higher-order derivatives of the passage time T with respect to edge weights in the two-point {a,b} environment (which removes non-differentiability issues), then derives an exact algebraic expression for Var(T) in terms of those derivatives. This follows directly from the min-over-paths structure of T and the discrete support assumption, without any reduction to fitted parameters, self-citations, or imported uniqueness results. No load-bearing step collapses to its own input by construction; the relation is a new identity obtained from the definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard probabilistic assumptions for first-passage percolation and the introduction of new derivative concepts.

axioms (1)

domain assumption Edge weights are i.i.d. random variables with support on {a, b}.
This is stated in the abstract as the setting for the model.

pith-pipeline@v0.9.0 · 5564 in / 1021 out tokens · 77556 ms · 2026-05-22T19:48:36.684862+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

var(f) = ∑_{M⊆W,M≠∅} (p(1-p))^{|M|} (E[∂_M f])^2 (Theorem 1); ∂_j ϕ = ϕ∘σ_b^j − ϕ∘σ_a^j (eq. 3)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Higher-order environment derivatives on discrete two-point support for variance bounds

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

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Random coalescing geodesics in first-passage percolation

D. Ahlberg, C. Hoffman. Random coalescing geodesics in first-passage percolation, (2019) arXiv: 1609.02447

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S. Armstrong, P. Cardaliaguet, P. Souganidis. Error estimates and convergence rates for the stochastic ho- mogenization of Hamilton-Jacobi equations.Journal of the American Mathematical Society,27 (2), (2014) 479–540

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A. Auffinger, M. Damron. A simplified proof of the relation between scaling exponents in first-passage percolation.Ann. Probab.42 (3)(2014): 1197–1211

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Auffinger, M

A. Auffinger, M. Damron, J. Hanson. 50 years of first-passage percolation,Amer. Math. Soc., 2017

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Bakhtin, D

Y . Bakhtin, D. Dow. Differentiability of limit shapes in continuous first passage percolation models. (2024) arXiv:2406.09652

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R. Basu, S. Ganguly, A. Sly. Upper tail large deviations in first passage percolationComm. Pure Appl. Math. 74 (8), (2021) 1577–1640

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R. Basu, V . Sidoravicius, A. Sly. Rotationally invariant first passage percolation: concentration and scaling relations. (2023) arXiv:2312.14143v1

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I. Benjamini, G. Kalai, and O. Schramm, First passage percolation has sublinear distance variance,Ann. Probab.31(2003), 1970–1978

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A. Bonami. Etude des coefficients de Fourier des fonctions deL p(G),Annales de l’Institut Fourier.20(2) (1970) 335–02

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Corwin, P

I. Corwin, P. Ghosal, A. Hammond. KPZ equation correlations in time.Ann. Probab.49 (2)(2021), 832– 876

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Cox and R

J.T. Cox and R. Durrett. Some limit theorems for percolation processes with necessary and sufficient condi- tions,Ann. Probab.9, (1981) 583–603. HIGHER-ORDER DERIV ATIVES OF FIRST-PASSAGE PERCOLATION 29

work page 1981
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Damron, J

M. Damron, J. Hanson, P. Sosoe. Sublinear variance in first-passage percolation for general distributions. Probab. Theory Related Fields,163 (1)(2015), 223–258

work page 2015
[20]

Damron, J

M. Damron, J. Hanson. Bigeodesics in first-passage percolation.Comm. Math. Phys.349(2), (2017) 753– 776

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Davini, E

A. Davini, E. Kosygina, A. Yilmaz. Stochastic homogenization of nonconvex viscous Hamilton-Jacobi equations in one space dimension.Commun. Partial Differ. Equ.,49, (2023) 698–734

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Dembin, D

B. Dembin, D. Elboim, R. Peled. Coalescence of geodesics and the BKS midpoint problem in planar first- passage percolation.Geometric and Functional Analysis,34, (2024) 733–797

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J.M. Hammersley, D.J.A. Welsh. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory.Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif, Springer-Verlag, New York, 1965, 61–110

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C. Hoffman. Geodesics in first passage percolation.Ann. Appl. Probab.18 (5), (2008) 1944–1969

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Howard, C.M

C.D. Howard, C.M. Newman. Euclidean models of first-passage percolationProbab. Theory Related Fields 108, (1997) 153–170

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Johansson

K. Johansson. Shape Fluctuations and Random Matrices,Comm. Math. Phys.209, (2000) 437–476

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H. Kesten. On the speed of convergence in first-passage percolation,Ann. Appl. Probab.3, (1993) 296–338

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[28]

Krishnan, F Rassoul-Agha, T

A. Krishnan, F Rassoul-Agha, T. Seppalainen. Geodesic length and shifted weights in first-passage perco- lation.Communications of the American Mathematical Society3 (05), (2023) 209–289

work page 2023
[29]

Matic, J

I. Matic, J. Nolen. A sublinear variance bound for solutions of a random Hamilton-Jacobi equation.Journal of Statistical Physics,149 (2), (2012), 342–361

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[30]

Matic, R

I. Matic, R. Radoicic, D. Stefanica. Almost sure bunds for higher-order derivatives of first-passage percola- tion with respect to the environment.in preparation, (2025)

work page 2025
[31]

Newman, M.S.T

C.M. Newman, M.S.T. Piza. Divergence of shape fluctuations in two dimensions.Ann. Probab.23, (1995) 977–1005

work page 1995
[32]

Przybylowski

T. Przybylowski. KKL theorem for the influence of a set of variables, (2024) arXiv: 2404.00084

work page arXiv 2024
[33]

Rezakhanlou, J.E

F. Rezakhanlou, J.E. Tarver. Homogenization for stochastic Hamilton-Jacobi equations.Arch. Ration. Mech. Anal.,151(4), (2000) 277–309

work page 2000
[34]

Seppalainen

T. Seppalainen. Existence, uniqueness and coalescence of directed planar geodesics: proof via the increment-stationary growth process.Ann. Inst. Henri Poincare Probab. Stat.56 (3), (2020) 1775–1791

work page 2020
[35]

A. Tal. Tight bounds on the fourier spectrum of AC 0,32nd Computational Complexity Conference (CCC 2017), Leibniz International Proceedings in Informatics (LIPIcs),79, (2017), 15:1–15:31

work page 2017
[36]

Talagrand

M. Talagrand. On Russo’s approximate zero-one law.Ann. Probab.22, (1994) 1576–1587

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[37]

K. Tanguy. Talagrand inequality at second order and application to Boolean analysis.J Theor Probab33, (202) 692–714

work page
[38]

Tracy, H

C.A. Tracy, H. Widom. Level spacing distributions and the Airy kernel.Commun. Math. Phys.159, (1994) 151–174

work page 1994
[39]

Zygouras

N. Zygouras. Directed polymers in a random environment: A review of the phase transitions.Stochastic Process. Appl.,177, (2024) 104–431 BARUCHCOLLEGE, CITYUNIVERSITY OFNEWYORK Email address:ivan.matic@baruch.cuny.edu BARUCHCOLLEGE, CITYUNIVERSITY OFNEWYORK Email address:rados.radoicic@baruch.cuny.edu BARUCHCOLLEGE, CITYUNIVERSITY OFNEWYORK Email address:d...

work page 2024

[1] [1]

Random coalescing geodesics in first-passage percolation

D. Ahlberg, C. Hoffman. Random coalescing geodesics in first-passage percolation, (2019) arXiv: 1609.02447

work page internal anchor Pith review Pith/arXiv arXiv 2019

[2] [2]

Alberts, K

T. Alberts, K. Khanin, J. Quastel. The intermediate disorder regime for directed polymers in dimension 1+1. Ann. Probab.42(2014), 1212–1256

work page 2014

[3] [3]

Alexander

K.S. Alexander. Geodesics, bigeodesics, and coalescence in first passage percolation in general dimension. Electron. J. Probab.28(2023), 1–83

work page 2023

[4] [4]

Armstrong, P

S. Armstrong, P. Cardaliaguet, P. Souganidis. Error estimates and convergence rates for the stochastic ho- mogenization of Hamilton-Jacobi equations.Journal of the American Mathematical Society,27 (2), (2014) 479–540

work page 2014

[5] [5]

Auffinger, M

A. Auffinger, M. Damron. A simplified proof of the relation between scaling exponents in first-passage percolation.Ann. Probab.42 (3)(2014): 1197–1211

work page 2014

[6] [6]

Auffinger, M

A. Auffinger, M. Damron, J. Hanson. 50 years of first-passage percolation,Amer. Math. Soc., 2017

work page 2017

[7] [7]

Bakhtin, D

Y . Bakhtin, D. Dow. Differentiability of limit shapes in continuous first passage percolation models. (2024) arXiv:2406.09652

work page arXiv 2024

[8] [8]

R. Basu, S. Ganguly, A. Sly. Upper tail large deviations in first passage percolationComm. Pure Appl. Math. 74 (8), (2021) 1577–1640

work page 2021

[9] [9]

R. Basu, V . Sidoravicius, A. Sly. Rotationally invariant first passage percolation: concentration and scaling relations. (2023) arXiv:2312.14143v1

work page arXiv 2023

[10] [10]

W. Beckner. Inequalities in Fourier analysis,Ann. of Math.102(1975), 159–182

work page 1975

[11] [11]

Bena ¨ım and R

M. Bena ¨ım and R. Rossignol. Exponential concentration for first passage percolation through modified Poincar´e inequalities.Ann. Inst. Henri Poincar ´e Probab. Stat.44(2008), 544–573. MR 2451057

work page 2008

[12] [12]

Benjamini, G

I. Benjamini, G. Kalai, and O. Schramm, First passage percolation has sublinear distance variance,Ann. Probab.31(2003), 1970–1978

work page 2003

[13] [13]

A. Bonami. Etude des coefficients de Fourier des fonctions deL p(G),Annales de l’Institut Fourier.20(2) (1970) 335–02

work page 1970

[14] [14]

Chatterjee

S. Chatterjee. The universal relation between scaling exponents in first-passage percolation.Ann. of Math. 177 (2)(2013), 663–697

work page 2013

[15] [15]

Chatterjee

S. Chatterjee. Superconcentration and related topics.Springer Monographs in Mathematics, Springer, Cham, 2014

work page 2014

[16] [16]

Chatterjee, P.S

S. Chatterjee, P.S. Dey. Multiple phase transitions in long-range first-passage percolation on square lattices. Comm. Pure Appl. Math.,69, (2016) 203–256

work page 2016

[17] [17]

Corwin, P

I. Corwin, P. Ghosal, A. Hammond. KPZ equation correlations in time.Ann. Probab.49 (2)(2021), 832– 876

work page 2021

[18] [18]

Cox and R

J.T. Cox and R. Durrett. Some limit theorems for percolation processes with necessary and sufficient condi- tions,Ann. Probab.9, (1981) 583–603. HIGHER-ORDER DERIV ATIVES OF FIRST-PASSAGE PERCOLATION 29

work page 1981

[19] [19]

Damron, J

M. Damron, J. Hanson, P. Sosoe. Sublinear variance in first-passage percolation for general distributions. Probab. Theory Related Fields,163 (1)(2015), 223–258

work page 2015

[20] [20]

Damron, J

M. Damron, J. Hanson. Bigeodesics in first-passage percolation.Comm. Math. Phys.349(2), (2017) 753– 776

work page 2017

[21] [21]

Davini, E

A. Davini, E. Kosygina, A. Yilmaz. Stochastic homogenization of nonconvex viscous Hamilton-Jacobi equations in one space dimension.Commun. Partial Differ. Equ.,49, (2023) 698–734

work page 2023

[22] [22]

Dembin, D

B. Dembin, D. Elboim, R. Peled. Coalescence of geodesics and the BKS midpoint problem in planar first- passage percolation.Geometric and Functional Analysis,34, (2024) 733–797

work page 2024

[23] [23]

Hammersley, D.J.A

J.M. Hammersley, D.J.A. Welsh. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory.Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif, Springer-Verlag, New York, 1965, 61–110

work page 1965

[24] [24]

C. Hoffman. Geodesics in first passage percolation.Ann. Appl. Probab.18 (5), (2008) 1944–1969

work page 2008

[25] [25]

Howard, C.M

C.D. Howard, C.M. Newman. Euclidean models of first-passage percolationProbab. Theory Related Fields 108, (1997) 153–170

work page 1997

[26] [26]

Johansson

K. Johansson. Shape Fluctuations and Random Matrices,Comm. Math. Phys.209, (2000) 437–476

work page 2000

[27] [27]

H. Kesten. On the speed of convergence in first-passage percolation,Ann. Appl. Probab.3, (1993) 296–338

work page 1993

[28] [28]

Krishnan, F Rassoul-Agha, T

A. Krishnan, F Rassoul-Agha, T. Seppalainen. Geodesic length and shifted weights in first-passage perco- lation.Communications of the American Mathematical Society3 (05), (2023) 209–289

work page 2023

[29] [29]

Matic, J

I. Matic, J. Nolen. A sublinear variance bound for solutions of a random Hamilton-Jacobi equation.Journal of Statistical Physics,149 (2), (2012), 342–361

work page 2012

[30] [30]

Matic, R

I. Matic, R. Radoicic, D. Stefanica. Almost sure bunds for higher-order derivatives of first-passage percola- tion with respect to the environment.in preparation, (2025)

work page 2025

[31] [31]

Newman, M.S.T

C.M. Newman, M.S.T. Piza. Divergence of shape fluctuations in two dimensions.Ann. Probab.23, (1995) 977–1005

work page 1995

[32] [32]

Przybylowski

T. Przybylowski. KKL theorem for the influence of a set of variables, (2024) arXiv: 2404.00084

work page arXiv 2024

[33] [33]

Rezakhanlou, J.E

F. Rezakhanlou, J.E. Tarver. Homogenization for stochastic Hamilton-Jacobi equations.Arch. Ration. Mech. Anal.,151(4), (2000) 277–309

work page 2000

[34] [34]

Seppalainen

T. Seppalainen. Existence, uniqueness and coalescence of directed planar geodesics: proof via the increment-stationary growth process.Ann. Inst. Henri Poincare Probab. Stat.56 (3), (2020) 1775–1791

work page 2020

[35] [35]

A. Tal. Tight bounds on the fourier spectrum of AC 0,32nd Computational Complexity Conference (CCC 2017), Leibniz International Proceedings in Informatics (LIPIcs),79, (2017), 15:1–15:31

work page 2017

[36] [36]

Talagrand

M. Talagrand. On Russo’s approximate zero-one law.Ann. Probab.22, (1994) 1576–1587

work page 1994

[37] [37]

K. Tanguy. Talagrand inequality at second order and application to Boolean analysis.J Theor Probab33, (202) 692–714

work page

[38] [38]

Tracy, H

C.A. Tracy, H. Widom. Level spacing distributions and the Airy kernel.Commun. Math. Phys.159, (1994) 151–174

work page 1994

[39] [39]

Zygouras

N. Zygouras. Directed polymers in a random environment: A review of the phase transitions.Stochastic Process. Appl.,177, (2024) 104–431 BARUCHCOLLEGE, CITYUNIVERSITY OFNEWYORK Email address:ivan.matic@baruch.cuny.edu BARUCHCOLLEGE, CITYUNIVERSITY OFNEWYORK Email address:rados.radoicic@baruch.cuny.edu BARUCHCOLLEGE, CITYUNIVERSITY OFNEWYORK Email address:d...

work page 2024