pith. sign in

arxiv: 2604.27519 · v1 · submitted 2026-04-30 · 🧮 math.PR

Quantitative homogenization of the maximal action of curves in a Brownian potential

Felix Otto , Matteo Palmieri This is my paper

Pith reviewed 2026-05-07 09:09 UTC · model grok-4.3

classification 🧮 math.PR
keywords quantitative homogenizationBrownian potentialmaximal actionvariational problemheight functionswhite noisescale invariance
0
0 comments X

The pith

The maximal action for curves in a Brownian potential equals a deterministic constant times ln L plus O(1) with overwhelming probability.

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

The paper proves a quantitative homogenization statement for a scale-invariant variational problem in one dimension. Piecewise-linear height functions on intervals of length 1 are used to maximize the difference between the white-noise area integral under the graph and the Dirichlet energy, with the functions fixed to zero at the endpoints 0 and L. With high probability the maximum value is exactly a fixed positive constant a times ln L, plus an additive error that remains bounded independently of L. A reader would care because the result supplies sharp large-scale control on optimal interfaces in random media, directly relevant to optimal matching and random-field Ising models, and it improves prior qualitative homogenization by achieving local uniformity in the boundary data.

Core claim

We consider the variational problem of maximizing the action given by the integral of white noise over the subgraph minus the Dirichlet integral of the continuum height function h(x), restricted to piecewise linear functions on intervals of size 1 vanishing at 0 and L. We show that with overwhelming probability the maximal action A satisfies A = a ln L + O(1) for a deterministic constant a in (0, ∞). This provides an optimally quantitative homogenization statement.

What carries the argument

The maximal action A of the variational problem over piecewise-linear height functions on the unit-scale cutoff, controlled through pointwise bounds that yield locally uniform estimates.

If this is right

  • The constant a is deterministic and lies in (0, ∞), independent of the particular noise realization.
  • The O(1) fluctuation bound holds uniformly in the boundary values of the height function.
  • The same scaling applies to related models including optimal matching problems and one-dimensional random-field Ising models.
  • The estimates sharpen earlier qualitative homogenization results by removing the need for global boundary-condition comparisons.

Where Pith is reading between the lines

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

  • Large-scale numerical approximations of such variational problems can safely replace the white-noise potential by a deterministic effective density.
  • The same quantitative scaling may extend to other small-scale cutoffs provided comparable pointwise optimizer bounds exist.
  • The constant a should admit a variational or ergodic characterization that does not require solving the full finite-L problem.

Load-bearing premise

Pointwise bounds on the optimizing height function are available that are sufficiently strong to produce locally uniform control on the action when boundary conditions change.

What would settle it

Numerical computation of the maximal action for successively larger interval lengths L, followed by checking whether A minus a candidate constant times ln L stays bounded.

read the original abstract

Motivated by an optimal-matching problem (Leighton-Shor) and the random-field Ising model (Aizenman-Wehr, Ding-Wirth), we consider a variational problem for graphs in $1+1$ dimension maximizing an action that is the difference of a field term given by integrating white noise over the subgraph on the one hand, and the Dirichlet integral of the (continuum) height function $h=h(x)$ on the other hand. This problem is scale-invariant in law, and requires a small-scale cut-off which we implement by restricting to $h$ that are piecewise linear on intervals of size $1$ and vanish at $x=0,L$. We show that with overwhelming probability, the maximal action $A$ satisfies $A=a\ln L+O(1)$ for a deterministic constant $a\in(0,\infty)$. This can be considered as a homogenization result that is quantitative in an optimal way. The present result sharpens a recent qualitative homogenization result by the authors with C. Wagner; it does so by finding bounds for the action that are locally uniform in the boundary conditions. Like the earlier result, the present one relies on pointwise bounds on the optimizer, as provided by Dembin-Elboim-Hadas-Peled in a more general setting. In the earlier work, the small-scale cut-off also involved an explicit discretization of the field term, yielding a Brownian potential that is i.i.d. in $x\in\mathbb{Z}$; this had the benefit of allowing for a comparison argument, but is inconvenient for the coarse-graining used here.

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.

Referee Report

1 major / 2 minor

Summary. The paper proves a quantitative homogenization result for the maximal action A of piecewise-linear height functions (with scale-1 cutoff) in a 1+1-dimensional Brownian potential on [0,L]. It shows that A = a ln L + O(1) holds with overwhelming probability for a deterministic constant a ∈ (0,∞). The argument proceeds by coarse-graining, using locally uniform upper and lower bounds on the variational functional that are derived from the pointwise optimizer controls of Dembin–Elboim–Hadas–Peled together with the fixed piecewise-linear cutoff; this sharpens the authors’ earlier qualitative homogenization result with Wagner.

Significance. If the central claim holds, the result supplies an optimal quantitative homogenization statement (error O(1) rather than o(ln L)) for a scale-invariant variational problem with direct links to the Leighton–Shor optimal matching problem and the Aizenman–Wehr/Ding–Wirth random-field Ising model. The technical advance—obtaining boundary-condition-uniform bounds from external pointwise estimates—strengthens the homogenization toolkit in this setting.

major comments (1)
  1. [coarse-graining argument (around the passage from pointwise to locally uniform estimates)] The O(1) error after coarse-graining over Θ(L) microscopic intervals is load-bearing for the main theorem. The manuscript must explicitly verify that the boundary-height/slope dependence in the error terms inherited from the Dembin–Elboim–Hadas–Peled pointwise bounds remains summable (without an extra logarithmic factor) when the local estimates are glued; otherwise the claimed additive O(1) bound fails with positive probability. This verification should appear in the coarse-graining section that converts pointwise controls into locally uniform action bounds.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should state the precise small-scale cutoff (piecewise linear on intervals of length 1, vanishing at endpoints) more explicitly when contrasting with the earlier i.i.d. discretization.
  2. [Introduction] Notation for the maximal action A and the deterministic constant a should be introduced with a forward reference to the theorem statement that asserts their existence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need for greater explicitness in the coarse-graining argument. The suggestion strengthens the presentation of the quantitative homogenization result. We address the major comment below and will revise the manuscript to incorporate the requested verification.

read point-by-point responses

  1. Referee: [coarse-graining argument (around the passage from pointwise to locally uniform estimates)] The O(1) error after coarse-graining over Θ(L) microscopic intervals is load-bearing for the main theorem. The manuscript must explicitly verify that the boundary-height/slope dependence in the error terms inherited from the Dembin–Elboim–Hadas–Peled pointwise bounds remains summable (without an extra logarithmic factor) when the local estimates are glued; otherwise the claimed additive O(1) bound fails with positive probability. This verification should appear in the coarse-graining section that converts pointwise controls into locally uniform action bounds.

    Authors: We agree that the transition from the Dembin–Elboim–Hadas–Peled pointwise optimizer controls to locally uniform action bounds requires an explicit summation argument to confirm that boundary-height and slope errors remain O(1) in total. The current draft derives the locally uniform bounds but does not spell out the accumulation over the Θ(L) intervals in full detail. In the revised manuscript we will insert a new paragraph (or short subsection) immediately after the statement of the locally uniform estimates. This paragraph will (i) recall the precise form of the boundary-dependent error terms from the pointwise bounds, (ii) invoke the global Lipschitz control on the maximizer (already established earlier in the paper) to bound the slopes uniformly on the overwhelming-probability event, and (iii) apply a standard union-bound / Borel–Cantelli argument showing that the sum of the individual errors is O(1) with overwhelming probability and without an extra logarithmic factor. We believe this addition will fully address the concern while preserving the existing proof structure. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to prior qualitative result; central quantitative claim relies on external bounds

specific steps
  1. self citation load bearing [Abstract]
    "The present result sharpens a recent qualitative homogenization result by the authors with C. Wagner; it does so by finding bounds for the action that are locally uniform in the boundary conditions. Like the earlier result, the present one relies on pointwise bounds on the optimizer, as provided by Dembin-Elboim-Hadas-Peled in a more general setting."

    The sentence explicitly cites the authors' own prior qualitative result, but the load-bearing ingredients for the quantitative statement (locally uniform bounds and the deterministic limit a) are supplied by the external Dembin et al. controls and the fixed piecewise-linear cutoff; the self-citation therefore remains incidental rather than forcing the main claim.

full rationale

The paper establishes the existence of deterministic a as a limit via coarse-graining and locally uniform bounds. It invokes external pointwise optimizer controls from Dembin-Elboim-Hadas-Peled (non-overlapping authors) together with standard white-noise properties. The only self-reference is to the authors' own prior qualitative homogenization work with Wagner, which is not load-bearing for the O(1) error or the value of a. No self-definitional reduction, no fitted parameter renamed as prediction, and no uniqueness theorem imported from the authors' own prior work appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claim rests on standard properties of white noise, the Dirichlet integral, and an external pointwise bound on optimizers; no new free parameters or invented entities are introduced.

axioms (3)
  • domain assumption White noise exists as a random distribution whose integrals over subgraphs are well-defined random variables
    Used to define the field term of the action.
  • standard math The Dirichlet integral is a lower-semicontinuous functional on piecewise-linear height functions vanishing at the endpoints
    Standard variational setting for the energy term.
  • domain assumption Pointwise bounds on the optimizer hold as stated by Dembin-Elboim-Hadas-Peled
    Invoked to control the height function in the coarse-graining argument.

pith-pipeline@v0.9.0 · 5584 in / 1412 out tokens · 57673 ms · 2026-05-07T09:09:26.360175+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Rounding effects of quenched randomness on first-order phase transitions.Comm

    Michael Aizenman and Jan Wehr. Rounding effects of quenched randomness on first-order phase transitions.Comm. Math. Phys.130(1990), no. 3, 489–528

    work page 1990

  2. [2]

    Uniform energy distribution for an isoperimetric problem with long-range interactions.Journal of the American Mathematical Society,22, 2, 569–605, (2009)

    Giovanni Alberti, Rustum Choksi, Felix Otto. Uniform energy distribution for an isoperimetric problem with long-range interactions.Journal of the American Mathematical Society,22, 2, 569–605, (2009)

    work page 2009

  3. [3]

    On optimal matchings.Combina- torica,4, 4, 259–264 (1984), Springer-Verlag Berlin/Heidelberg

    Mikl´ os Ajtai, J´ anos Koml´ os, G´ abor Tusn´ ady. On optimal matchings.Combina- torica,4, 4, 259–264 (1984), Springer-Verlag Berlin/Heidelberg

    work page 1984

  4. [4]

    Quantitative stochastic homogenization of con- vex integral functionals.Annales scientifiques de l’Ecole normale sup´ erieure,49, 2, 423–481, (2016)

    Scott Armstrong, Charles Smart. Quantitative stochastic homogenization of con- vex integral functionals.Annales scientifiques de l’Ecole normale sup´ erieure,49, 2, 423–481, (2016)

    work page 2016

  5. [5]

    A PDE approach to a 2- dimensional matching problem.Probability Theory and Related Fields,173, 1, 433–477 (2019), Springer

    Luigi Ambrosio, Federico Stra, Dario Trevisan. A PDE approach to a 2- dimensional matching problem.Probability Theory and Related Fields,173, 1, 433–477 (2019), Springer

    work page 2019

  6. [6]

    Bogachev.Gaussian measures

    Vladimir I. Bogachev.Gaussian measures. Mathematical Surveys and Monographs

    work page

  7. [7]

    American Mathematical Society, Providence, RI, 1998

    work page 1998

  8. [8]

    Rotationally invariant first passage percolation: Breaking then/lognvariance barrier.arXiv preprint 2604.01214, 2026

    Riddhipratim Basu, Vladas Sidoravicius, and Allan Sly. Rotationally invariant first passage percolation: Breaking then/lognvariance barrier.arXiv preprint 2604.01214, 2026

    work page arXiv 2026

  9. [9]

    Xiaopeng Cheng, Felix Otto and Matteo Palmieri.In preparation

    work page

  10. [10]

    The directed landscape

    Duncan Dauvergne, Janosch Ortmann, and B´ alint Vir´ ag. The directed landscape. Acta Mathematica229(2022), no. 2, 201–285

    work page 2022

  11. [11]

    Minimal surfaces in random environment.arXiv preprint2401.06768, 2024

    Barbara Dembin, Dor Elboim, Daniel Hadas, and Ron Peled. Minimal surfaces in random environment.arXiv preprint2401.06768, 2024

    work page arXiv 2024

  12. [12]

    Minimal surfaces in strongly cor- related random environments.arXiv preprint2504.10379, 2025

    Barbara Dembin, Dor Elboim, and Ron Peled. Minimal surfaces in strongly cor- related random environments.arXiv preprint2504.10379, 2025

    work page arXiv 2025

  13. [13]

    Correlation length of the two-dimensional random field Ising model via greedy lattice animal.Duke Math

    Jian Ding and Mateo Wirth. Correlation length of the two-dimensional random field Ising model via greedy lattice animal.Duke Math. J.172(2023), no. 9, 1781–1811

    work page 2023

  14. [14]

    Correlation length of the two-dimensional random field Ising model via greedy lattice animal.arXiv preprint2011.08768v3, 2022

    Jian Ding and Mateo Wirth. Correlation length of the two-dimensional random field Ising model via greedy lattice animal.arXiv preprint2011.08768v3, 2022. 30 FELIX OTTO, MATTEO PALMIERI

    work page arXiv 2022

  15. [15]

    Shirshendu Ganguly.private communication

    work page

  16. [16]

    Last passage perco- lation in hierarchical environments.arXiv preprint2411.08018, 2024

    Shirshendu Ganguly, Victor Ginsburg, and Kyeongsik Nam. Last passage perco- lation in hierarchical environments.arXiv preprint2411.08018, 2024

    work page arXiv 2024

  17. [17]

    Almost sharp rates of conver- gence for the average cost and displacement in the optimal matching problem.The Abel Symposium, 93–103 (2023), Springer

    Michael Goldman, Martin Huesmann, Felix Otto. Almost sharp rates of conver- gence for the average cost and displacement in the optimal matching problem.The Abel Symposium, 93–103 (2023), Springer

    work page 2023

  18. [18]

    Tight bounds for minimax grid matching with applications to the average case analysis of algorithms.Combinatorica9(1989), 161–187

    Thomas Leighton and Peter Shor. Tight bounds for minimax grid matching with applications to the average case analysis of algorithms.Combinatorica9(1989), 161–187

    work page 1989

  19. [19]

    On minimizing curves in a Brownian potential.Probability Theory and Related Fields, 1–62 (2026), Springer

    Felix Otto, Matteo Palmieri, and Christian Wagner. On minimizing curves in a Brownian potential.Probability Theory and Related Fields, 1–62 (2026), Springer

    work page 2026

  20. [20]

    Domain branching in uniaxial ferromagnets: as- ymptotic behavior of the energy.Calculus of variations and partial differential equations,38, 1, 135–181 (2010), Springer

    Felix Otto, Thomas Viehmann. Domain branching in uniaxial ferromagnets: as- ymptotic behavior of the energy.Calculus of variations and partial differential equations,38, 1, 135–181 (2010), Springer

    work page 2010

  21. [21]

    Scaling hy- pothesis for the Euclidean bipartite matching problem.Physical Review E,90, 1, 012118 (2014), APS

    Sergio Caracciolo, Carlo Lucibello, Giorgio Parisi, Gabriele Sicuro. Scaling hy- pothesis for the Euclidean bipartite matching problem.Physical Review E,90, 1, 012118 (2014), APS

    work page 2014

  22. [22]

    Large scale regularity and correlation length for almost length-minimizing random curves in the plane.arXiv preprint 2412.17625, 2024

    Tobias Ried, and Christian Wagner. Large scale regularity and correlation length for almost length-minimizing random curves in the plane.arXiv preprint 2412.17625, 2024

    work page arXiv 2024

  23. [23]

    Michel Talagrand.Upper and lower bounds for stochastic processes: decomposition theorems. Vol.60. Springer Nature, 2022

    work page 2022