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Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs

T0 review · 0 major / 3 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read Dynamical optimal transport on Z^d-periodic graphs converges to a continuum limit even with linear energy growth.

desk verdict This closes the linear growth gap left open by the 2023 dynamical OT paper. read the letter →

arxiv 2311.17284 v1 pith:W7PEKVWZ submitted 2023-11-28 math.OC cs.NAmath.APmath.NA

Lorenzo Portinale , Filippo Quattrocchi This is my paper
classification math.OCcs.NAmath.APmath.NA
keywords optimaltransportdiscrete-to-continuumlimitperiodicgraphslineargrowthdynamicalWassersteindistancegraphgeometry
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

The paper proves discrete-to-continuum convergence for dynamical optimal transport problems on Z^d-periodic graphs when the energy density grows only linearly at infinity. Earlier work established the result only under the stronger superlinear growth assumption, leaving open the linear case that includes scaling limits of 1-Wasserstein transport. The limit object is a continuous dynamical transport problem whose cost functional is shaped by the periodic geometry of the graph. A reader would care because this supplies rigorous justification for using discrete graph models to approximate a broader family of linear-cost transport problems that arise in applications.

What carries the argument

Discrete-to-continuum convergence of the dynamical optimal transport functionals on Z^d-periodic graphs under linear growth of the energy density.

What would settle it

A sequence of Z^d-periodic graphs on which the discrete transport costs fail to approach the predicted continuum functional as the mesh size tends to zero.

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Extended reading notes

Core claim

We prove that for Z^d-periodic graphs the discrete dynamical optimal transport functionals with linear growth energy densities converge to a continuum dynamical transport problem. The result resolves an open question from prior work that required superlinear growth and directly covers scaling limits of 1-Wasserstein problems. The geometry of the underlying periodic graph determines the structure of the effective continuum cost, as illustrated by explicit examples and visual representations.

Load-bearing premise

The graphs are required to be periodic with respect to the integer lattice Z^d.

Editorial extensions

If this is right

  • The convergence holds for discrete boundary-value dynamical transport problems.
  • The limit cost functional depends on the geometry of the periodic graph.
  • The result includes scaling limits of 1-Wasserstein transport as a special case.
  • Graph geometry influences the continuum cost in a manner comparable to the quadratic-growth setting.

Reading between the lines

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

  • Discrete periodic-graph models could serve as computable approximations to continuous linear-cost transport problems.
  • Similar convergence arguments might extend to graphs that are only locally periodic or statistically periodic.
  • Linear growth may permit different concentration or flow behaviors in the limit than those seen under superlinear growth.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The manuscript proves discrete-to-continuum convergence for dynamical optimal transport on Z^d-periodic graphs when the energy density has linear growth at infinity. This extends the superlinear-growth results of Gladbach-Kopfer-Maas-Portinale (2023) and applies in particular to scaling limits of 1-Wasserstein problems; the final section examines how the graph geometry shapes the limit cost, with accompanying visual representations.

Significance. If the convergence statement holds, the work closes a recognized gap in the literature on dynamical optimal transport and broadens the range of admissible growth conditions. The explicit treatment of graph geometry in the limit functional is a concrete contribution that parallels known quadratic-case phenomena.

minor comments (3)
  1. The abstract and introduction cite Gladbach et al. (Calc. Var. PDE 2023) but do not specify the precise theorem number or statement being extended; adding this reference would clarify the exact open problem resolved.
  2. Section 5 (or the final section) presents visual representations of the limit cost; the captions should explicitly state the underlying graph, the value of the linear-growth parameter, and the discretization scale used for each figure.
  3. Notation for the discrete energy density and its continuum counterpart is introduced without a consolidated table; a short notation table would improve readability for readers comparing the linear and superlinear cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report lists no specific major comments under the MAJOR COMMENTS section.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: convergence proof against external continuum target

full rationale

The paper establishes a discrete-to-continuum convergence theorem for dynamical optimal transport on Z^d-periodic graphs under linear growth. Periodicity is an explicit modeling hypothesis used to enable homogenization, not derived from the result itself. The limit cost is defined via an independent continuum formulation (standard in the literature on Wasserstein-type problems), and the proof proceeds by compactness, lower semicontinuity, and recovery sequences without reducing any quantity to a fit or self-referential definition. No self-citation chain is load-bearing for the central statement, and the result is a standard Gamma-convergence style argument rather than a renaming or ansatz smuggling. This is the normal, non-circular outcome for a mathematical convergence paper.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The proof relies on periodicity of the graph and standard assumptions of dynamical optimal transport; no free parameters or new entities are mentioned in the abstract.

assumptions (1)
  • domain assumption The underlying graphs are Z^d-periodic
    Required for the discrete-to-continuum passage on a lattice-like structure.

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Cite this review

Pith. "Pith review of Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs." pith.science (2026). https://pith.science/paper/W7PEKVWZ

@misc{pith2026231117284,
  author       = {Pith},
  title        = {Pith review of: Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7PEKVWZ}},
  note         = {Machine review of arXiv:2311.17284}
}
abstract

We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with energy density having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of 1-Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.

Figures

Figures reproduced from arXiv: 2311.17284 by the authors.

Figure 1
Figure 1. Example of Z d -periodic graph embedded in R d satisfying X x∈X ∩[0,1)d m(x) = ρ , div J = 0 , and Eff(J) := 1 2 X (x,y)∈E x∈[0,1)d J(x, y)(y − x) = j . The result covers several examples, both for what concerns the geometric properties of the graph (such as isotropic meshes of T d , or the simple nearest-neighbors interaction on the symmetric grid) as well as the choice of the cost functionals (including discretisa… view at source ↗
Figure 2
Figure 2. Examples of graphs in R 2 and corresponding unit balls for fhom [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗

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