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arxiv: 2604.15183 · v1 · submitted 2026-04-16 · 🧮 math.AP

Combined effect of homogenization and dimension-reduction in the random Neumann sieve problem

Mert Ba\c{s}tu\u{g} This is my paper

Pith reviewed 2026-05-10 09:58 UTC · model grok-4.3

classification 🧮 math.AP
keywords Neumann sievestochastic homogenizationthin domainsdimension reductionrandom perforationsPoisson equationmarked point processasymptotic analysis
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The pith

In thin randomly perforated domains, Neumann sieve solutions for the Poisson equation converge according to three scaling-dependent regimes under optimal hole-radius integrability.

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

The paper analyzes the limiting behavior of solutions to a Poisson equation with Neumann boundary conditions inside a thin domain that contains randomly placed small holes. The holes are distributed according to a stationary marked point process, which can include clusters. By varying how the domain thickness scales relative to the typical hole size, three different limiting problems are obtained as the scales go to zero. The analysis also pins down the weakest moment condition on the random hole sizes that still permits the homogenization to go through.

Core claim

We prove that the combined homogenization and dimension-reduction analysis of the random Neumann sieve problem in a thin domain produces three distinct limiting regimes that depend on the scaling between the thickness of the domain and the typical size of the holes. The holes are modeled by a stationary marked point process, and we determine the optimal stochastic integrability requirement on the hole radii that ensures the homogenization result holds even in the presence of clustering.

What carries the argument

The relative scaling between domain thickness and hole radii, used in conjunction with the stationarity of the marked point process that generates the random perforations.

If this is right

  • The solutions converge to a homogenized limit problem whose form changes with the scaling regime.
  • Stochastic homogenization succeeds under the identified minimal integrability condition on radii despite clustering.
  • Dimension reduction interacts with the random perforations to produce effective models that may or may not retain the sieve effect.

Where Pith is reading between the lines

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

  • This approach may extend to other boundary value problems or to domains with more complex geometries.
  • The sharpness of the integrability condition could be tested by constructing examples where lower moments lead to failure of homogenization.
  • Practical computations of solutions in thin perforated materials could use the limiting regimes to reduce dimensionality.

Load-bearing premise

The perforations must arise from a stationary marked point process with hole sizes scaling in a controlled way relative to the domain thickness.

What would settle it

A numerical or analytical counterexample demonstrating that homogenization fails when the hole radii have moments below the optimal threshold identified in the paper.

Figures

Figures reproduced from arXiv: 2604.15183 by Mert Ba\c{s}tu\u{g}.

Figure 1
Figure 1. Figure 1: Illustrations of the domain Uε and the cross-section U ′ \ T ′ ε . interfacial energy term, which results from the penalization of debonding at the interface, in agreement with the phenomenological model introduced in [5]. In our paper, we consider a simplification of the problem by replacing the nonlinear elastic energy with the Dirichlet energy, as we are mainly interested in observing the effect of rand… view at source ↗
read the original abstract

We investigate the asymptotic behavior of the solutions to the Neumann sieve problem for the Poisson equation in a thin, randomly perforated domain. The perforations (sieve-holes) are generated by a stationary marked point process. According to the scaling between the domain thickness and the typical hole size, three distinct limiting regimes emerge. We also identify the optimal stochastic integrability condition on the random hole radii that guarantees stochastic homogenization, even in the presence of clustering holes.

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

0 major / 2 minor

Summary. The manuscript investigates the asymptotic behavior of solutions to the Neumann sieve problem for the Poisson equation in a thin randomly perforated domain. The perforations are generated by a stationary marked point process. Depending on the scaling between the domain thickness and the typical hole size, three distinct limiting regimes emerge. The paper also identifies the optimal stochastic integrability condition on the random hole radii that guarantees stochastic homogenization even in the presence of clustering holes.

Significance. If the results hold, this contributes to homogenization theory by unifying dimension reduction and stochastic homogenization in random thin domains. The explicit delineation of three scaling regimes and the optimal integrability condition (allowing clustering) are notable advances, as they rely on variational methods and properties of stationary marked point processes rather than ad-hoc assumptions. This provides a precise framework with potential implications for modeling in porous media and materials with random perforations.

minor comments (2)
  1. The abstract states that three regimes emerge but does not name or briefly characterize them; adding one sentence summarizing the regimes would improve immediate readability without lengthening the abstract substantially.
  2. In the setting section, the precise relation between the thickness parameter and the typical hole radius (used to distinguish the regimes) should be stated as an explicit scaling assumption with a reference to the subsequent limit theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript on the combined homogenization and dimension-reduction effects in the random Neumann sieve problem, including the accurate identification of the three scaling regimes and the optimal integrability condition allowing for clustering holes. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity in scaling-based homogenization limits

full rationale

The paper establishes three distinct limiting regimes for the Neumann sieve problem in a thin randomly perforated domain by analyzing the interplay between domain thickness scaling and typical hole size, together with an optimal integrability condition on random radii that ensures stochastic homogenization under clustering. These limits are obtained from the variational formulation, stationary marked point process properties, and direct estimates controlling the perforated measure and capacity; no step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The derivation remains self-contained under the stated scaling assumptions and point-process hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions from stochastic homogenization theory and dimension reduction techniques for PDEs in perforated domains.

axioms (1)
  • domain assumption Perforations generated by a stationary marked point process
    Invoked to model the random holes in the thin domain.

pith-pipeline@v0.9.0 · 5359 in / 1061 out tokens · 29324 ms · 2026-05-10T09:58:14.508935+00:00 · methodology

discussion (0)

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Reference graph

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