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arxiv: 2606.17968 · v1 · pith:4UVKZ722new · submitted 2026-06-16 · 🧮 math.AP · math.PR

Diffuse Interface Energies with Microscopic Heterogeneities II: Rare Events

Peter S. Morfe , Christian Wagner This is my paper

Pith reviewed 2026-06-26 23:54 UTC · model grok-4.3

classification 🧮 math.AP math.PR
keywords Allen-Cahn functionalsdiffuse interfacesrare eventshomogenizationlarge deviationsrandom checkerboardstationary ergodic coefficientsmicroscopic heterogeneities
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The pith

If the ratio of heterogeneity scale to interface width vanishes too slowly, the limiting energy of Allen-Cahn functionals falls below the homogenized value.

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

The paper establishes that Allen-Cahn functionals with stationary ergodic coefficients converge to a smaller effective surface energy than the homogenized limit when the ratio ε^{-1} δ vanishes too slowly. This occurs in the rare events regime because atypical configurations of the coefficients dominate the energy minimization. For the one-dimensional random checkerboard, the limiting energy depends nontrivially on the limit of ε^{-1} δ |log ε|. The authors also construct examples in any dimension where rare events matter at algebraic scale ratios δ ≈ ε^{1+α} and show that almost periodic media can produce analogous effects.

Core claim

When the ratio ε^{-1} δ vanishes too slowly, the limit of the functional may actually be smaller than the homogenized energy. In the case of the random checkerboard in dimension one, the limiting energy depends in a nontrivial way on the limit of ε^{-1} δ |log ε|. Rare events become relevant at algebraic scales δ ≈ ε^{1+α} for arbitrary α > 0, and atypical configurations play the same role in almost periodic examples.

What carries the argument

Large deviations principle for stationary ergodic random fields, which quantifies the cost of atypical configurations that lower the energy when the scale ratio vanishes slowly.

If this is right

  • The limiting energy for the one-dimensional checkerboard is modified by a nontrivial dependence on lim ε^{-1} δ |log ε|.
  • Rare events dominate the limit at polynomial scale separations δ ≈ ε^{1+α} for any α > 0.
  • Almost periodic media exhibit energy reduction equivalent to that from random rare events.
  • The homogenized energy from the companion paper is not the correct Gamma-limit when the ratio vanishes slowly.

Where Pith is reading between the lines

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

  • Physical models of interfaces in heterogeneous media may require sampling rare configurations to obtain correct energy scaling at intermediate scale ratios.
  • The dependence on the scale ratio limit could take different forms in dimensions greater than one or in other phase-field models.
  • Numerical approximation schemes for such energies would need to incorporate large-deviation sampling to avoid underestimating the reduction.

Load-bearing premise

The coefficients are stationary ergodic random fields for which a large deviation principle holds with a good rate function allowing atypical configurations to be quantified.

What would settle it

Numerical computation of the minimal energy for the one-dimensional random checkerboard Allen-Cahn functional across sequences where ε^{-1} δ |log ε| approaches different finite limits, checking whether the value matches the predicted modified energy rather than the homogenized one.

Figures

Figures reproduced from arXiv: 2606.17968 by Christian Wagner, Peter S. Morfe.

Figure 1
Figure 1. Figure 1: Theorem 1 shows that on the one hand, if ϵ −1 δ vanishes faster than 1/| log ϵ|, then the relevant effective interfacial energy den￾sity is σW √ ¯θ. This is the homogenization regime. On the other hand if ϵ −1 δ vanishes slower than 1/| log ϵ|, the minimum value θ∗ of θ deter￾mines the macroscopic energy. Between these two behaviors there is a continuous transition w.r.t. to the limit κ of ϵ −1 δ(ϵ)| log ϵ… view at source ↗
Figure 2
Figure 2. Figure 2: (2) The image {[y + tη] | t ∈ R} is a dense subset of T 2 , see [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Projection of the rational line {(x, y) | y = 5 6 x − 1 2 } ⊆ R 2 on the torus T 2 1 [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: √ Projection of (parts of) the irrational line {(x, y) | y = 3x} ⊆ R 2 on T 2 and a possible choice of boxes {En}n∈N It is not hard to show that situation (1) occurs if and only if there is an integer k ∈ Z 2 such that η = k |k| . In that case, we say that η is a rational direction; otherwise, we say that η is an irrational direction. Evidently, the set of rational directions is dense in S 1 . Thus, given … view at source ↗
read the original abstract

We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $\delta$ of the heterogeneities is much smaller (microscopic) than the interface width $\epsilon$ (mesoscopic). In a companion paper, we show that if the ratio $\epsilon^{-1} \delta$ vanishes fast enough as $\epsilon \to 0$, then the functionals converge to an effective surface energy where the energy density is determined by homogenization effects originating at microscopic scales. Here we prove that if the ratio $\epsilon^{-1} \delta $ vanishes too slowly, the limit of the functional may actually be smaller than this homogenized energy. We refer to this as the rare events regime. In the case of the random checkerboard in dimension one, we use large deviations techniques to give a complete description of the rare events regime, showing that the limiting energy depends in a nontrivial way on the limit of $\epsilon^{-1} \delta | \log \epsilon |$. We further construct, in any dimension, examples of random media in which rare events become relevant at algebraic scales $\delta \approx \epsilon^{1 + \alpha}$ for an arbitrary $\alpha > 0$, as well as almost periodic examples in which atypical configurations play the same role as rare events.

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 analyzes Allen-Cahn functionals with stationary ergodic coefficients in the regime δ ≪ ε. The companion paper establishes Γ-convergence to a homogenized surface energy when ε^{-1}δ vanishes sufficiently fast. This paper shows that when the ratio vanishes too slowly the Γ-limit is strictly smaller (rare-events regime). For the 1D random checkerboard, large-deviation techniques yield a complete description in which the limiting energy depends nontrivially on lim ε^{-1}δ |log ε|. Explicit constructions are given in any dimension for random media where rare events dominate at algebraic scales δ ≈ ε^{1+α} (α>0 arbitrary) and for almost-periodic media in which atypical configurations play an analogous role.

Significance. If the results hold, the work completes the asymptotic picture for diffuse-interface energies with microscopic heterogeneities by identifying and characterizing the rare-events regime that can lower the effective energy below the homogenized value. The explicit 1D large-deviation analysis and the general constructions at algebraic scales constitute concrete, falsifiable contributions that build directly on established ergodic theory and large-deviation principles.

minor comments (2)
  1. The abstract refers to 'the companion paper' without a citation; adding the arXiv number or title in the introduction would improve readability.
  2. Notation for the scaling ratio ε^{-1}δ and the logarithmic correction is introduced clearly in the abstract; ensure the same symbols are used consistently in all statements of the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which accurately summarizes the scope of both papers in the series. The recommendation of minor revision is noted; we will incorporate any editorial or typographical suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external LDP and ergodicity independently

full rationale

The paper establishes the rare-events regime by applying standard large-deviation principles for stationary ergodic random fields (from prior probability theory) to the Allen-Cahn functional when ε^{-1}δ vanishes slowly. The companion paper supplies only the baseline homogenized energy for the fast-vanishing case; the slower-vanishing claim, the explicit 1D checkerboard description via lim ε^{-1}δ|log ε|, and the algebraic-scale constructions in any dimension are derived directly from LDP rate functions and recovery-sequence arguments without reducing to fitted parameters, self-definitional loops, or load-bearing self-citations. The argument remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from homogenization theory (stationarity and ergodicity of coefficients) and probability (large deviation principle for the random media), with no free parameters fitted to data and no new postulated entities.

axioms (2)
  • domain assumption The random coefficients are stationary and ergodic.
    Invoked throughout to apply the homogenized energy from the companion paper and to justify averaging in the standard regime.
  • domain assumption A large deviation principle holds for the probability measures on the random media with a good rate function.
    Required to quantify the probability and energy cost of atypical configurations that dominate when ε^{-1}δ vanishes slowly.

pith-pipeline@v0.9.1-grok · 5755 in / 1600 out tokens · 43948 ms · 2026-06-26T23:54:32.930162+00:00 · methodology

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