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arxiv: 2604.11236 · v2 · submitted 2026-04-13 · ❄️ cond-mat.mtrl-sci · cond-mat.stat-mech· physics.data-an

Surface correlation functions of dead-leaves models

Cedric J. Gommes This is my paper

Pith reviewed 2026-05-10 16:05 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.stat-mechphysics.data-an
keywords dead-leaves modelspore-surface correlationsurface-surface correlationBoolean modelrandom mediasmall-angle scatteringporous materialstwo-point statistics
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The pith

Exact expressions for pore-surface and surface-surface correlation functions are derived for dead-leaves models in any dimension and for arbitrary grain shapes.

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

Dead-leaves models generate two-phase pore and solid structures by sequentially dropping grains that overwrite everything already present. The paper obtains closed-form expressions for the pore-surface and surface-surface correlation functions that characterize these structures. These functions enter both the theory of small-angle scattering and models of material transport and mechanics. A reader following the derivation would see that the same probabilistic counting of overlapping grains works uniformly across all grain geometries and all spatial dimensions. The work also supplies a general formula for the Boolean model as a direct consequence of the same counting.

Core claim

Within dead-leaves models a two-phase structure is built by sequential random placement of pore-like or solid-like grains that completely overlap any pre-existing material. The pore-surface and surface-surface correlation functions are obtained by enumerating the probabilities that two points lie on particular combinations of pore and solid boundaries. The resulting formulas are exact, contain no further approximations, and remain valid for grains of any shape in any dimension. As a byproduct the same counting argument supplies closed expressions for the Boolean model.

What carries the argument

The dead-leaves process of sequential random grain placement that completely overlaps prior structure, from which correlation functions follow by direct probabilistic geometry.

If this is right

  • The expressions apply without change to grains of any shape or to any spatial dimension.
  • A general closed-form result for the Boolean model follows immediately from the same derivation.
  • Dead-leaves realizations engineered to have exponential two-point correlations differ from numerically reconstructed Debye media in their pore-surface functions while sharing nearly identical surface-surface functions.
  • Grain size distributions can be chosen to produce structures whose correlation functions match any prescribed target.

Where Pith is reading between the lines

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

  • The analytic forms could be inserted directly into scattering inversion routines to test whether a given experimental pattern is consistent with a dead-leaves geometry.
  • Because the pore-surface function distinguishes dead-leaves models from other random-media constructions, measurements sensitive to that function could discriminate among competing microstructural hypotheses.
  • The same probabilistic counting may extend without modification to related overlapping-grain models that add a finite lifetime or a growth rule to each grain.

Load-bearing premise

The only rule governing the structure is that each new grain is placed at random and erases every part of the previous structure it covers.

What would settle it

Generate a large three-dimensional dead-leaves realization with spherical grains of known radius, measure the pore-surface and surface-surface correlation functions by direct counting on the voxelized structure, and compare the numerical values to the closed-form expressions.

Figures

Figures reproduced from arXiv: 2604.11236 by Cedric J. Gommes.

Figure 1
Figure 1. Figure 1: FIG. 1. Two-dimensional dead-leaves model, whereby pore-like (black) or solid-like (white) disks [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Definition of the covariograms [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Realizations of dead-leaves models from monodispersed spherical grains, with decreasing [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Pore-surface correlation functions [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Surface-surface correlation functions [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Debye random medium realized as a dead-leaves structure: (a) spherical grain size distribu [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Pore-surface correlation functions [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Steiner-offset analysis of the surface curvature of dead-leaves structures with exponential [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Surface-surface correlation functions [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Realizations of a Boolean model of interpenetrating hollow spheres, with pore volume [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Pore-surface correlation functions of the Boolean model of interpenetrating hollow spheres, [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Surface-surface correlation functions of the Boolean model of interpenetrating hollow [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
read the original abstract

The pore-surface and surface-surface correlation functions are structural characteristics that play an important role in theoretical materials science and in small-angle scattering theory. Exact analytical expressions for the surface correlation functions are available only for very few models, and we here derive such expressions for the general class of dead-leaves models. Within these models, a two-phase pore/solid structure is created by sequentially and randomly filling space with pore-like or solid-like grains that overlap any pre-existing structure, in the same way as dead leaves fall on the ground. The obtained mathematical expressions are valid for any grain shape, in arbitrary dimension. The results are illustrated with monodispersed spherical grains,as well as with a dead-leaves realization of a Debye random medium. In the latter case, the size distribution of the grains is designed to produce a structure having exponential two-point correlation function. Compared to Debye random media obtained by numerical reconstruction, the dead-leaves structure has almost identical surface-surface correlation function, but distinctly different pore-surface correlation function. As a byproduct of our analysis, we also submit a general expression for the pore-surface and surface-surface correlation functions of the Boolean model, valid for arbitrary grains.

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 / 3 minor

Summary. The manuscript derives exact analytical expressions for the pore-surface and surface-surface correlation functions of general dead-leaves models, which generate two-phase pore/solid structures via sequential random grain placement with complete overlap of pre-existing material. The expressions are stated to hold for arbitrary grain shapes and dimensions without approximations, illustrated for monodisperse spheres and a dead-leaves realization of a Debye random medium (with grain size distribution chosen to yield exponential two-point correlations). A general expression for the Boolean model is obtained as a byproduct.

Significance. If the derivations are correct, this provides closed-form, parameter-free results for key structural descriptors used in materials modeling and small-angle scattering, extending the small set of models with known exact surface correlations. The direct probabilistic derivation from the dead-leaves process definition, validity for arbitrary grains/dimensions, and explicit contrast with numerical Debye reconstructions (identical surface-surface but different pore-surface functions) are notable strengths.

minor comments (3)
  1. The abstract and introduction could more explicitly reference prior exact results for Boolean models to clarify the incremental novelty of the dead-leaves generalization.
  2. In the Debye-medium illustration, the quantitative difference between the dead-leaves and reconstructed pore-surface functions should be reported with a specific metric (e.g., integrated absolute deviation) rather than described qualitatively.
  3. Notation for the grain-size distribution and the resulting two-point function in the Debye case should be cross-referenced to the general expressions to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its strengths (including the direct probabilistic derivation, validity for arbitrary grains and dimensions, and the explicit contrast with numerical Debye reconstructions), and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; direct probabilistic derivation

full rationale

The paper's central results are exact analytical expressions for the pore-surface and surface-surface correlation functions of dead-leaves models, obtained directly from the model's definition as a sequential random placement process in which grains completely overlap prior structure. The derivation applies to arbitrary grain shapes and dimensions with no fitted parameters, no predictions of quantities that are inputs by construction, and no load-bearing self-citations or uniqueness theorems invoked from prior author work. The Boolean model appears only as a special case. The provided abstract and description contain no steps that reduce the claimed results to tautological inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the probabilistic geometry of the dead-leaves process; no free parameters are introduced and no new entities are postulated.

axioms (1)
  • domain assumption A two-phase structure is formed by sequentially placing grains that overlap any pre-existing structure in a random manner.
    This is the defining property of dead-leaves models invoked to obtain the correlation functions.

pith-pipeline@v0.9.0 · 5504 in / 1165 out tokens · 67892 ms · 2026-05-10T16:05:21.706646+00:00 · methodology

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