Mixing Properties of Random Laguerre Tessellations

Martina \v{S}varc Petr\'akov\'a; Zbyn\v{e}k Pawlas

REVIEW 2 major objections 3 minor 35 references

Mixing Properties of Random Laguerre Tessellations

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Random Laguerre tessellations inherit ergodicity, mixing, and α-mixing from the stationary marked point processes that generate them, under a nearly optimal moment bound on negative weights.

desk verdict Solid paper on mixing preservation for Laguerre tessellations, with one real but repairable notation gap in the α-mixing proof. read the letter →

arxiv 2608.13235 v1 pith:YEKEDZE7 submitted 2026-08-13 math.PR

Zbyněk Pawlas , Martina Švarc Petráková This is my paper · ORCID

classification math.PR MSC 60D0537A2560G55

keywords randomLaguerretessellationergodicitymixingalpha-mixingmarkedpointprocesstemperedconfigurationspowerdistance

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

Random Laguerre tessellations—weighted generalizations of Voronoi diagrams in which each cell is defined by power distance from a nucleus with a weight—are shown to be well defined and to inherit the asymptotic-independence structure of the point process that generates them. The sufficient condition is a moment bound on the negative part of the weights: very large negative weights must be sufficiently rare. Under this condition, an ergodic marked point process produces an ergodic Laguerre tessellation, a mixing process produces a mixing tessellation, and an α-mixing process produces an α-mixing tessellation. This matters for applications in materials science and biology, where Laguerre tessellations model foams, polycrystals, and cellular structures and where one wants spatial averages to converge and far-apart regions to behave independently.

What carries the argument

The machinery has three parts. The Laguerre cell of a generator $(x,m)$ is the set of points $z$ whose power distance $\|z-x\|^2 + m$ is no larger than the power distance to every other generator; the Laguerre diagram is the collection of cells with non-empty interior, and for suitable configurations it is a locally finite partition of $\mathbb{R}^d$ into compact convex cells. Tempered configurations, defined by a bound of the form $\sum_{(x,m)\in\varphi\cap(kB^d\times\mathbb{R})}(1+|m|^{d+\delta})\le l\,k^d$ for all $k$, provide the control of unbounded negative weights that makes the tessellation well defined. The paper also proves the measurability of the mapping that sends a weighted configuration to its Laguerre diagram, which is what allows the mixing properties of the generator to be transferred to the tessellation via $\sigma$-algebras of cells intersecting balls.

What would settle it

Check the $\alpha$-mixing transfer at the endpoint: take an $\alpha$-mixing marked point process whose typical mark has density proportional to $|m|^{-1-d/2}/\log^2 |m|$ for $m\le -e$. This mark distribution satisfies $E[M_-^{d/2}]<\infty$ but $E[M_-^{(d+\delta)/2}]=\infty$ for every $\delta>0$. If simulated Laguerre tessellations from such generators still have $\alpha$-mixing coefficients tending to zero, the strict $\delta>0$ in Theorem 1 is an artifact of the proof; if not, it is essential.

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Extended reading notes

Core claim

The central result, Theorem 1, states that if $\eta$ is a stationary marked point process on $\mathbb{R}^d \times \mathbb{R}$ whose typical mark $M$ satisfies $E[M_-^{(d+\delta)/2}]<\infty$ for some $\delta>0$, then (a) ergodicity of $\eta$ implies $L(\eta)$ is almost surely a tessellation of $\mathbb{R}^d$ and is ergodic; (b) mixing of $\eta$ implies $L(\eta)$ is mixing; and (c) $\alpha$-mixing of $\eta$ implies $L(\eta)$ is $\alpha$-mixing. The proof works by showing that $\eta$ almost surely belongs to the set of tempered configurations, which controls the growth of large negative weights, and by proving that the Laguerre mapping $\varphi\mapsto L(\varphi)$ is measurable, so that the tessellation's $\sigma$-algebras can be compared with the generator's. For marked Poisson processes the moment threshold can be improved: $L(\eta)$ is almost surely a tessellation if and only if $E[M_-^{d/2}]<\infty$.

Load-bearing premise

The theorem requires the typical negative weight to have a moment of order $(d+\delta)/2$ for some strictly positive $\delta$; if the negative weights are only integrable at the critical order $d/2$, or not at all, the proof's control of far-away cells and the $\alpha$-mixing transfer does not go through.

Editorial extensions

If this is right

Poisson–Laguerre tessellations are well defined exactly when $E[M_-^{d/2}]<\infty$; when this $d/2$-moment is infinite, the origin is almost surely not covered by any cell, so the diagram is not space-filling.
For any ergodic stationary marked point process meeting the moment bound, spatial averages over the tessellation converge almost surely, so ergodic theorems apply to cell-based statistics.
Long-range independence of the generator passes to the geometry: far-apart windows of the tessellation become independent at the same qualitative rate class, mixing or $\alpha$-mixing.
The measurability theorem makes random Laguerre tessellations legitimate random elements in a Fell-topology space, so mixing and ergodicity can be defined and checked through cells rather than through cell boundaries.
The examples show that the results cover Poisson, Cox, cluster, geostatistically marked, and some Gibbs and determinantal generators, widening the non-Poissonian theory beyond bounded weights.

Reading between the lines

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

The strict $\delta>0$ margin in the moment condition looks technical: for well-definedness alone the endpoint condition $E[M_-^{d/2}]<\infty$ already suffices by a Campbell-formula argument for every stationary marked point process, and the $\delta$ is used only in the tempered-configuration estimates behind the $\alpha$-mixing transfer. If those estimates can be sharpened, the theorem may hold at
A similar measurable-mapping plus tempered-configuration route could transfer mixing properties to other weighted diagrams, such as power diagrams with higher-order power distances or Johnson–Mehl tessellations with random birth times.
For the Poisson case, the sharp threshold suggests a testable prediction: simulations with negative-weight tails just below the $d/2$-moment threshold should show large empty holes growing with the box, while tails just above it should tessellate.
The paper itself notes that no rate of decay of the mixing coefficients follows from the proof; closing that gap would require distributional control of the random tempered-configuration index $l(\hat\eta)$, not just its almost-sure finiteness.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is a genuine step forward for random Laguerre tessellations: it gives the first general preservation results for ergodicity, mixing, and α-mixing under the Laguerre map, and it proves a measurability theorem for the Laguerre mapping that earlier work had silently assumed. The tempered-configuration machinery for unbounded weights is substantial and well motivated, and the vacancy lemmas in Section 4.3 are clever. The moment condition (2) is cleanly stated, and the paper is honest that the δ>0 is a technical price beyond the sharp d/2 moment in the Poisson case.

The stress-test note about Lemma 12 is on target. In the second case, the proof establishes that any generator with t≤r^2 lies outside the ball centered at r x_j of radius β_l r, but the lemma states the exclusion for the ball centered at β_l r x_j with radius β_l r. Those are different balls; the stated version is false, while the proof's version is correct. Lemma 14 then uses the stated version, so the written proof of Theorem 1(c) does have a genuine gap at that point. The good news is that the repair is immediate: replace β_l r B_j by B(r x_j, β_l r) in Lemma 12, the proof, and Lemma 14. Lemma 13 still applies because the radius is a fixed multiple of r, so the asymptotic argument survives unchanged. The central theorem is not in danger, but the manuscript as submitted is not fully correct in this section.

A smaller issue: the abstract calls the moment condition “nearly optimal,” but the paper itself shows in Lemma 17 that d/2 suffices for well-definedness of a stationary marked point process, so the (d+δ)/2 condition is not close to necessary in general. That is a minor overstatement, not a flaw in the mathematics.

This paper is for stochastic geometers and point process theorists, and it definitely deserves a serious referee. I would not desk-reject it. My recommendation: send to referee, and specifically ask the referee to check the ball-centering in Lemma 12 and Lemma 14. With that removed, the paper is a solid contribution.

Referee Report

2 major / 3 minor

Summary. The paper studies random Laguerre tessellations generated by a stationary marked point process on R^d x R, with possibly unbounded weights. Its main result, Theorem 1, states that under the moment condition E[M_-^{(d+δ)/2}]<∞ for some δ>0, ergodicity, mixing, and α-mixing of the generating marked point process are each preserved by the Laguerre tessellation map, and that the generating process is then almost surely an admissible generator. Section 3 introduces tempered configurations and proves measurability of the Laguerre mapping (Theorem 4). Section 4 proves the main theorem, with the α-mixing preservation argument based on geometric lemmas controlling large cells via empty regions. Section 5 gives the Poisson case, where the optimal moment condition E[M_-^{d/2}]<∞ is obtained, and discusses several non-Poissonian examples.

Significance. If the proof gaps described below are repaired, the paper would be a valuable contribution: it extends mixing preservation results beyond Poisson and Gibbs-type generators to general stationary marked point processes, provides the first formal measurability proof for the Laguerre diagram map, reproduces the needed argument from the authors' earlier note [26] in full, and gives a clean optimal moment condition in the Poisson case (Lemma 17). The paper is largely self-contained and its main claims are concrete and falsifiable. However, two load-bearing proof steps currently fail as written, so the manuscript is not yet ready for acceptance.

major comments (2)

[§4.3, Lemma 12] The proof of Lemma 12 contains a scaling error in the second case. From z∈rB_j and ||z−y||>√(C_l^2−1)r one cannot conclude that y∉β_l r B_j, because β_l r B_j = B(β_l r x_j, β_l r) is not centered at z; it may contain points whose distance from z is much larger than √(C_l^2−1)r. The valid triangle-inequality conclusion is instead y∉B(r x_j, β_l r), i.e. y∉r(β_l B_j), which is a different ball. Since Lemma 14 uses exactly the stated conclusion to bound P(η∈E^3_{r,K3}) by a sum of empty-ball probabilities, the proof of Theorem 1(c) is incomplete as submitted. The argument appears repairable by replacing β_l r B_j with r(β_l B_j) in the statement and proof of Lemma 12 and adapting Lemma 14 accordingly, but this must be carried out explicitly.
[§4.1, proof of Theorem 4] The measurability proof discards the sentinel values B(0,1) and B(0,2) in the definition of κ, but these closed balls can themselves be genuine Laguerre cells with non-empty interior. For example, a generator at the origin together with suitably weighted generators in directions dense on the sphere can produce an origin cell that is exactly a ball. For such configurations the equality L(φ)=∪_n L_n^1(φ) is false, because the genuine cell is removed. Therefore the proof of Theorem 4, which is invoked in the proofs of Theorem 1(a) and (b), is not valid as written. The issue is local and can be fixed, for instance by choosing dummy values that cannot coincide with any non-empty-interior Laguerre cell, or by carrying an indicator of whether the value is a dummy.

minor comments (3)

[Abstract and §1] The phrase "nearly optimal" is overstated. The 'if' direction of Lemma 17 uses only Campbell's formula and does not require the Poisson assumption, so E[M_-^{d/2}]<∞ already suffices for well-definedness for arbitrary stationary marked point processes; the stronger moment (2) in Theorem 1 is needed for the tempered-configuration estimates used in the α-mixing part, not for well-definedness itself.
[§4.2, after proof of Theorem 1(b)] The sentence introducing (10) is a side remark and could be integrated more cleanly with the surrounding proof, since the proof of part (b) has already been completed at that point.
[§5.1, Lemma 17] In the 'only if' direction, the notation ρ(0,x) is used before being explicitly defined in the Laguerre formalism section; a brief reminder of the definition of power distance would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation relies on standard point-process ergodic theorems, measurability and equivariance of the Laguerre map, and direct approximation lemmas; the only self-citation supplies a proof idea that is restated with an external standard reference.

full rationale

The paper's central claims are not circular. Theorem 1(a) and (b) follow from Lemma 5 (tempered configurations imply the regularity conditions), the standard marked-point-process ergodic theorem applied to h(m)=1+(sqrt(m_-))^{d+delta} (cited to [5, Corollary 12.2.V(b)]), and the measurability plus equivariance of the Laguerre map proved in Theorem 4. The one self-citation, [26], is used only as 'the same idea' in the proof of Theorem 1(a), but the needed convergence statement is reproduced in the text and rests on the external ergodic theorem, so it is not a load-bearing unverified premise. The alpha-mixing proof in Section 4.3 uses the approximation lemmas 14-16, whose estimates are derived from the definitions of the events E^1, E^2, E^3 and from tempered-configuration bounds; no target quantity is inserted into its own definition, and no fitted parameter is renamed as a prediction. No uniqueness theorem from the authors is invoked to force a choice. A reviewer-identified gap in Lemma 12 is a geometric scaling correctness issue, not a circularity: even if the claimed implication fails as written, the theorem is not derived by assuming its conclusion. The paper is therefore self-contained against standard point-process theory, with no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof rests on standard point process and stochastic geometry results. The only paper-specific assumptions are the stationarity and non-degeneracy of η and the moment condition (2); no free parameters or new entities are introduced.

assumptions (6)

domain assumption η is a stationary, simple, non-degenerate marked point process on R^d×R with locally finite ground intensity
Standing assumption in Section 2; ensures the intensity measure factorizes as γλ_d⊗Q and the convex hull of the ground process is R^d a.s.
domain assumption There exists δ>0 such that E[M_-^{(d+δ)/2}]<∞
Assumption (2) in Theorem 1; used to force P(η-hat ∈ M_temp,δ)=1 via the ergodic theorem and hence enter the tempered-configuration framework.
standard math Campbell's formula and the ergodic theorem for stationary point processes
Used in Section 4.2 to show a.s. convergence of the empirical average of h(m)=1+(√(m_-))^{d+δ}; cited from Daley-Vere-Jones Corollary 12.2.V(b).
standard math Measurable enumeration of atoms of locally finite counting measures
Lemma 6, quoted from Last-Penrose Proposition 6.3, used in the proof of Theorem 4.
standard math Fell topology measurability results for closed sets and unions of sets
Schneider-Weil Theorems 12.2.1, 12.2.3 and 12.2.6, used in Lemma 8 and Theorem 4.
standard math Alpha-mixing coefficient properties and the semi-algebra reduction
Lemma 9, cited as analogous to Schneider-Weil Lemma 9.3.1; needed in Section 4.3.

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Pith. "Pith review of Mixing Properties of Random Laguerre Tessellations." pith.science (2026). https://pith.science/paper/YEKEDZE7

@misc{pith2026260813235,
  author       = {Pith},
  title        = {Pith review of: Mixing Properties of Random Laguerre Tessellations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YEKEDZE7}},
  note         = {Machine review of arXiv:2608.13235}
}

abstract

In this paper, we study the random Laguerre tessellation, a weighted generalization of the Voronoi tessellation, generated by a general stationary marked point process. We first derive a nearly optimal sufficient condition on the generating marked point process which ensures that the resulting random Laguerre tessellation is well-defined, utilising the concept of tempered configurations to handle potentially unbounded weights. We then investigate how the three mixing properties - ergodicity, mixing and $\alpha$-mixing - of the generating marked point process are preserved for the corresponding random Laguerre tessellation. Our approach combines standard approximation arguments with the properties of tempered configurations and the measurability of the Laguerre mapping.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [26]

Petr´ akov´ a and Z

M. Petr´ akov´ a and Z. Pawlas (2025). Some notes about Poisson–Laguerre tessellation with unbounded weights. Proceedings of the 23rd European Young Statisticians Meeting (ed. A. Srakar), Ljubljana, Slovenia, 81–86. PDF on ResearchGate

work page 2025

[2] [1]

C. A. N. Biscio and R. Waagepetersen (2019). A general central limit theorem and a sub- sampling variance estimator forα-mixing point processes.Scand. J. Statist.46(4), 1168–1190. https://doi.org/10.1111/sjos.12389 17

work page doi:10.1111/sjos.12389 2019

[3] [2]

R. C. Bradley (2007). Introduction to Strong Mixing Conditions Kendrick Press, Heber City

work page 2007

[4] [3]

Flimmel, Z

D. Flimmel, Z. Pawlas and J. E. Yukich (2020). Limit theory for unbiased and con- sistent estimators of statistics of random tessellations.J. Appl. Probab.57(2), 679–702. https://doi.org/10.1017/jpr.2020.4

work page doi:10.1017/jpr.2020.4 2020

[5] [4]

Daley and D

D. Daley and D. Vere-Jones (2003). An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. Second Edition. Springer, New York

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Daley and D

D. Daley and D. Vere-Jones (2008). An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure. Second Edition. Springer, New York

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Gusakova and M

A. Gusakova and M. in Wolde-L¨ ubke (2025). Poisson–Laguerre tessellations.Electron. J. Probab.30, 1–48. https://doi.org/10.1214/25-EJP1320

work page doi:10.1214/25-ejp1320 2025

[8] [7]

Gusakova and M

A. Gusakova and M. in Wolde-L¨ ubke (2026). Limits of Poisson–Laguerre tessellations. Preprint. https://arxiv.org/abs/2602.06906

work page arXiv 2026