arXiv:2608.13235v1

Mixing Properties of Random Laguerre Tessellations

Zbyněk Pawlas, Martina Švarc Petráková

math.PR60D0537A2560G55

Paper metadata

Original submission: August 13, 2026
Claimed MSC: 60D05, 37A25, 60G55
arXiv categories: math.PR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 19 pages

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 $α$ -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.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsContains wrong statements

The moment condition does correctly make the stationary Laguerre generator admissible, and the subsequent localization scheme gives substantial support for preservation of ergodicity, mixing, and $\alpha$ -mixing. Theorem 4 is nevertheless false on its printed domain: for a locally finite weighted configuration the displayed family of positive-dimensional cells need not be closed in the Fell hyperspace, so $L(\varphi)$ need not even belong to the asserted codomain. Because all three mixing implications invoke that theorem, their proofs require a measurable-map result restricted to regular or tempered configurations before those implications can be fully verified.

Theorem 1(a), admissibility clauseCorrect

The moment condition yields a well-defined Laguerre tessellation

Pages 4–6 and 10 · Theorem 1(a), Lemmas 2 and 5 · arXiv:2608.13235v1

Applying the tempered-configuration estimate to the transformed marks $\sqrt{M_-}$ gives eventual control $m\geq-(\lVert x\rVert-k)^2$ for distant generators. This makes every sublevel set of the power distance locally finite. Stationarity and nondegeneracy give $\operatorname{conv}(\eta')=\mathbb R^d$ almost surely, and the spatial ergodic theorem places the transformed configuration in $\mathcal M_{\mathrm{temp},\delta}$ under $\mathbb E M_-^{(d+\delta)/2}<\infty$ . Proposition 3 then supplies compact, locally finite Laguerre cells covering space.

Theorem 4Incorrect

The Laguerre map does not take every simple marked configuration into the claimed hyperspace

Page 6 · Theorem 4 · arXiv:2608.13235v1

Let $d\geq2$ , $e_1=(1,0,\ldots,0)$ , $x_n=ne_1$ , $b_1=0$ , $b_n=-2\sum_{j=1}^{n-1}j^{-1}$ , and $m_n=b_n-n^2$ . The configuration $\varphi=\{(x_n,m_n):n\geq1\}$ has a simple locally finite ground set and hence lies in the theorem's domain. After subtracting the common term $\lVert z\rVert^2$ , its power functions are the affine functions $-2nz_1+b_n$ . Consecutive functions cross at $z_1=-1/n$ , so the positive-interior Laguerre cells are $(-\infty,-1]\times\mathbb R^{d-1}\quad\text{and}\quad[-1/(n-1),-1/n]\times\mathbb R^{d-1}\quad(n\geq2).$ The latter cells converge in the Fell topology to $\{0\}\times\mathbb R^{d-1}$ , which is not one of the cells. Thus $L(\varphi)$ is not closed in $\mathcal F'_d$ and is not an element of $\mathcal F(\mathcal F'_d)$ , contrary to Theorem 4. A repair must restrict the domain, for example to configurations whose diagrams are locally finite tessellations, and prove measurability on that restricted measurable set; merely changing the proof's two displayed symbols does not repair the printed theorem.

Reviewed manuscript, version 1 ↗

Theorem 1(a)–(c), mixing clausesNot able to verify

The factor and localization conclusions need a valid restricted measurability theorem

Pages 4 and 10–15 · Theorem 1 and its proof · arXiv:2608.13235v1

Ergodicity and ordinary mixing would follow immediately if $L$ were a measurable translation-equivariant map on an almost-sure set containing the tempered configurations. The $\alpha$ -mixing proof similarly localizes inner and outer cell events and then applies measurability to truncated diagrams. Theorem 4 does not provide the required premise because it is false on its stated domain, and the paper does not define and prove measurability of a repaired restriction. The localization estimates themselves do not furnish that missing measurable-factor step. No counterexample to the three implications was found, but the supplied argument does not formally establish them until this gap is repaired.

Lemma 17Correct

The Poisson moment threshold is exact

Pages 15–16 · Lemma 17 · arXiv:2608.13235v1

For a marked Poisson process, $\mathbb E M_-^{d/2}<\infty$ makes the number of generators whose power ball reaches a fixed compact set finite, verifying (R1), while stationarity verifies (R2). If that moment diverges, the Poisson void formula gives $\inf_{x\in\eta}\rho(0,x)=-\infty$ almost surely; hence no cell contains the origin and the diagram cannot be space filling. This proves both directions of the stated threshold.

02Proofs6 reported findingsContains incorrect or incomplete proofs

The temperedness and void-probability arguments are correct, and the geometric localization lemmas support the intended $\alpha$ -mixing reduction. The proof of Theorem 4 fails at the level of its codomain and also contains two reversed extrema and a sentinel collision. Consequently the measurable-factor steps in all three mixing proofs are incomplete. The reversed extrema are reported as typos because each has a uniquely determined local correction, but those corrections alone do not fix the theorem.

Proof of Theorem 4Incorrect as written

The proof assumes the cell family is closed and uses possible cells as exceptional sentinels

Pages 7–10 · Lemmas 7–8 and proof of Theorem 4 · arXiv:2608.13235v1

The construction treats $L(\varphi)$ as an element of $\mathcal F(\mathcal F'_d)$ before proving that its set of cells is closed; the explicit configuration in the statements finding shows that this can fail. In addition, $\Lambda$ replaces empty-interior cells by $B(0,2)$ and $\widetilde\Lambda$ replaces non-atoms by $B(0,1)$ , after which $\kappa$ deletes both values. A genuine Laguerre cell can equal either ball: countably many locally finite distant generators can provide a dense family of tangent half-spaces. Thus the construction can delete a real cell. Nonconvex closed sentinels would repair that local collision, but a domain restriction and a proof that the restricted cell family is hyperspace-valued are still required. Repair classification: no complete repair is supplied.

Lemma 7Typo

A supremum is printed where an infimum is required

Page 7 · proof of Lemma 7 · arXiv:2608.13235v1

For $g(z)=\rho(z,x)-\min_i\rho(z,y_i)$ on a compact $C$ , the condition $g(z)>0$ for every $z\in C$ is equivalent to $\inf_{z\in C}g(z)>0$ , not to $\sup_{z\in C}g(z)>0$ as printed. Replacing `sup' by `inf' gives the intended measurable continuous minimum and repairs this step mechanically.

Proof of Theorem 4Typo

The finite partial union uses the wrong extremum

Pages 9–10 · definition of $L_1^n$ · arXiv:2608.13235v1

The upper index of the purported finite partial union is printed as $\max\{n,\varphi(\mathbb R^d\times\mathbb R)\}$ . It must be $\min\{n,\varphi(\mathbb R^d\times\mathbb R)\}$ : with the printed maximum it is already infinite whenever the configuration has infinitely many atoms, so the claimed measurability of the finite union is circular. The following identity $L=\bigcup_nL_1^n$ determines the correction uniquely.

Proof of Theorem 1(a) and (b)Incomplete as written

The ergodic and mixing factor argument lacks its measurable map

Page 10 · proof of Theorem 1(a) and (b) · arXiv:2608.13235v1

After correctly proving almost-sure temperedness, the proof substitutes $L^{-1}(A)$ and $L^{-1}(B)$ into the definitions of ergodicity and mixing. That substitution requires the restricted Laguerre map to be measurable. The only cited justification is Theorem 4, which is false on its printed domain. A valid repair is plausible—prove measurability on the measurable set of regular or tempered configurations and extend the map arbitrarily off that set—but neither the measurable domain nor the extension argument is supplied.

Proof of Theorem 1(c)Incomplete as written

The localization estimate is conditional on the same missing measurability repair

Pages 10–15 · Lemmas 9–16 and proof of Theorem 1(c) · arXiv:2608.13235v1

Lemmas 10–14 correctly turn an abnormally long or displaced cell into a growing vacant region, whose probability tends to zero by stationarity. Lemmas 15–16 then identify the inner and outer diagrams on the high-probability localization events, yielding the displayed bound by $\alpha_\eta$ . The probabilities and preimages in this chain are asserted measurable through Theorem 4. Since that theorem fails and no restricted replacement is proved, the final supremum argument has an unresolved formal prerequisite. The geometric and probabilistic estimates otherwise match the stated definition $\alpha_\eta(c,\infty;\Delta)$ .

Lemma 18Minor formal correction

The separation radius is described by an insufficient condition

Page 16 · proof of Lemma 18 · arXiv:2608.13235v1

The proof chooses $a_0$ only so that $\operatorname{dist}(B_1,B_2)\leq a_0$ , then uses independence of the two marking families for $z\notin B(0,a_0)$ . That inequality does not ensure $(B_1+z)\cap B_2=\varnothing$ . Choose instead $a_0>\sup\{\lVert y-x\rVert:x\in B_1,\ y\in B_2\}$ , which is finite because both sets are bounded. The omitted integral over the fixed ball then vanishes after normalization, and the rest of the argument is unchanged.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.13235v1
Authors listed: Zbyněk Pawlas, Martina Švarc Petráková
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

is proved for every sufficiently small

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P

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P(delta)

holds for every

delta>0

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t>0

and subsequently applied at

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Mixing Properties of Random Laguerre Tessellations

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The moment condition yields a well-defined Laguerre tessellation

The Laguerre map does not take every simple marked configuration into the claimed hyperspace

The factor and localization conclusions need a valid restricted measurability theorem

The Poisson moment threshold is exact

The proof assumes the cell family is closed and uses possible cells as exceptional sentinels

A supremum is printed where an infimum is required

The finite partial union uses the wrong extremum

The ergodic and mixing factor argument lacks its measurable map

The localization estimate is conditional on the same missing measurability repair

The separation radius is described by an insufficient condition

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