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arxiv: 2606.30517 · v1 · pith:4TB2WPO7new · submitted 2026-06-29 · 🧮 math.PR

Convergence towards Ideal Poisson--Voronoi tessellations with a focus on Diestel--Leader graphs

Matteo D'Achille , Ali Khezeli This is my paper

Pith reviewed 2026-06-30 04:33 UTC · model grok-4.3

classification 🧮 math.PR
keywords ideal Poisson-Voronoi tessellationshorocompactificationregular variationDiestel-Leader graphsCayley graphssymmetric spacesmetric measure spacesvolume growth
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The pith

Necessary and sufficient conditions ensure convergence to a unique ideal Poisson-Voronoi tessellation on any proper pointed measured metric space.

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

The paper establishes necessary and sufficient conditions for convergence of Poisson-Voronoi tessellations toward a unique ideal version on general proper pointed measured metric spaces. The conditions are that the volume function composed with log is regularly varying and that the uniform probability measure on large balls converges in the horocompactification. These criteria are applied to prove convergence on higher-rank symmetric spaces and to show that Diestel-Leader graphs yield an IPVT whose cells are distinguishable, the first such Cayley graph. Versions for graphs introduce a parameter ξ whose independence is established under mild assumptions.

Core claim

On any proper pointed measured metric space, convergence toward a unique ideal Poisson-Voronoi tessellation holds if and only if the volume function composed with log is regularly varying and the uniform probability measure on large balls admits a limit in the horocompactification. The same criterion is used to resolve convergence for higher-rank symmetric spaces and to establish that the IPVT on Diestel-Leader graphs has distinguishable cells.

What carries the argument

The two conditions of regular variation of the log-volume function together with existence of the horocompactification limit of uniform ball measures, which together characterize uniqueness and convergence of the ideal Poisson-Voronoi tessellation.

If this is right

  • Convergence to a unique IPVT holds for all higher-rank symmetric spaces.
  • The IPVT on graphs is independent of the parameter ξ in a specific sense under mild assumptions.
  • Diestel-Leader graphs give the first Cayley graph whose IPVT cells are distinguishable.
  • The theorem extends to edge-measured graphs with the same independence property.

Where Pith is reading between the lines

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

  • The same pair of conditions may be verifiable on additional spaces with regular volume growth to obtain further convergence results.
  • Distinguishability of cells on Diestel-Leader graphs could permit finer statistical analysis of the random tessellation.
  • Parameter independence suggests the limiting tessellation is stable across different natural choices of discretization.

Load-bearing premise

The uniform probability measure on large balls must converge to some limit in the horocompactification.

What would settle it

A proper pointed measured metric space in which the uniform ball measures fail to converge in the horocompactification yet the tessellations still converge to a unique IPVT would falsify necessity of the second condition.

Figures

Figures reproduced from arXiv: 2606.30517 by Ali Khezeli, Matteo D'Achille.

Figure 1.1
Figure 1.1. Figure 1.1: Portrait of a Poisson–Voronoi tessellation of [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Portrait of the horoboundary of DL(2, 2) described in Propositions 6.3 to 6.5. To study the IPVT of Diestel-Leader graphs, we first identify their horoboundary as follows. Proposition 6.3. (Horoboundary of DL(p, q)) The horoboundary of DL(p, q) consists of the following horofunctions: ρω(y, y′ ) := d(y, ω) given w ∈ ∂T, ρ ′ ω′(y, y′ ) := d(y ′ , ω′ ) given w ′ ∈ ∂T′ , ρx0 (y, y′ ) := −2h(y ∧ x0) − [PITH… view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Validity region defined before (6.1). So, p −r [PITH_FULL_IMAGE:figures/full_fig_p043_6_2.png] view at source ↗
read the original abstract

We provide necessary and sufficient conditions for convergence towards a unique IPVT on any proper pointed measured metric space. The conditions are that the volume function, when composed with $\log$, is regularly varying and that the limit of the uniform probability measure on a large ball exists in the horocompactification. As an application we prove convergence towards a unique IPVT for higher rank symmetric spaces, which solves an open problem of \cite{MiMe23}. Versions of this theorem are provided for graphs and edge-measured graphs, where a natural parameter $\xi$ appears. We prove independence on $\xi$ in a specific sense under mild assumptions, which answers an open problem of~\cite{IPVT}. As a main example, we show that the latter holds for the IPVT of Diestel-Leader graphs. We also focus on further properties of this example, in particular, that its IPVT cells are distinguishable, providing the first Cayley graph with this property.

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

2 major / 2 minor

Summary. The manuscript establishes necessary and sufficient conditions for convergence of a Poisson point process toward a unique ideal Poisson-Voronoi tessellation (IPVT) on any proper pointed measured metric space: the composition of the volume function with log must be regularly varying, and the uniform probability measures on large balls must converge in the horocompactification. Versions are given for graphs and edge-measured graphs. Applications prove the convergence for higher-rank symmetric spaces (resolving an open problem of MiMe23) and for Diestel-Leader graphs, where independence on the parameter ξ holds under mild assumptions (resolving an open problem of IPVT) and the cells are distinguishable (first such Cayley graph).

Significance. The general necessary-and-sufficient criterion on arbitrary proper pointed measured metric spaces is a strong structural result. If the verifications of the horocompactification limit are complete, the applications resolve two stated open problems and supply the first Cayley-graph example with distinguishable IPVT cells. The graph-theoretic versions and the independence-on-ξ statement add concrete value for discrete settings.

major comments (2)
  1. [§4] §4 (higher-rank symmetric spaces): the argument establishing existence of the horocompactification limit (condition (ii)) for these spaces is load-bearing for the claimed resolution of MiMe23. The passage from regular variation of vol ∘ log to the existence of the limit must be spelled out explicitly; it is not immediate that the same compactness or continuity properties used in rank 1 extend without additional justification.
  2. [§5] §5 (Diestel-Leader graphs): the proof that the horocompactification limit exists and is independent of ξ (used for both the convergence theorem and the distinguishability claim) must verify that the limit measure does not depend on the choice of basepoint or on the edge-measure parameter. The current sketch appears to invoke regular variation directly; an explicit construction of the limit measure or a reference to a prior compactness result in the horocompactification topology is needed.
minor comments (2)
  1. [§2] The definition and topology of the horocompactification should be recalled in §2 before the statement of the main theorem, rather than deferred to an appendix.
  2. [Notation throughout] Notation for the uniform probability measure on the ball of radius r should be introduced once and used consistently; the transition between continuous and discrete (graph) versions is occasionally unclear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the significance of the general criterion and the resolutions of the open problems. We address the two major comments below and will revise the manuscript to strengthen the explicitness of the arguments in §§4 and 5.

read point-by-point responses

  1. Referee: [§4] §4 (higher-rank symmetric spaces): the argument establishing existence of the horocompactification limit (condition (ii)) for these spaces is load-bearing for the claimed resolution of MiMe23. The passage from regular variation of vol ∘ log to the existence of the limit must be spelled out explicitly; it is not immediate that the same compactness or continuity properties used in rank 1 extend without additional justification.

    Authors: We agree that the transition from regular variation of vol ∘ log to the horocompactification limit in higher-rank symmetric spaces needs to be made fully explicit. In the revised version we will expand the relevant paragraph in §4, detailing the extension of the compactness and continuity arguments via the root-space decomposition and the action on the Furstenberg boundary, thereby confirming that condition (ii) holds independently of the rank-1 case. revision: yes

  2. Referee: [§5] §5 (Diestel-Leader graphs): the proof that the horocompactification limit exists and is independent of ξ (used for both the convergence theorem and the distinguishability claim) must verify that the limit measure does not depend on the choice of basepoint or on the edge-measure parameter. The current sketch appears to invoke regular variation directly; an explicit construction of the limit measure or a reference to a prior compactness result in the horocompactification topology is needed.

    Authors: We accept that the independence statement in §5 requires a more detailed verification. The revision will include either an explicit construction of the limiting measure on the horocompactification (showing invariance under basepoint change and under the edge-measure parameter ξ) or a direct citation of the relevant compactness theorem in the horocompactification topology, thereby confirming that the limit is well-defined under the stated mild assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: conditions are external and independent of IPVT

full rationale

The paper states necessary and sufficient conditions (regular variation of vol ∘ log, plus existence of uniform ball measure limit in horocompactification) for convergence to a unique IPVT on any proper pointed measured metric space. These inputs are defined independently of the target IPVT and are not obtained by fitting, self-definition, or renaming. Applications to symmetric spaces and Diestel-Leader graphs consist of verifying the stated external conditions rather than reducing the conclusion to a parameter fit or self-citation chain. No load-bearing self-citation or ansatz smuggling is exhibited in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the two stated conditions (regular variation after log and existence of the horocompactification limit) together with standard background facts about proper pointed measured metric spaces and horocompactifications; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)
  • domain assumption The underlying space is a proper pointed measured metric space
    Invoked at the outset as the setting for the convergence theorem.
  • standard math Standard properties of horocompactification and regular variation hold
    Used to formulate the necessary and sufficient conditions.

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