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arxiv: 2604.05760 · v1 · submitted 2026-04-07 · 🧮 math.PR

The volume of hyperbolic Poisson zero cells: critical divergence and exact second moment

Tillmann B\"uhler , Christoph Th\"ale This is my paper

Pith reviewed 2026-05-10 18:50 UTC · model grok-4.3

classification 🧮 math.PR
keywords hyperbolic geometryPoisson hyperplane processzero cellvolume momentscritical phenomenaMeijer G-functionphase transition
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The pith

In hyperbolic space the second volume moment of Poisson zero cells diverges like R cubed at the critical intensity.

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

The authors study the volume of the zero cell formed by a Poisson hyperplane process in hyperbolic space. They prove that at the critical intensity where the zero cell becomes bounded almost surely, the second moment of the volume inside a large ball grows exactly like the cube of the radius, independently of dimension. Above this intensity the unrestricted second moment is finite and they give its exact value in terms of a Meijer G-function. This establishes a sharp phase transition and supplies explicit formulas that can be compared to the Euclidean setting and to percolation theory.

Core claim

At the critical intensity γ_c^{(d)} the second volume moment of the zero cell restricted to a hyperbolic ball of radius R diverges at the universal rate R^3 as R tends to infinity in any dimension. For intensities above criticality the second moment of the full zero cell is finite and equals an explicit expression involving the Meijer G-function. The paper also determines the asymptotic behavior of this moment as the intensity tends to infinity and as it approaches the critical value from above.

What carries the argument

The zero cell of the Poisson hyperplane tessellation in hyperbolic space, whose volume moments are computed using harmonic analysis to produce closed-form Meijer G-function expressions.

If this is right

  • The divergence at criticality occurs at the same rate in all dimensions.
  • The second moment remains finite and has a closed form when the intensity exceeds the critical threshold.
  • As the intensity increases without bound the second moment approaches a value comparable to the Euclidean case.
  • The behavior near criticality matches expectations from the mean-field universality class of percolation theory.

Where Pith is reading between the lines

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

  • This phase transition is absent in Euclidean space, suggesting curvature plays a key role in bounding the zero cell.
  • The exact Meijer G expression opens the possibility of studying higher-order moments or other geometric functionals of the zero cell with the same methods.
  • Similar techniques might apply to Poisson processes on other non-Euclidean spaces to reveal analogous critical phenomena.

Load-bearing premise

The Poisson hyperplane process admits a well-defined stationary version on hyperbolic space for which a finite critical intensity exists that makes the zero cell almost surely bounded.

What would settle it

Numerical simulation of the Poisson hyperplane process in hyperbolic space at the critical intensity, measuring the growth rate of the second volume moment of the restricted zero cell as the ball radius increases.

Figures

Figures reproduced from arXiv: 2604.05760 by Christoph Th\"ale, Tillmann B\"uhler.

Figure 1
Figure 1. Figure 1: Simulations of Poisson zero cells Zo for different intensities γ in the Poincaré disc model for H2 . Left: γ = 0.25 (subcritical). Middle: γ = π/2 ≈ 1.57 (critical). Right: γ = 2 (supercritical). is defined as the almost surely unique cell containing a fixed reference point o ∈ Hd , also called the origin. Let γ (d) c := √ π(d − 1)Γ( d+1 2 ) Γ( d 2 ) (2) denote the critical intensity in dimension d. It is … view at source ↗
Figure 2
Figure 2. Figure 2: Blue: Exact value of EH2 (Zo) 2 (left) and EH3 (Zo) 2 (right) as a function of γ. Red: The Euclidean value 4π 6 7γ 4 (left) and 14336π 2 γ 6 (right) as a function of γ. an exact analytic expression for this moment, which naturally involves the Meijer G-function. Recall that the Meijer G-function is a classical special function in the theory of hypergeometric functions and integral transforms, see [17]. It … view at source ↗
Figure 3
Figure 3. Figure 3: The path of integration L = iR separates the left pole families from the right pole families. Let h(a; s) be the integrand in (24). Since the Gamma function is meromorphic with simple poles at 0, −1, −2, . . . and has no zeros, the reciprocal 1/Γ(·) is an entire function. Consequently, possible poles of h(a; s) can only come from the Gamma factors in the numerator. These occur at s = −a − n, s = a + n, s =… view at source ↗
read the original abstract

We investigate the second volume moment of the zero cell $Z_o$ of a Poisson hyperplane tessellation with intensity $\gamma$ in the $d$-dimensional hyperbolic space. We focus on the phase transition at the critical intensity $\gamma_c^{(d)}$, the minimum value for which $Z_o$ is almost surely bounded. In the critical regime $\gamma=\gamma_c^{(d)}$, we show that the second volume moment of the restricted zero cell $Z_o \cap B_R$, where $B_R$ is a hyperbolic ball of radius $R$ centred at $o$, diverges in any dimension at the universal rate $R^3$ as $R \to \infty$. In the supercritical case $\gamma > \gamma_c^{(d)}$, we prove that the full second volume moment is finite. Using tools from harmonic analysis in hyperbolic space, we derive an exact expression for this moment in terms of the Meijer $G$-function. Furthermore, we determine the asymptotic behaviour of the second moment as $\gamma \to \infty$ and as $\gamma \downarrow \gamma_c^{(d)}$, facilitating a direct comparison with the corresponding Euclidean values as well as the mean-field universality class of percolation theory.

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 investigates the second volume moment of the zero cell Z_o of a Poisson hyperplane tessellation with intensity γ in d-dimensional hyperbolic space. It identifies the critical intensity γ_c^{(d)} at which Z_o becomes almost surely bounded, proves that at criticality the second moment of vol(Z_o ∩ B_R) diverges universally as R^3 as R→∞, and shows that for γ > γ_c^{(d)} the unrestricted second moment is finite with an exact closed-form expression in terms of the Meijer G-function. Asymptotics as γ→∞ and γ↓γ_c^{(d)} are derived to compare with Euclidean counterparts and percolation mean-field behavior.

Significance. If the central claims hold, the work supplies rare exact results in hyperbolic stochastic geometry, including a universal critical divergence rate and a Meijer-G expression obtained via harmonic analysis. These permit direct comparison with Euclidean Poisson hyperplane tessellations and with percolation universality classes, strengthening the bridge between hyperbolic random geometry and critical phenomena.

minor comments (2)
  1. The abstract and introduction refer to 'tools from harmonic analysis in hyperbolic space' without naming the specific integral representations or transforms employed; adding a brief outline of the key steps (e.g., the relevant spherical functions or Fourier inversion formula) would improve readability.
  2. Notation for the restricted cell Z_o ∩ B_R and the ball radius R is introduced clearly, but the dependence of γ_c^{(d)} on dimension is stated without an explicit formula or reference to its derivation; a short paragraph recalling its existence proof would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary accurately captures our main results on the phase transition for the second volume moment of the zero cell at the critical intensity, the universal R^3 divergence, and the Meijer G-function expression in the supercritical regime. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations rely on external harmonic analysis

full rationale

The paper defines the critical intensity γ_c^{(d)} as the threshold where the zero cell becomes a.s. bounded and then applies standard integral representations and tools from hyperbolic harmonic analysis to obtain the R^3 divergence of the restricted second moment and the exact Meijer G-function expression for the supercritical case. These steps do not reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the claimed asymptotics and exact formulas follow from external machinery without internal reduction to the inputs. The abstract and structure indicate self-contained use of known harmonic-analysis techniques, consistent with a score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes the standard definition of a Poisson hyperplane tessellation on hyperbolic space and the existence of a critical intensity, but introduces no fitted numerical parameters, no new postulated entities, and no ad-hoc axioms beyond the usual background of stochastic geometry and hyperbolic harmonic analysis.

axioms (2)
  • domain assumption Poisson hyperplane tessellation with intensity γ is well-defined in d-dimensional hyperbolic space
    Invoked to define the zero cell Z_o and the critical intensity γ_c^{(d)}.
  • standard math Harmonic analysis on hyperbolic space yields an exact integral representation convertible to a Meijer G-function
    Used to obtain the closed-form expression for the supercritical second moment.

pith-pipeline@v0.9.0 · 5515 in / 1609 out tokens · 52760 ms · 2026-05-10T18:50:26.593401+00:00 · methodology

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