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Non-vanishing uniqueness threshold for hyperbolic Poisson-Voronoi percolation in dimension at least three

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Pith's one-line read A lower bound holds for the uniqueness threshold in hyperbolic Voronoi percolation

desk verdict This paper resolves an open question of Grebík and Recke with a genuinely new reduced-path technique, and the proof largely holds up under scrutiny. read the letter →

arxiv 2607.17764 v1 pith:VNOMTCNX submitted 2026-07-20 math.PR

Matthias Irlbeck , Tobias Müller This is my paper · ORCID
classification math.PR MSC 60K3582B4360D05
keywords Poisson-VoronoipercolationhyperbolicspaceuniquenessthresholdDelaunaygraphGabrielballphasetransitionrandomconnectionmodelreducedpaths
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

The paper proves that in hyperbolic space of dimension at least three, the uniqueness threshold p_u(λ) — the smallest black-colouring probability needed to have positive chance of exactly one unbounded cluster — stays bounded below by a dimension-dependent constant, uniformly over all intensities λ>0. This answers a question left open by recent work showing the threshold vanishes in product spaces and tends to 1 on the hyperbolic plane, establishing that high-dimensional hyperbolic space behaves differently from both. The proof shows that for p below a constant, the probability that two far-apart points are joined by black Voronoi cells decays to zero, which rules out a unique infinite cluster. The argument works by comparing the dependent Delaunay graph with a tractable independent-edge model and controlling the comparison through a geometric fact special to hyperbolic space: equal balls overlap exponentially little as their centres separate.

What carries the argument

The key object is the 'reduced path': a subsequence of a black Delaunay path that is shortcut-free and whose 'normal' edges have Gabriel balls with controlled overlap. The Gabriel ball of two points is the ball with the geodesic midpoint as centre and radius equal to half their distance; a Delaunay edge requires this ball to contain no Poisson point. The machinery works by showing the expected number of reduced paths in the dependent graph is no larger than the expected number of shortcut-free paths in an independent-edge model, using the exponential overlap decay of equal hyperbolic balls to redistribute costs over the path. The independent model's edge probabilities decay as a power of a g

What would settle it

Find, for some d≥3, an intensity λ and a p smaller than the claimed uniform lower bound such that the connection probability between two points at hyperbolic distance h does not tend to zero as h→∞ — for instance by simulating the model and measuring whether a fixed point reaches arbitrarily distant targets with positive probability at small p. A single such pair (λ,p) would contradict Theorem 1. Alternatively, compute p_u(λ) exactly at a fixed small λ and show it is zero.

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Extended reading notes

Core claim

The central claim is Theorem 1: inf_{λ>0} p_u(λ)>0 for Poisson-Voronoi percolation on H^d with d≥3. More precisely there is a p0=p0(d)>0 such that for all p≤p0 and all λ>0 the connection probability between two far-apart points decays to zero with their distance, and consequently there is almost surely no unique unbounded black cluster. The paper establishes this by bounding the probability that a pair of points are adjacent in the Delaunay graph by a doubly exponential function of their distance, introducing an independent-edge random connection model whose edge probabilities dominate this bound and are exactly computable, and showing via specially constructed 'reduced paths' that expected

Load-bearing premise

The central geometric input is that in hyperbolic space the volume of the overlap of two equal balls decays exponentially in their separation; if that overlap decayed only polynomially (as it does in Euclidean space), the bound transferring independent-edge paths to dependent Delaunay paths would fail.

Editorial extensions

If this is right

  • For every intensity λ>0 and every p below a dimension constant p0(d), the event of exactly one unbounded black cluster has probability zero.
  • Connection probabilities between far-apart points in the Delaunay graph decay to zero uniformly in λ for p≤p0, so the uniqueness threshold satisfies p_u(λ)≥p0 for all λ.
  • The hyperbolic plane (where the threshold tends to 1) and product spaces (where it tends to 0) are both separated from H^d with d≥3, which has a threshold uniformly bounded away from 0.
  • The proof yields effective, though not sharp, constants, so a quantitative lower bound on p_u(λ) could in principle be extracted for any fixed dimension.

Reading between the lines

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

  • The exponential overlap estimate is the load-bearing geometric input: a space whose equal-ball intersections decay only polynomially (Euclidean space, product spaces) would not admit this comparison, suggesting the uniform positive threshold is a genuinely negatively-curved phenomenon.
  • A natural next target is determining whether lim_{λ→0} p_u(λ) exists; the techniques here give upper bounds on connection probabilities but not the asymptotic value, and could be sharpened by choosing the independent-edge probabilities closer to the true Delaunay edge probabilities.
  • The reduced-path construction is a general scheme for handling path dependencies in geometric graphs; it could plausibly be transferred to site percolation on Delaunay complexes of other negatively curved spaces or to non-amenable Cayley graphs built from hyperbolic isometries, though that transfer would require re-proving the overlap controls.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. This paper proves that for Poisson-Voronoi percolation on hyperbolic space H^d, d≥3, the uniqueness threshold satisfies inf_{λ>0} p_u(λ)>0. The proof uses the standard characterization: for a small dimension-dependent p_0 and all λ≤1, the connection probability between two far-away points tends to zero, so p_0≤p_u(λ); for λ≥1, a separate coarse-graining argument gives a uniform positive lower bound on p_c(λ), hence on p_u(λ). The main technical work is an independent-edge model with edge probabilities P(z,w) given by (23), a bound on the expected number of shortcut-free paths (Proposition 10), and a reduced-path construction (Section 3.5) showing that any black Delaunay path can be compressed so that the expected number of reduced paths is at most the independent-edge path count. The reduction uses the exponential decay of overlap volume for equal hyperbolic balls (Lemma 9). All auxiliary lemmas are proved, including a self-contained proof of the Gracar–Lüchtrath–Mörters-type bound (Lemma 12) and a new elementary proof of the positive high-intensity percolation threshold (Proposition 23).

Significance. The result answers an open question of Grebík and Recke and establishes a strong contrast: whereas the uniqueness threshold vanishes at small intensities for Poisson-Voronoi percolation on products of hyperbolic spaces, it has a positive uniform lower bound in the rank-one case H^d, d≥3. The proof is original and essentially self-contained; the reduced-path technique for transferring estimates from an independent-edge model to the dependent Delaunay graph is the most novel contribution and may be of independent interest. The paper also contains a self-contained proof of a positive percolation threshold for all sufficiently large intensities, replacing a black-box use of a deep result. I found no load-bearing errors; the main chain Lemma 6 → Proposition 10 → Lemma 15/Proposition 14 → Corollaries 20–21 → Theorem 1 is sound.

minor comments (5)
  1. [Lemma 16] In the proof of Lemma 16, the displayed chain ending in '=14x' appears to be a numerical typo; the lower bound is actually 16.5x. The subsequent conclusion dist(c_ℓ,c_i)>14x remains valid, so this is purely presentational.
  2. [Abstract/§1] The statement attributing p_u(λ)≤1/2 to an unpublished argument of D'Achille and Curien is not verifiable from the cited literature. Since this result is not used in the proof, please either replace it by a public reference or clearly mark it as context not needed for the theorem.
  3. [Appendix A.6, Proposition 23] In the paragraph 'We claim there is no infinite path...', the sentence 'Letting n→∞ and using (47), this probability is zero' is terse. Spell out that an infinite bad cluster contains, for every n, a chordless bad path of length n, whose even-indexed vertices have independent badness events.
  4. [Corollary 8] The first inequality in the proof uses that E and the E_i are decreasing in the white point configuration; adding one sentence on this monotonicity would improve readability.
  5. [Section 3.5 / Corollary 20] The definition of R_n(h) counts reduced paths whose internal vertices all have height <h; in the proof of Corollary 20 it is implicitly used that a connection to a point of height ≥h can be truncated at the first time the height exceeds h. This is correct, but it may be worth stating explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is self-contained and does not reduce to its inputs by construction.

full rationale

The derivation chain for Theorem 1 is: Lemma 6/Corollary 8 bound Delaunay edge probabilities; Lemma 9 supplies the hyperbolic ball-overlap decay; the independent-edge model with kernel P is analyzed in Proposition 10; the reduced-path comparison (Lemma 15, Proposition 14) transfers the bound back to the Delaunay graph; Corollaries 20 and 21 convert the expectation bound into vanishing connection probabilities; and Propositions 22 and 23 turn this into the stated lower bound on p_u(λ). Every load-bearing auxiliary result is proved in the paper, including Lemma 12 (with a self-contained proof in Appendix A.3 despite the attribution to Gracar et al.), Lemma 9, Proposition 14, Proposition 22, and Proposition 23 (with an elementary proof in Appendix A.6). The kernel P(z,w) is chosen to dominate the true edge probability and is not fitted to the desired conclusion; the constant p_0 is selected only after the bound S_n(h) ≤ (c_1 p)^n P(Po(c_2 ln h) < n) is established. External results (Grebík–Recke, d'Achille et al., D'Achille–Curien, Bühler et al., Benjamini–Schramm) are contextual or non-load-bearing; in particular the unpublished D'Achille–Curien bound p_u ≤ 1/2 is not used in the proof. The only self-citation of Hansen–Müller serves to recall the Mecke formula, which is a standard external tool and not the target result. No equation or fitted parameter is shown to be equivalent by construction to the claimed theorem, so the derivation is not circular.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The proof rests on standard probabilistic tools (Chernoff, Wendel, Mecke/Palm) and standard hyperbolic geometry, all cited with proofs or references. The hand-chosen constants s0, x0, δ, A are proof parameters, not fitted values, and none encode the conclusion. The independent-edge model and reduced paths are auxiliary mathematical constructions, not new physical entities. No circularity burden arises from the cited literature because the paper proves the key transfer lemmas itself.

free parameters (4)
  • s0 = chosen sufficiently large to satisfy (38); exact value unspecified
    Cutoff parameter in the definition of φ and of the independent-edge probability P; controls where P(z,w) is identically 1. It enters the constants c1(s0), c3, c5 and ultimately p0.
  • x0 = max(x1(δ), 10(d-1))
    Integer threshold in the reduced-path definition (32); must be large enough for Corollary 17 and Lemma 16 to control overlaps of Gabriel balls.
  • delta = sufficiently small; not given numerically
    Overlap fraction in Corollary 8/Corollary 17; chosen small enough that the conditional edge probability retains its exponential bound.
  • A (Appendix A.6) = large enough that Δ_A e^{-A/4}<1
    Used in the elementary proof of Proposition 23 to make the covering argument work for all λ≥λ0.
assumptions (6)
  • standard math Chernoff bound for Poisson and binomial random variables (Lemma 2).
    Used in Lemma 6 and in the proof of Proposition 23.
  • standard math Wendel's formula for i.i.d. rotationally invariant points (Lemma 4).
    Used in Lemma 6 to bound the probability that random points in a ball avoid a halfspace.
  • standard math Mecke formula and Palm calculus for Poisson processes (Theorem 5).
    Used throughout to count paths of black Poisson points by integrals.
  • standard math Hyperbolic geometry facts: volume of balls is (1+o(1)) ω_{d-1} e^{(d-1)r}/2^{d-1}(d-1), balls in Poincaré models are Euclidean balls, hyperbolic cosine rule (Section 2.2).
    Provides the volume estimates (8)-(10) and the overlap computations in Lemma 9.
  • domain assumption Meester-Roy ergodicity: the event of exactly one unbounded cluster has probability 0 or 1.
    Used in Proposition 22 to convert a positive-probability uniqueness event into an almost-sure event.
  • domain assumption Harris-FKG inequality for the two-type Poisson process (increasing in black, decreasing in white).
    Used in the final step of Proposition 22 to lower-bound the probability of two points connecting.

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Pith. "Pith review of Non-vanishing uniqueness threshold for hyperbolic Poisson-Voronoi percolation in dimension at least three." pith.science (2026). https://pith.science/paper/VNOMTCNX

@misc{pith2026260717764,
  author       = {Pith},
  title        = {Pith review of: Non-vanishing uniqueness threshold for hyperbolic Poisson-Voronoi percolation in dimension at least three},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNOMTCNX}},
  note         = {Machine review of arXiv:2607.17764}
}
abstract

We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space $\mathbb{H}^d$ for $d\geq 3$. By recent results of Greb\'ik and Recke and d'Achille et al., this "uniqueness threshold" $p_u(\lambda)$ tends to zero as the intensity $\lambda$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products $\mathbb{H}^{d_1}\times\dots\times\mathbb{H}^{d_k}$ with $k,d_1,\dots,d_k\geq 2$. In contrast, for Poisson-Voronoi percolation on the hyperbolic plane $\mathbb{H}^2$, Benjamini and Schramm have shown that $p_u(\lambda)$ tends to one as $\lambda$ tends to zero, and $p_u(\lambda)>1/2$ for all $\lambda>0$. An unpublished argument of D'Achille and Curien shows that for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$, the uniqueness threshold satisfies $p_u(\lambda)\leq 1/2$ for all $\lambda>0$. Here we will show that $\inf_{\lambda>0}p_u(\lambda)>0$ for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$. This answers a question of Greb\'ik and Recke.

Figures

Figures reproduced from arXiv: 2607.17764 by the authors.

Figure 1
Figure 1. Computer simulation of Poisson-Voronoi percolation on [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the sets A1, cone(y, α). This gives µ(A1) ≤ µ  B(0, r) ∩ cone(y, 3 3/2 e −s/2 )  = µ(B(0, r)) · P(U ∈ cone(y, 3 3/2 e −s/2 )), where U is uniform on B(0, r), i.e. chosen according to the renormalized hyperbolic volume measure on B(0, r). By the (rotational) symmetries of the Poincar´e ball model, we can invoke Lemma 3, to obtain: µ(A1) = µ(A2) ≤ µ(B(0, r)) · c · 3 3(d−1)/2 · e −((d−1)/2)·s , (21) w… view at source ↗

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