{"id":"6da95f2b-0ec3-4017-905b-cac65435b06f","arxiv_id":"2607.17764","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For Poisson-Voronoi percolation on H^d, d≥3, the uniqueness threshold satisfies inf_{λ>0} p_u(λ) > 0.","lead":"This paper proves that for Poisson-Voronoi percolation on hyperbolic space of dimension at least three, the threshold for having exactly one unbounded black cluster stays bounded away from zero even when the point process is made extremely sparse. It settles a question of Grebík and Recke and shows hyperbolic space behaves differently from product hyperbolic spaces, where the same threshold was recently shown to vanish.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the exponential-overlap Lemma 9 — the reader's weakest assumption — is correct and the reduced-path comparison survives scrutiny.","rationale":"Stress-test pass found no load-bearing flaw. The central claim inf p_u>0 rests on showing connection probabilities vanish for small p uniformly in λ≤1; the mechanism is the reduced-path comparison. I verified the geometric core: Lemma 9 correctly establishes exponential decay of overlap of equal hyperbolic balls; the proof's reduction to s≤r/2 and use of Wendel/Chernoff are sound. Corollary 17 then makes the overlapping Gabriel balls' contribution controllable, and Lemma 15's conditional-probability arguments are internally consistent. I also checked Lemma 12's appendix proof: the Catalan recursion and the integral bound hold (the exponent algebra in the φ estimate is correct after restoring 2(d-1) exponents; the displayed 4^{d-1} factor is consistent). Proposition 23's direct proof of inf_{λ>λ0} p_c>0 is valid: the bad-vertex coarse-graining yields an infinite geodesic of bad vertices with probability zero. The only issues are minor: a typographical inequality in Lemma 16's proof and reliance on unpublished work for context, neither affecting the theorem. Thus the reader's ACCEPT stands unchanged; the identified weakest assumption, while central, does not fail.","tokens_in":33584,"tokens_out":49361,"duration_ms":451574,"concrete_test":"Independently verify Lemma 9 numerically for d=3,4: in the halfspace model with x=(0,0,1), y=(0,0,e^s), compute the ratio μ(B(x,r)∩B(y,r))/μ(B(x,r)) for r=10,20 and s=1,2,5,10 using high-precision quadrature of (1/z_3^d) over the intersection of the two hyperbolic balls. Confirm the ratio decays exponentially in s with a rate independent of r. This is the exact input to Corollary 17; if the decay were only polynomial, the convergence of the sum in (35) and hence R_n(h)≤S_n(h) would fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the transfer from the independent-edge model to the Delaunay graph via reduced paths, which rests on the exponential overlap decay of equal hyperbolic balls (Lemma 9). I checked this chain. Lemma 9 is correct: the proof uses the hyperbolic cosine rule to show that points in A1 subtend an angle ≤ C e^{-s/2} at x, and the rotational invariance of the Poincaré ball then gives volume fraction ≤ C e^{-(d-1)s/2}; the reduction from s≤2r to s≤r/2 is valid. Corollary 17's use of Lemma 9 to make the tail Σ(16x^4+1)K e^{-c(x-1)} small is legitimate. Lemma 15's greedy selection and conditional-probability bound via Corollary 8 are consistent; the displayed inequality in Lemma 16 has a harmless typo (the computed lower bound is actually 16.5x, still >x). Lemma 12's induction is self-contained and the algebra checks out. Proposition 23's elementary proof of a positive high-intensity percolation threshold is also sound. No internally inconsistent step or hidden λ-dependence was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":33905,"tokens_out":27310,"duration_ms":244901,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Lemma 16"},{"comment":"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.","section":"Abstract/§1"},{"comment":"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.","section":"Appendix A.6, Proposition 23"},{"comment":"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.","section":"Corollary 8"},{"comment":"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.","section":"Section 3.5 / Corollary 20"}],"recommendation":"minor_revision","confidential_remarks":"For the editor: the manuscript is a strong, essentially self-contained resolution of an open problem. The only external unverified input (D'Achille–Curien) is not used. I recommend minor revision; the requested changes are cosmetic/typographical. No concerns about novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper answers the explicit open question of Grebík and Recke: for Poisson-Voronoi percolation on H^d with d≥3, inf_{λ>0} p_u(λ)>0. That is the headline. It separates H^d from product hyperbolic spaces, where p_u(λ) tends to zero as λ→0.\n\nWhat is actually new: the reduced-path technique in Section 3.5. It is a clever way to control dependencies when transferring bounds from an independent-edge model to the Delaunay graph. The key geometric input is Lemma 9: two equal hyperbolic balls have exponentially small volume overlap as their centers separate. That is correct — the cosine-rule argument and the cone estimate check out. The stress-test note is right; this is not a soft spot. The main probabilistic chain—Lemma 6, Proposition 10, Lemma 15, Corollaries 20–21, Propositions 22–23—is present and coherent. The proof is largely self-contained; the appendix supplies the postponed arguments, including an elementary proof of Proposition 23 and a simplified proof of the Gracar et al. lemma. That is real work, and credit is due.\n\nThe soft spots are mostly cosmetic. Lemma 16 has a harmless typo in the displayed inequality; the constants are dense but manageable; and the paper cites an unpublished D'Achille–Curien statement for context only. The proof is intricate and not machine-checked, so there is residual correctness risk, but I found no step that fails. The dependence on hyperbolic geometry is structural, as it should be: the exponential overlap decay is what makes the reduced-path comparison work, and that is exactly what separates this setting from Euclidean space.\n\nFor whom: specialists in percolation and stochastic geometry, especially anyone working on hyperbolic or dependent percolation. The writing is clear enough for a PhD student with background in Poisson processes and percolation to follow the main ideas, though the details require patience.\n\nMy verdict: it deserves a serious referee and should be published. I would cite it in my own work and would bring it to a reading group if the group tolerates long technical sections.\n\nRecommendation: send to peer review at a top probability journal; expect a solid but lengthy referee report.","headline":"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.","tokens_in":34374,"tokens_out":1938,"would_cite":true,"duration_ms":21043,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B43","60D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A lower bound holds for the uniqueness threshold in hyperbolic Voronoi percolation","keywords":["Poisson-Voronoi percolation","hyperbolic space","uniqueness threshold","Delaunay graph","Gabriel ball","percolation phase transition","random connection model","reduced paths"],"falsifier":"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.","tokens_in":33498,"feed_emoji":"📐","tokens_out":4788,"duration_ms":47680,"temperature":0.7,"pith_summary":"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.","feed_headline":"Unique-cluster threshold cannot vanish on H^d, d≥3","feed_subtitle":"A uniform lower bound holds for every Poisson intensity, setting hyperbolic space apart from product spaces.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Hyperbolic Voronoi percolation: uniqueness threshold cannot vanish","For hyperbolic Voronoi percolation, p_u never drops to zero","In H^d (d≥3), uniqueness threshold has uniform positive bound","Hyperbolic dimension 3+: uniqueness threshold stays away from zero","Poisson-Voronoi on H^d (d≥3): p_u infimum is positive"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hyperbolic Voronoi percolation: uniqueness threshold cannot vanish","For hyperbolic Voronoi percolation, p_u never drops to zero","In H^d (d≥3), uniqueness threshold has uniform positive bound","Hyperbolic dimension 3+: uniqueness threshold stays away from zero","Poisson-Voronoi on H^d (d≥3): p_u infimum is positive"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000827,"raw_usage":{"total_tokens":3498,"prompt_tokens":841,"completion_tokens":2657,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":2570}},"tokens_in":585,"tokens_out":2657,"duration_ms":18368,"temperature":1.0,"reasoning_tokens":2570,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T17:05:38.056612+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}