{"id":"25e5df08-8292-48b5-90fa-00b29da74775","arxiv_id":"2506.10511","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the critical long-range percolation metric, geodesics are almost surely unique and every geodesic has Hausdorff dimension equal to the distance exponent theta.","lead":"This paper proves that the random distance function attached to critical long-range percolation has a unique shortest path between any two fixed points, and that every such shortest path has fractal dimension exactly equal to the known distance growth exponent theta. The results settle the basic geodesic geometry of this limiting metric, completing a picture that had been open in the recent construction of Ding, Fan, and Huang.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.2's transfer argument is not demonstrated: conditioning on the fixed exterior edge set E(v) changes the law of the metric inside B_r(v), and Eq. (4.6) swaps D(x,y;B_r(v)^c) for D(x,y;E without the edge <u,v>) without justification. This gap directly supports the uniqueness theorem.","rationale":"The reader and I converge on the same gap: Lemma 4.2 is the critical bridge that is not self-contained, and the step from (4.5) to (4.6) switches metrics without justification. The paper otherwise is carefully argued, with the Sperner machinery of Sections 2 and 3 giving plausible support for Theorem 1.2, and the renormalization estimates of Section 5 providing a solid upper bound on the geodesic dimension. I see no evidence of fraud, no circularity, and no invented entities. The issue is one of an unverified conditional version of a delicate argument and a notational metric switch, both plausibly fixable with a longer write-up. Hence the appropriate verdict remains CONDITIONAL, and the recommendation is to condition acceptance on a complete proof of Lemma 4.2 and a careful justification of the (4.5)-(4.6) transition.","tokens_in":18715,"tokens_out":1789,"duration_ms":18660,"concrete_test":"Re-derive Lemma 4.2 without the sentence in question: write the conditional law of D on B_r(v) given E(v), verify that Theorem 1.2's Sperner construction (Section 3.3, especially Lemma 3.14) goes through under this conditional law with the modified edge intensity, and prove the inequality/equality bridging D(x,y;B_r(v)^c) and D(x,y;E\\{<u,v>}) in (4.6). If (4.3) fails for some z or the (4.5)-(4.6) switch is not valid, Theorem 1.4 has no proof in the present form.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central structural claim is Theorem 1.4, and the only place where multiple geodesics are ruled out is Section 4. The proof hinges on Lemma 4.2, which asserts that a continuum analogue of the Sperner-based continuity result (Theorem 1.2) holds inside the ball B_r(v) after conditioning on the deterministic edge set E(v) that fixes all edges outside the ball and the long edge <u,v>. The lemma is discharged with the sentence 'according to the proof of Theorem 1.2 by replacing 0, x and M_i with v, z and |v-z|M_i, we can obtain (4.3) immediately.' But the Sperner construction in Section 3 crucially uses scale/translation covariance of the metric law and the tightness axioms, and it is not immediate that these axioms survive conditioning on E(v), which fixes a specific deterministic boundary set and a deterministic edge entering v. The proof of Theorem 1.2 also uses independence of edges in annuli; in the conditional setting the edge set inside B_r(v) has a Poisson law with a modified intensity (edges to B_r(v)^c are censored), and the distribution of boundary distances is no longer the unconditional D-law. A second, related gap is the silent switch from (4.5), which concerns D(x,y;B_r(v)^c), to (4.6), which concerns D(x,y;E\\{<u,v>}) for arbitrary pairs of edges; the displayed inequality asserts equality of a length with the metric computed in a different restricted edge set, and no monotonicity or projection argument is supplied. Until both transfers are written out, the uniqueness theorem rests on an unverified reduction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the limiting random metric D for critical long-range percolation on Z^d, recently constructed in [5], and proves two main results: Theorem 1.4, that there is almost surely a unique D-geodesic between any two fixed points, and Theorem 1.5, that every D-geodesic has Hausdorff dimension equal to the distance-growth exponent θ. The proof of continuity of the metric distribution (Theorem 1.2) is based on a generalized Sperner theorem, and this continuity is then used to rule out branching of geodesics in Section 4. The dimension result is proved by a lower bound via a mass distribution principle combined with a uniform continuity estimate, and an upper bound via a renormalization argument on the graph of small cubes.","tokens_in":18989,"tokens_out":3498,"duration_ms":39575,"significance":"If the results are correct, they are significant: they give the first proof of geodesic uniqueness for the critical long-range percolation metric and pin down the geodesic dimension exactly as the exponent θ. The paper contains substantial original ingredients, especially the Sperner-based continuity argument in Section 3 and the detailed renormalization upper bound in Section 5.2, and it builds in a transparent way on the established metric axioms and exponent estimates from [5] and [2]. However, the uniqueness theorem currently rests on a conditional-transfer step in Lemma 4.2 that is not proved, and the step from (4.5) to (4.6) requires additional justification. These issues affect the central claim of Theorem 1.4 and therefore the paper is not yet ready for acceptance.","major_comments":[{"comment":"The proof of Lemma 4.2 is not self-contained: the statement 'according to the proof of Theorem 1.2 by replacing 0, x and M_i with v, z and |v−z|M_i, we can obtain (4.3) immediately' is a deferral, not an argument. Conditioning on the edge set E(v) in (4.1) fixes a deterministic set of edges outside B_r(v) and the edge ⟨u,v⟩; this changes the law of the metric inside B_r(v), since edges to B_r(v)^c are censored and the Poisson intensity is modified. The proof of Theorem 1.2 in Section 3 uses independence of edges in disjoint annuli, translation and scale covariance, and the tightness axioms; none of these are verified for the conditional law, and the Sperner construction in Lemma 3.14 conditions on E\\E_J, which is not the same conditioning as here. This is load-bearing: (4.3) is the only place where the continuity of the conditional distribution is invoked, and without it the uniqueness proof has no basis.","section":"§4, Eqs. (4.5)–(4.6)"},{"comment":"The transition from (4.5) to (4.6) is not justified. Equation (4.5) proves, for each fixed r and z, that len(Q;D) ≠ D(x,y;B_r(v)^c) almost surely, where D(·,·;B_r(v)^c) is the metric computed using only edges in the complement of the ball B_r(v). Equation (4.6), however, asserts that almost surely len(Q;D) ≠ D(x,y;E\\{⟨u,v⟩}) for some edge ⟨u,v⟩ and some Q, where D(x,y;E\\{⟨u,v⟩}) is the metric in the entire graph with that single edge removed. These are different restricted metrics, and no monotonicity or projection argument is supplied to show that an equality with one transfers to the other. In addition, the implication 'multiple geodesics imply the event in (4.6)' is only sketched; one must prove that any pair of distinct geodesics can be related to a path Q_uvz as in Definition 4.1 for some edge ⟨u,v⟩ and some z∈Z_r(v), and that the ball-restricted metric in (4.5) can be replaced by the edge-removed metric in (4.6). Until both transfers are written out, Theorem 1.4 is unproved.","section":"§4, Eqs. (4.5)–(4.6)"}],"minor_comments":[{"comment":"There is a typo: 'we will choose oly one geodesic' should read 'we will choose only one geodesic'.","section":"Definition 4.1"},{"comment":"The word 'emcompass' in the introduction should be 'encompass'.","section":"Abstract/Introduction"},{"comment":"The phrase 'condition on A^c∩F, on the value of J and on E\\E_J' is grammatically awkward; it should be 'conditioning on ...'.","section":"Lemma 3.14"},{"comment":"The mass distribution ζ_P is defined on subsets of Range(P), but Lemma 5.1 requires a mass distribution on a metric space; since Range(P) is compact, this is fine, but a brief remark on measurability would improve readability.","section":"§5.1"},{"comment":"In the display following (5.4), the conditional expectation E[1_{M_i^c} Σ ... | G] is written without defining G as the renormalized graph edge set; this should be clarified.","section":"Eq. (5.4)"}],"recommendation":"major_revision","confidential_remarks":"The paper's uniqueness theorem (Theorem 1.4) is the main advertised novelty, but the proof of Lemma 4.2 is a deferral to Theorem 1.2 in a different conditioning regime, and the step from (4.5) to (4.6) is not justified. These are not cosmetic gaps: they are the load-bearing parts of the uniqueness argument. The dimension theorem (Theorem 1.5) appears to be proved in detail and may well be correct independently. I recommend asking the authors for a complete proof of the conditional continuity statement, or a different route to uniqueness, before acceptance. The reliance on the companion paper [5] is heavy; the authors should state explicitly which axioms and lemmas from [5] are used in the conditional setting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main results are real and the paper is worth reading. The continuity of the metric distribution (Thm 1.2) and the Hausdorff dimension of geodesics (Thm 1.5) are new, and the Sperner–Firework argument in Section 3 is a neat piece of work. Section 5's upper bound via renormalization and the lower bound via mass distribution both look coherent, given the axioms from [5]. The paper does not fit data or engage in circular reasoning; the dependence on [5] is heavy, but that is the natural foundation.\n\nThe soft spot is precisely where the reader's report says it is: Section 4. Lemma 4.2 is the only place where multiple geodesics are ruled out, and it is not proved. Saying 'according to the proof of Theorem 1.2 by replacing 0, x and M_i with v, z and |v–z|M_i' is not a proof, because the Sperner construction in Section 3 uses translation/scale covariance and independence of edges in annuli, and after conditioning on E(v) the law inside B_r(v) has a deterministic boundary edge set and a modified intensity near the boundary. It is not immediate that the same argument yields continuity of the conditional distribution of the internal distance. The subsequent step from (4.5) to (4.6) also needs work: (4.5) concerns D(x,y;B_r(v)^c), while (4.6) concerns D(x,y;E\\{<u,v>}), and these are different metrics. The inequality does not follow from monotonicity alone. This is not a typo-level concern; it is a load-bearing gap in the uniqueness theorem.\n\nI would not call the paper incoherent—the overall strategy is clear and the remaining sections are credible—but the current version is not self-contained on its central claim. A serious referee should be able to see whether the transfer argument can be repaired, but the authors need to write it out. The paper deserves peer review, not desk rejection, because the results are important and the likely fix is substantial but plausible.\n\nFor a reading group, the Sperner section alone is worth a session. I would cite it if I needed the continuity result, but I would hedge on uniqueness until Section 4 is fixed.","headline":"Strong new results on geodesics of the critical long-range percolation metric, but the uniqueness proof rests on a compressed transfer argument in Section 4 that needs to be written out.","tokens_in":19630,"tokens_out":7678,"would_cite":true,"duration_ms":89336,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B27","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"The critical long-range percolation scaling limit has a unique geodesic between any two fixed points, and every geodesic has Hausdorff dimension equal to $\\theta$.","keywords":["long-range percolation","random metric","geodesic","geodesic uniqueness","Hausdorff dimension","Sperner family","continuity of distribution","scaling limit"],"falsifier":"Check the step from (4.5) to (4.6): find an edge configuration where $D(x,y;B_r(v)^c)$ differs from $D(x,y;E\\setminus\\{\\langle u,v\\rangle\\})$ on the relevant event. If there is an admissible configuration in which removing the single edge $\\langle u,v\\rangle$ changes the restricted distance, the implication used in Theorem 1.4 is not justified and the uniqueness claim collapses.","tokens_in":18422,"feed_emoji":"📏","tokens_out":7077,"duration_ms":84769,"temperature":0.7,"pith_summary":"This paper studies the random metric $D$ obtained as the scaling limit of distances in critical long-range percolation on $\\mathbb{Z}^d$. It proves that for any fixed pair of distinct points, almost surely there is exactly one $D$-geodesic between them, and that every $D$-geodesic has Hausdorff dimension $\\theta$, the same exponent controlling typical distance growth. To get uniqueness, the authors first establish that $D$ has a continuous distribution, meaning no single distance value carries positive probability, and then transfer that continuity to conditional laws inside small balls. A separate renormalization argument produces matching upper and lower bounds on the fractal dimension of geodesics. If correct, the limiting metric has a well-defined, fractal shortest-path geometry determined only by $d$ and $\\beta$.","feed_headline":"Critical long-range percolation geodesics are unique and fractal","feed_subtitle":"Every pair of points in the limiting metric has one shortest path, with Hausdorff dimension exactly theta.","key_machinery":"The load-bearing machinery is a generalized Sperner theorem for families of subsets of $\\{1,\\dots,n\\}$ with upward or downward instability, giving $P(\\mathcal{A}_n)\\le C/\\sqrt{n}$ for Bernoulli$(p)$ configurations. It controls the event that the rescaled metric $D(0,x)$ falls in an $\\varepsilon$-interval by encoding, across $K$ geometrically spaced annuli, whether long edges are sparse or present, thereby reducing the distance event to a Sperner family of Bernoulli outcomes. For the dimension upper bound, a renormalized lattice whose vertices are $\\delta L$-cubes, together with 'good' cubes having separated crossing edges, gives a bound on the number of cubes a geodesic can visit; the mass distribution principle gives the lower bound.","core_discovery":"The central claim is that the critical long-range percolation scaling limit is geodetically rigid: after taking the unique subsequential scaling limit constructed by the authors, almost surely every pair of distinct points $x,y$ is joined by exactly one shortest path, and every such path occupies a set of Hausdorff dimension $\\theta\\in(0,1)$. The dimension result identifies the fractal size of any geodesic with the universal distance-growth exponent of the model, rather than with a new exponent. The uniqueness result follows from a new continuity property of the law of $D$: the probability that $D(0,x)$ lands in an interval of width $\\varepsilon$ tends to zero with $\\varepsilon$, established through a Sperner-family estimate on dependent Bernoulli configurations. This continuity is then transferred to conditional laws inside small balls to show that two competing geodesics would have to split at an edge and produce a path whose length exactly matches the ambient distance, an event of probability zero.","pith_inferences":["The uniqueness result should extend from fixed pairs to countably many pairs simultaneously, since the proof's events are countable; the paper states it only for a fixed pair.","The conditional-continuity strategy could plausibly apply to scaling limits of other edge models with independent long edges, provided the boundary-transfer step can be checked.","The exponential tail in (5.6) suggests the upper bound on the number of visited cubes could be sharpened beyond $\\delta^{-\\theta-\\varepsilon}$ to $\\delta^{-\\theta}(\\log 1/\\delta)^C$, something the Borel-Cantelli argument does not extract.","One can ask whether the dimension-identity result holds for geodesics between points on the original lattice at the discrete level in a scaling-invariant sense, not only for the continuum limit."],"forward_implications":["If $D$ exists as constructed, then every pair of distinct points in $\\mathbb{R}^d$ is joined by a unique shortest path almost surely, so geodesics are well-defined objects for the scaling limit.","The Hausdorff dimension of every $D$-geodesic equals $\\theta$, so the fractal size of shortest paths is read off from the distance-growth exponent instead of a new parameter.","The continuity of the law of $D$ implies that no positive-probability atom sits at any particular distance value, which is the quantitative input that makes the uniqueness proof go through.","The conditional version of continuity extends the argument to balls with fixed boundary edge sets, supporting later statements about paths that split at a single edge.","The renormalization upper bound shows a $D$-geodesic visits at most $\\delta^{-\\theta-\\varepsilon}$ cubes of side $\\delta$, so geodesics are quantitatively thin at small scales."],"supporting_citations":[{"why":"Supplies the polynomial distance-growth exponent $\\theta$ and the existence of subsequential scaling limits for the graph distance.","marker":"[2]"},{"why":"Constructs the unique limiting metric $D$ and its axioms (locality, scale covariance, tightness), the object this paper studies.","marker":"[5]"},{"why":"Provides the lower bound matching the earlier upper bound for the Hausdorff dimension of the metric, fixing the exponent $\\theta$ used here.","marker":"[3]"},{"why":"Supplies the generalized Sperner theorem whose proof is adapted to all $p\\in(0,1)$ in Proposition 2.2.","marker":"[9]"},{"why":"Provides the renewal/rumour-process proposition used to estimate the Firework spreading in Lemma 3.6.","marker":"[10]"},{"why":"Gives the concentration inequality applied in Proposition 3.3 to turn the Firework estimate into the large-deviation event $E_1$.","marker":"[14]"},{"why":"Supplies the mass distribution principle used for the lower bound on the Hausdorff dimension of geodesics.","marker":"[13]"},{"why":"Supplies the BK inequality used in Lemma 5.6 to bound the probability that a connected renormalized cluster is bad.","marker":"[15]"}],"fun_headline_variants":["Geodesics in critical long-range percolation are unique and fractal","Unique geodesics and exact fractal dimension for critical long-range percolation","Critical long-range percolation: geodesic uniqueness and Hausdorff dimension","One geodesic per pair: uniqueness and dimension for critical long-range percolation","Unique geodesics and exact Hausdorff dimension for critical long-range percolation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of geodesic uniqueness depends on the continuity argument for the metric distribution working after conditioning on all edges outside a small ball, with a deterministic boundary edge set; if that conditional continuity fails in that boundary setting, the uniqueness theorem lacks a proof.","fun_headline_variants_meta":{"raw":{"variants":["Geodesics in critical long-range percolation are unique and fractal","Unique geodesics and exact fractal dimension for critical long-range percolation","Critical long-range percolation: geodesic uniqueness and Hausdorff dimension","One geodesic per pair: uniqueness and dimension for critical long-range percolation","Unique geodesics and exact Hausdorff dimension for critical long-range percolation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001292,"raw_usage":{"total_tokens":5201,"prompt_tokens":800,"completion_tokens":4401,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":416,"completion_tokens_details":{"reasoning_tokens":4300}},"tokens_in":416,"tokens_out":4401,"duration_ms":38662,"temperature":1.0,"reasoning_tokens":4300,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:26:21.576591+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the step from (4.5) to (4.6): find an edge configuration where $D(x,y;B_r(v)^c)$ differs from $D(x,y;E\\setminus\\{\\langle u,v\\rangle\\})$ on the relevant event. If there is an admissible configuration in which removing the single edge $\\langle u,v\\rangle$ changes the restricted distance, the implication used in Theorem 1.4 is not justified and the uniqueness claim collapses.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the polynomial distance-growth exponent $\\theta$ and the existence of subsequential scaling limits for the graph distance."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the unique limiting metric $D$ and its axioms (locality, scale covariance, tightness), the object this paper studies."},{"cited_title":"The polynomial growth of the infinite long-range percolation cluster","cited_arxiv_id":"2311.14352","evidence_quote":"Provides the lower bound matching the earlier upper bound for the Hausdorff dimension of the metric, fixing the exponent $\\theta$ used here."},{"cited_title":"Ding and C.K","cited_arxiv_id":null,"evidence_quote":"Supplies the generalized Sperner theorem whose proof is adapted to all $p\\in(0,1)$ in Proposition 2.2."},{"cited_title":"Gallo, N.L","cited_arxiv_id":null,"evidence_quote":"Provides the renewal/rumour-process proposition used to estimate the Firework spreading in Lemma 3.6."},{"cited_title":"Russell and K","cited_arxiv_id":null,"evidence_quote":"Gives the concentration inequality applied in Proposition 3.3 to turn the Firework estimate into the large-deviation event $E_1$."},{"cited_title":"Cambridge Series in Statistical and Probabilistic Math- ematics","cited_arxiv_id":null,"evidence_quote":"Supplies the mass distribution principle used for the lower bound on the Hausdorff dimension of geodesics."},{"cited_title":"van den Berg and H","cited_arxiv_id":null,"evidence_quote":"Supplies the BK inequality used in Lemma 5.6 to bound the probability that a connected renormalized cluster is bad."}],"review_version":1}