Uniqueness and dimension for the geodesic of the critical long-range percolation metric

Jian Ding; Lu-Jing Huang; Zherui Fan

REVIEW 2 major objections 5 minor 15 references

Uniqueness and dimension for the geodesic of the critical long-range percolation metric

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The critical long-range percolation scaling limit has a unique geodesic between any two fixed points, and every geodesic has Hausdorff dimension equal to $\theta$.

desk verdict 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. read the letter →

arxiv 2506.10511 v1 pith:VRI55VIK submitted 2025-06-12 math.PR

Jian Ding , Zherui Fan , Lu-Jing Huang This is my paper · ORCID

classification math.PR MSC 60K3582B2782B43

keywords long-rangepercolationrandommetricgeodesicuniquenessHausdorffdimensionSpernerfamilycontinuityofdistributionscalinglimit

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

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 $\beta$.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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\{}), 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.

I 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.

For 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[§4, Eqs. (4.5)–(4.6)] 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.
[§4, Eqs. (4.5)–(4.6)] 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.

minor comments (5)

[Definition 4.1] There is a typo: 'we will choose oly one geodesic' should read 'we will choose only one geodesic'.
[Abstract/Introduction] The word 'emcompass' in the introduction should be 'encompass'.
[Lemma 3.14] 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 ...'.
[§5.1] 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.
[Eq. (5.4)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: geodesic uniqueness and dimension are proved as theorems about the previously constructed LRP metric, with no fitted parameter renamed as a prediction.

full rationale

The central claims are Theorems 1.4 and 1.5, which concern the already-defined limiting metric D constructed by Bäumler [2] and the authors' prior work [5]. The paper does not define D, its exponent θ, or the geodesic dimension in terms of the desired conclusions. Theorem 1.2 is proved directly in Section 3 via Sperner-family estimates and edge-sparsity events; Theorem 1.4 then uses Theorem 1.2 and a path-perturbation argument in Section 4. Theorem 1.5 uses the mass distribution principle together with regularity estimates from [5], while θ is the previously established growth exponent from (1.1), not a parameter fitted to match the Hausdorff dimension. No equation equates a predicted quantity with an input by construction, and no fitted input is renamed as an output. The citations to [5] are foundational prior work about existence, uniqueness, and axioms of the metric D, not about geodesic uniqueness or geodesic dimension; invoking them is normal cumulative mathematics and does not make the new theorems circular. One caveat, which is a rigor concern rather than circularity: Lemma 4.2 transfers the unconditional continuity proof to the conditional law given E(v) with the terse justification 'according to the proof of Theorem 1.2 by replacing 0, x and M_i with v, z and |v−z|M_i', and the step from (4.5) to (4.6) silently changes the restricted edge set from B_r(v)^c to E \ {<u,v>}. If those transfers fail, Section 4 has a proof gap, but the gap does not reduce the theorem to its inputs, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No free parameters are introduced by this paper: theta is imported from [2], and delta, K, M_i, N are auxiliary constants chosen as functions of epsilon and fixed model parameters. The axioms list the external theorems and the definition of D on which the proof rests. No new physical or mathematical entities are postulated.

assumptions (9)

domain assumption The random metric D on R^d and its full set of axioms from [5, Theorem 1.1 and Sections 1.3-1.4] exist, including locality, translation and scale covariance, tightness, and the uniform continuity estimate of [5, Proposition 1.13].
Invoked throughout the paper: in (3.1), (3.16), Lemma 5.2, and the tightness bound in Section 5. If [5] is false, Theorems 1.2, 1.4, and 1.5 have no object.
domain assumption The distance exponent theta in (0,1) satisfies n^theta approximately equal in probability to d(0,n1) and to E[d(0,n1)] for all n, as in (1.1) from [2].
Imported from Bäumler [2]. Used to set the scales M_i in Section 3 and to define the expected dimension in Theorem 1.5.
domain assumption The critical LRP edge probability is 1 minus exp of the integral kernel beta over |u-v|^{2d}, as defined at the start of Section 1.
Model definition; all probability estimates in Sections 3 and 5 are computed with this kernel.
standard math Generalized Sperner theorem from [9, Theorem 4.2] bounding the sum of a_k over binomial coefficients by 4 for Sperner families.
Used in Lemma 2.3 and Proposition 2.2 to bound the probability of a Sperner event by C divided by sqrt(n).
standard math Firework process tail estimate from [10, Proposition 1] for renewal-type processes.
Used in Lemma 3.6 to bound P[M_k >= k] leq kappa^k for the spreading procedure.
standard math Concentration inequality for dependent Bernoulli sequences from [14, Theorem 1], see also [1, Theorem 3.4].
Used in Proposition 3.3 to obtain the large deviation estimate on the number of uncovered annuli.
standard math Mass distribution principle from [13, Theorem 4.19].
Used in Section 5.1 to convert the almost sure Holder-type regularity of D into the lower bound on Hausdorff dimension.
standard math Expected number of connected subsets of size k in the renormalized graph satisfies E[|CS_k(i)|] leq (4 mu_beta)^k from [2, Lemma 3.2].
Used in Section 5.2 to bound the probability that large connected subsets of boxes are bad.
standard math BK inequality from [15] for positively associated events.
Used in Lemma 5.6 to upper bound the probability that a connected set of vertices has many bad cubes.

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Pith. "Pith review of Uniqueness and dimension for the geodesic of the critical long-range percolation metric." pith.science (2026). https://pith.science/paper/VRI55VIK

@misc{pith2026250610511,
  author       = {Pith},
  title        = {Pith review of: Uniqueness and dimension for the geodesic of the critical long-range percolation metric},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VRI55VIK}},
  note         = {Machine review of arXiv:2506.10511}
}

read the original abstract

By recent works of B\"aumler [2] and of the authors of this paper [5], the (limiting) random metric for the critical long-range percolation was constructed. In this paper, we prove the uniqueness of the geodesic between two fixed points, for which an important ingredient of independent interest is the continuity of the metric distribution. In addition, we establish the Hausdorff dimension of the geodesics.

Figures

Figures reproduced from arXiv: 2506.10511 by the authors.

**Figure 1.** The illustration for the events Ai and Ei,1. The event Ai indicates the presence of red long edges in the picture, while the event Ei,1 indicates the presence of blue edges in the picture. ≤ 1 − exp ( − Z Bri−1 (0) Z BMi/8(u) c β |u − v| 2d dudv ) = 1 − exp −βec1(d)r d i−1 Z +∞ Mi/8 1 s d+1 ds ≤ βec2(d) ri−1 Mi d ≤ ec3(d, β)N −d for some constants ec1(d), ec2(d) > 0 (depending only on d) and ec3(d, β) > 0 (de… view at source ↗

**Figure 2.** The illustration for the definition of Ll . The event {Ll = s} is equivalent to the event that there exists at least one long edge in EAri l+1 −Mi l+1 /8,ri l −Mi l /8×Bri l+s (0) c (the red curve), while no long edge exists in EAri l+1 −Mi l+1 /8,ri l −Mi l /8×Bri l+s+1 (0) c (the red dashed curve). For the distribution of Ll , we have the following property. Lemma 3.5. For all d ≥ 1 and all β > 0, there exists a c… view at source ↗

**Figure 3.** Illustration for Definition 4.1. The blue lines represent the edges in E≥q including ⟨u, v⟩. The black line represents the edge ⟨z, w⟩ with z ∈ Zr(v). The red lines represent three D-geodesics taken in Definition 4.1. It is obvious that (4.2) holds trivially when b ≥ a. In the rest of the proof, we assume that b < a. We now turn to show that (4.3) lim ε→0+ P [len(Quvz(x, y) ∩ Br(v); D) ∈ (a − b − ε, a − b + ε) |E(v)… view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [5]

J. Ding, Z. Fan, and L.-J. Huang. Uniqueness of the critical long-range percolation metrics. To appear in Mem. Amer. Math. Soc

work page

[2] [2]

J. Bäumler. Distances in 1 |x−y|2d percolation models for all dimensions.Comm. Math. Phys., 404:1495–1570, 2023

work page 2023

[3] [1]

Alessandro and S

P. Alessandro and S. Aravind. Randomized distributed edge coloring via an extension of the Chernoff- Hoeffding bounds.SIAM J. Comput., 26(2):350–368, 1997

work page 1997

[4] [3]

J. Bäumler. The polynomial growth of the infinite long-range percolation cluster. 2023. ArXiv preprint arXiv:2311.14352

work page Pith review arXiv 2023

[5] [4]

Coppersmith, D

D. Coppersmith, D. Gamarnik, and M. Sviridenko. The diameter of a long-range percolation graph.Math- ematics and Computer Science, II:147–159, 2002

work page 2002

[6] [6]

J. Ding, Z. Fan, and L.-J. Huang. Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation.Comm. Pure Appl. Math., 78(7):1251–1284, 2025

work page 2025

[7] [7]

Ding and E

J. Ding and E. Gwynne. Uniqueness of the critical and supercritical Liouville quantum gravity metrics. Proc. Lond. Math. Soc., 126(1):216–333, 2023

work page 2023

[8] [8]

Distances in critical long range percolation

J. Ding and A. Sly. Distances in critical long range percolation. 2013. ArXiv preprint arXiv:1303.3995

work page Pith review arXiv 2013