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Critical long-range percolation II: Low effective dimension

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Pith's one-line read Long-range percolation below its upper critical dimension has closed-form critical exponents and cluster-size tails.

desk verdict A strong conditional paper: the main LR-LD exponents are proven only under an unverified correlation-length condition, but the machinery and unconditional alpha<1 results are genuinely new and worth refereeing. read the letter →

arxiv 2508.18808 v1 pith:Y5LKPYFK submitted 2025-08-26 math.PR math-phmath.MP

Tom Hutchcroft This is my paper
classification math.PRmath-phmath.MP MSC 60K3582B4382B27
keywords long-rangepercolationcriticalexponentshyperscalingcorrelationlengthconditionk-pointfunctionsconformalinvariancefractaldimension
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 aims to pin down the critical behavior of long-range percolation on Z^d in the long-range, low-dimensional regime d/3 < alpha < alpha_c(d), where the model is below its upper critical dimension. It proves that, under a precise correlation-length condition, the probability that the cluster of the origin has size at least n decays like n^{-(d-alpha)/(d+alpha)} and that the p-th moment of the cluster inside a ball of radius r grows like r^{alpha+(p-1)(d+alpha)/2}. These two-sided estimates identify the volume-tail exponent delta and fractal dimension d_f as (d+alpha)/(d-alpha) and (d+alpha)/2. The same machinery gives the two-point exponent eta = 2 - alpha throughout the long-range regime, subcritical scaling relations, and an up-to-constants formula for the k-point function in terms of a Mobius-covariant geometric functional S. A cautious reader should note that the correlation-length condition is proven unconditionally only for alpha < 1; for larger alpha it is an explicit assumption, and if it fails the exponent identities do not follow from this paper.

What carries the argument

The central object is the correlation length condition (CL): the L^p radius of gyration xi_p(beta_c,r) of the critical cluster, in the model with all edges longer than r removed, must grow at most like C_p r. This condition defines 'effectively long-range' behavior and is used to control two-point functions and cluster-volume moments. A weaker condition (wCL) suffices for several hyperscaling theorems and controls only the first four truncated moments. The second main tool is a family of higher-order Gladkov inequalities, which bound the k-point function by products of two-point functions over connected multigraphs, together with the associated sweep/spread functional S whose Mobius covarian

What would settle it

Take d = 3 and alpha = 3/2, a parameter set where the theorems require (CL) but the paper proves it only for alpha < 1. Simulate critical long-range percolation with all edges longer than r removed and estimate the L^2 radius of gyration xi_2(beta_c,r). If xi_2(beta_c,r)/r is unbounded as r -> infinity, then (CL) fails at this parameter and the exponent identities proven here do not apply there.

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Extended reading notes

Core claim

The paper's central discovery is that, in the effectively long-range, low-dimensional regime, critical cluster sizes and multi-point connection probabilities are determined up to multiplicative constants by explicit powers of Euclidean quantities. Under the correlation length condition (CL), P_{beta_c}(|K| >= n) ~ n^{-(d-alpha)/(d+alpha)} and E_{beta_c}|K cap B_r|^p ~_p r^{alpha+(p-1)(d+alpha)/2}, so delta = (d+alpha)/(d-alpha) and d_f = (d+alpha)/2. For k points, tau_{beta_c}(x_1,...,x_k) ~_k S(x_1,...,x_k)^{-(d-alpha)/2}, where S is defined by a recursive three-way partition formula and is Mobius-covariant. The same framework yields the two-point function exponent eta = 2 - alpha throughou

Load-bearing premise

The whole low-dimensional exponent picture rests on the correlation length condition (CL): at criticality, the cluster of the origin in the model with all edges longer than r removed must have typical radius of order r, and this is proven unconditionally only for alpha < 1, while for 1 <= alpha < alpha_c(d) in dimensions 2 through 5 it remains an unverified assumption.

Editorial extensions

If this is right

  • If (CL) holds and d < 3alpha, the critical cluster volume tail satisfies P_{beta_c}(|K| >= n) ~ n^{-(d-alpha)/(d+alpha)}, so the exponent delta is exactly (d+alpha)/(d-alpha).
  • The fractal dimension of a critical cluster is d_f = (d+alpha)/2: the p-th volume moment in a ball of radius r grows like r^{alpha+(p-1)(d+alpha)/2} and the largest cluster in the ball has size of order r^{(d+alpha)/2}.
  • Across the entire long-range regime defined by (CL), the critical two-point function obeys P_{beta_c}(x <-> y) ~ ||x-y||^{-d+alpha}, confirming eta = 2 - alpha, and this also covers high- and critical-dimensional cases with alpha < 2.
  • Slightly subcritical scaling relations gamma = (2-eta)nu and Delta = nu d_f hold, with d_f = min{(d+alpha)/2, 2alpha}, whenever any of gamma, nu, or Delta is well-defined.
  • The k-point function matches the Mobius-covariant functional S up to constants: tau_{beta_c}(x_1,...,x_k) ~_k S(x_1,...,x_k)^{-(d-alpha)/2}, exhibiting a concrete form of conformal covariance at the level of connection probabilities.

Reading between the lines

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

  • If (CL) fails for some 1 <= alpha < alpha_c(d), the exponent identities proven here would not apply in that range, though the paper's author conjectures the model then behaves like the nearest-neighbour model; testing (CL) numerically in dimensions 3 or 4 is the sharpest way to locate the true crossover.
  • The fact that only the weaker condition (wCL) suffices for the hyperscaling theorems suggests the dichotomy is robust: any definition of 'effectively long-range' that makes the cut-off cluster feel its cutoff at scale r should yield the same exponents.
  • The Mobius covariance of S, combined with the k-point estimates, makes a full conformal-invariance result for the scaling limit plausible; if established, it would give a route to computing higher-point functions in long-range models in three dimensions.
  • The contradiction argument shows the hydrodynamic condition and (CL) cannot coexist when d < 3alpha; extending this type of 'no mean-field ODE' argument may mark the precise boundary at which hyperscaling begins.
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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

2 major / 4 minor

Summary. This paper is the second in a three-part series on critical long-range Bernoulli bond percolation on Z^d with kernel |J'(r)| ~ r^{-d-α-1}. It introduces a correlation-length condition (CL) as an axiomatic definition of the effectively long-range regime and proves (CL) for α < 1. Under (CL), and in particular for α < 1, the paper establishes pointwise two-point bounds, up-to-constants estimates for cluster volume tails and moments, k-point function estimates, and the scaling relations γ = (2−η)ν, Δ = ν d_f. In the low-dimensional range d < 3α, the main results identify δ = (d+α)/(d−α) and d_f = (d+α)/2, and give τ(x_1,...,x_k) ≍ S(x_1,...,x_k)^{-(d-α)/2} for an explicit Möbius-covariant functional S. Theorems II.1.1 and II.1.2 are unconditional for α < 1; the rest of the LR-LD results for d ≥ 3 are conditional on (CL), which is proved only for α < 1. The proofs are detailed and internally coherent, and the paper is transparent about its main hypothesis.

Significance. If the assumptions are accepted, the paper delivers the first rigorous, non-perturbative computation of genuine non-mean-field exponents (δ and d_f) and of k-point scaling for long-range percolation, together with a new pointwise two-point theorem under (CL). The higher-order Gladkov inequalities and the identification of the geometric functional S are interesting in their own right. The proof is not circular: (CL) is a geometric condition on the cut-off model and is used to derive, rather than fit, the exponent identities. The main weakness is scope: for d = 3,4,5 the condition d < 3α forces α > 1, and in that range (CL) is an unverified hypothesis. Thus the advertised LR-LD exponent identities for these dimensions are conditional, not established. The unconditional results cover only d = 1,2 in the LR-LD regime.

major comments (2)
  1. [§II.1.1, Definitions II.1.3, Theorem II.1.5, §II.2.2] The paper's central theorems II.1.10, II.1.11, and II.1.13 are all conditional on (CL), while Theorem II.1.5 proves (CL) only for α < 1. The proof of Theorem II.1.5 via Proposition II.2.7 and Lemma II.2.8 uses a bound of order (r/N)^{1−α}; this degenerates at α = 1. Since d < 3α for d = 3,4,5 forces α > 1, there is no unconditional instance of the claimed LR-LD exponents in exactly the dimensions d = 3,4,5. Remark II.1.6 explicitly leaves open whether (CL) holds for α ≥ 1 and even whether it depends only on d and α. This is a load-bearing limitation: if (CL) fails for some α ≥ 1, the exponent identities in Theorems II.1.10, II.1.11, and II.1.13 do not follow from this manuscript. I recommend that the abstract and introduction state prominently that the d ≥ 3 LR-LD results are conditional on the unverified condition (CL), rather than presenting them as theorems about the full LR-LD regime
  2. [§II.6, Corollary II.1.4, Lemma II.6.6] Several results stated as theorems here rely on results from unpublished companion papers, in particular Theorem III.1.11 from the third paper and Theorem III.1.2. For example, Corollary II.1.4 in the case d = 3α depends on Theorem III.1.11, and Lemma II.6.6 / Theorem II.1.15 invoke Theorem III.1.2 and Theorem III.1.11 in the critical-dimensional case. The footnote says the paper is intended to be readable independently, but the dependence on [49,50] is substantial. This is not a flaw if the series is treated as a whole, but for a standalone submission the exact dependencies should be listed explicitly so that the referee and reader can verify what is established here versus in the companion papers.
minor comments (4)
  1. [Theorem II.1.10] The statement says δ = (d+α)/(d−α) with no exclusion of α = d. For α = d the formula has zero denominator, and Corollary II.2.4 shows the model has a discontinuous transition, so the tail probability does not decay. Please state the δ identity for α < d and treat α = d separately.
  2. [Proof of Lemma II.5.11] In the proof, the sentence 'it follows by Markov's inequality and (II.5.14)' refers to an equation that is only labelled later in the proof of Proposition II.5.9. The displayed bound inside Lemma II.5.11 should be labelled with its own number, or the cross-reference corrected.
  3. [Figure 3 and Section II.1.1] The green 'Proven LR' region in Figure 3 is only α < 1. The caption and the surrounding text could be more explicit that for d ≥ 3 the LR-LD exponent theorems require the unproven condition (CL), so that the figure is not read as evidence of an unconditional result.
  4. [Throughout] The paper uses many lemmas from the first paper of the series (e.g. Corollary I.4.8, Lemma I.2.1, Proposition I.3.1) without restating them. A short appendix or dependency table listing which numbered results are proven in this paper and which are imported would improve verifiability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the exponent identities are conditional on the (CL) correlation-length condition, which is not the target statement; reliance on same-author companion theorems is a verification gap, not a circular reduction.

full rationale

The paper's main results are conditional theorems: given (CL) (Definition II.1.3) and d<3α, it derives up-to-constants two-point, volume-tail, and k-point estimates. (CL) is a geometric correlation-length condition on the cut-off model (ξ_p(β_c,r) ≤ C_p r); it does not assert any of the exponent identities being derived. Both upper and lower bounds are obtained through differential inequalities, Gladkov-type inequalities, and contradiction arguments, so the exponents δ=(d+α)/(d−α) and d_f=(d+α)/2 are not fitted parameters and do not enter by construction. The paper explicitly states its own limitation: for 1 ≤ α < α_c(d) in d=2,3,4,5, (CL) is not proved—Theorem II.1.5 verifies it only for α<1, while Corollary II.1.4 covers d≥3α. The α=1 boundary is indeed where the proof of Proposition II.2.7 degenerates via the (r/N)^{1−α} bound in Lemma II.2.8, but this is an unverified assumption/technical gap, not circularity. Likewise, some boundary results for d=3α and d≥3α rely on companion-paper theorems (e.g., Theorem III.1.2 and III.1.11) that are not proved in this paper; Remark II.1.6 concedes that even whether (CL) depends only on d and α is open. These are missing-support issues and heavy same-author citations, but they do not reduce any prediction to its own input. The derivation chain is self-contained conditional on the stated assumptions and on the cited external theorems, which are parameter-free results with assumptions not including the target exponent identities.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numeric parameters are fitted to data; alpha, d, and the kernel are inputs. The principal load-bearing ingredients are the correlation length conditions (CL) and (wCL), which are proven only in limited parameter ranges and otherwise assumed, plus heavy reliance on prior and companion papers by the same author. The Mobius-covariant functional S is a mathematical construction, not a new physical entity.

assumptions (7)
  • domain assumption Kernel normalization: |J'(r)| = (1+delta_r) r^{-d-alpha-1} with delta_r -> 0 and integrable error, and the unit ball of the norm has unit Lebesgue measure.
    Assumed throughout the paper, for example in the display marked (*) and in Lemma II.4.9; it fixes the scale of the kernel and beta_c without changing critical exponents.
  • ad hoc to paper The correlation length condition (CL): xi_p(beta_c,r) <= C_p r for every p>0 and r>=1.
    Definition II.1.3 is the main assumption under which the LR and LR-LD theorems are stated. It is proven for alpha<1 in Theorem II.1.5, but for the full claimed regime it remains an unverified postulate.
  • ad hoc to paper The weak correlation length condition (wCL): E_{beta_c,r}|K|^p <= (1+epsilon) E_{beta_c,r}|K cap B_{C r}|^p for p=1,2,3,4.
    Definition II.4.1 is used to prove Theorems II.4.2, II.5.2, and II.5.7; it is implied by (CL) in the low-dimensional regime but is a separate added assumption for the weaker results.
  • domain assumption The regime condition d < 3alpha.
    All low-dimensional hyperscaling claims are stated under d < 3alpha, which is the LR-LD part of the phase diagram in Figure 1.
  • domain assumption Prior pointwise two-point estimate P_{beta_c}(x<->y) is comparable to ||x-y||^{-d+alpha} for alpha<1.
    Cited as [51, Theorem 1.4] and used in the proof of Theorem II.1.5 via Proposition II.2.7; it is an external published result, not re-proven here.
  • domain assumption Spatially averaged upper bound: sum over x in [-r,r]^d of P_{beta_c}(0<->x) is at most C r^{-d+alpha}.
    This is the main theorem of [55], used repeatedly, for example in (II.1.6), Lemma II.2.3, and the proof of Theorem II.4.2.
  • domain assumption Universal tightness theorem of [53] and several results from companion papers I and III, including Theorem III.1.11.
    Used at multiple points, e.g. the inequality (II.4.16), Lemma II.4.9, and Corollary II.1.4. Some of these are not yet published as standalone papers and are only available as parts of the series.

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Pith. "Pith review of Critical long-range percolation II: Low effective dimension." pith.science (2026). https://pith.science/paper/Y5LKPYFK

@misc{pith2026250818808,
  author       = {Pith},
  title        = {Pith review of: Critical long-range percolation II: Low effective dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y5LKPYFK}},
  note         = {Machine review of arXiv:2508.18808}
}
abstract

In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta \geq 0$ is a parameter. As $d$ and $\alpha$ vary, the model is conjectured to exhibit eight qualitatively different second-order critical behaviours, with a transition between mean-field and low-dimensional regimes when $d=\min\{6,3\alpha\}$, a transition between long- and short-range regimes at a crossover value $\alpha_c(d)$, and with various logarithmic corrections at the boundaries between these regimes. This is the second of three papers developing a rigorous theory of the model's critical behavior in five of these eight regimes, including all long-range (LR) and high-dimensional (HD) regimes. We focus on the long-range low-dimensional (LR-LD) regime $d/3<\alpha<\alpha_c(d)$, where the model is below its upper critical dimension. Since computing $\alpha_c(d)$ for $2<d<6$ appears to be beyond the scope of current techniques, we give an axiomatic definition of the LR regime which we prove holds for $\alpha <1$. Using this, we prove up-to-constants estimates for the critical and slightly subcritical two-point function in the LR regime and for the volume tail and $k$-point function in the LR-LD regime. We deduce that the critical exponents satisfy the identities \[ \eta = 2-\alpha, \qquad \gamma = (2-\eta)\nu, \qquad \text{ and } \qquad \Delta = \nu d_f \] in the LR regime (if $\gamma$, $\nu$, or $\Delta$ is well-defined) and that $\delta$ and $d_f$ follow the hyperscaling identities \[ \delta = \frac{d+\alpha}{d-\alpha} \qquad \text{ and } \qquad d_f = \frac{d+\alpha}{2} \] in the LR-LD regime. Our results are suggestive of conformal invariance in the LR-LD regime, with the critical $k$-point function matching an explicit M\"obius-covariant function up-to-constants.

Figures

Figures reproduced from arXiv: 2508.18808 by the authors.

Figure 1
Figure 1. Schematic illustration of the different regimes of critical behaviour for long-range percolation. LR, SR, HD, LD, and CD stand for “Long Range”, “Short Range”, “High Dimensional”, “Low Dimensional”, and “Critical Dimensional” respectively, while mSR stands for “marginally Short Range”. Here we ignore the special behaviours occuring when d = 1 (where there is either no phase transition when α > 1 or a discontinuous p… view at source ↗
Figure 2
Figure 2. Sak’s predicted value of 2 − η (left) for d = 2 and its consequences for δ (center) and df (right) assuming the validity of the hyperscaling relations for deff < 6 (right). The predicted crossover value αc(2) = 43/24 arises as 2 − ηSR with ηSR = 5/24, δSR = 91/5, and df,SR = 91/48. and df discussed in more detail in Section I.1, one obtains the prediction that if 1 < d < 6 then (δ, df ) =     2, 2α… view at source ↗
Figure 3
Figure 3. Schematic illustration of the values of α and d where the model is proven to satisfy (CL) (green), proven not to satisfy (CL) (pink), or neither (white). The coloured ribbons and vertices represent our knowledge on the boundary points of these regions, with the red vertex at d = α = 1 representing the discontinuous phase transition that is conjectured to occur only at this point [6, 30]. (The line d = α < 1 is left … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Schematic illustration of the disjoint occurrence in the proof of the Gladkov inequality. γ2 in each of the two continuations of the cluster connecting this tree to a; these two paths γ1 and γ2 may be taken to use only edges that are revealed after time T, when the two…
Figure 5
Figure 5. Figure 5: An example of the arboresence attaining the minimum in the definition of the sweep on a set of six points in the plane. Each point x ∈ A is associated to a disc that bounds the diameter of its associated set Ax in the arboresence (dotted circles); the sweep is the prod…

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  1. Super-Brownian limits and the $k$-point function for high-dimensional percolation

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    High-dimensional critical percolation clusters rescale to super-Brownian excursion, verifying the 1984 Aizenman–Newman k-point conjecture under lace-expansion hypotheses.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.