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Critical long-range percolation III: The upper critical dimension

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Pith's one-line read At d=3α, long-range percolation has volume-tail correction (log n)^{1/4}/√n, and the paper proves the hydrodynamic condition that makes the computation rigorous.

desk verdict Genuine major step—hydrodynamic condition at d=3α plus exact log corrections—but the headline (log n)^{1/4} rests on a deferred second-order error analysis that needs referee scrutiny. read the letter →

arxiv 2508.18809 v1 pith:FXE7NTAJ submitted 2025-08-26 math.PR math-phmath.MP

Tom Hutchcroft This is my paper
classification math.PRmath-phmath.MP MSC 60K3582B4382B2782B28
keywords long-rangepercolationuppercriticaldimensionlogarithmiccorrectionshydrodynamicconditionsuperprocessscalinglimitmean-fieldbehaviourvolumetailthree-pointfunction
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 proves that critical long-range percolation on Z^d at the upper critical dimension d=3α<6 behaves as a mean-field system with explicit logarithmic corrections. The central technical step is a proof of the hydrodynamic condition, which states that the largest cluster inside a block is asymptotically smaller than the universal upper bound; this unlocks a renormalization-group analysis developed in earlier papers. Using that analysis to second order, the paper computes the critical volume tail as const (log n)^{1/4}/√n and shows the critical cluster, scaled by r^{2α}(log r)^{-1/2}, converges to an integrated α-stable superprocess excursion. These logarithmic corrections match hierarchical long-range percolation and differ from the conjectured (log n)^{2/7} for nearest-neighbour percolation on Z^6, making the long-range model the first critical-dimensional percolation model with a rigorous, non-perturbative determination of its logarithmic corrections.

What carries the argument

The hydrodynamic condition is the central object: it says Mr = o(r^{(d+α)/2}), i.e. the largest cluster in a ball is much smaller than the universal tightness bound. This condition upgrades the moment equations of the first paper from conditional to unconditional statements. The second-order computation then hinges on two vertex factors: V_r = E|K|^2/(E|K|)^3, which measures how strongly the different arms of a cluster interact, and a second factor ̃V_r that arises in the error terms D^{(1)}_r and D^{(2)}_r describing interactions between two clusters. The paper proves the asymptotic equality V_r ∼ ̃V_r by writing moment ODEs for the error terms and solving them with the triple-interaction l

What would settle it

Compute the ratio ̃V_r/V_r directly from the cut-off model at d=3α<6: the proof requires this ratio to tend to 1, and any nonzero limiting deviation would falsify Proposition III.6.2 and the (log n)^{1/4} tail. A complementary check is to measure the critical volume tail: if P_{β_c}(|K|≥n)√n/(log n)^{1/4} does not tend to a positive constant, the central asymptotic formula fails.

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Extended reading notes

Core claim

The paper establishes that for d=3α the hydrodynamic condition holds: the edian Mr, the typical size of the largest cluster in a ball of radius r under the cut-off critical measure, is o(r^{(d+α)/2}). When d=3α<6 this makes the first paper's RG analysis unconditional, yielding the same superprocess scaling limits as in high dimension after slowly-varying corrections are included. The paper then analyzes the RG flow to second order and proves E_{β_c,r}|K|^2 ∼ (α/β_c) A r^{3α}/√(log r), from which it derives the critical volume tail P_{β_c}(|K|≥n) ∼ C (log n)^{1/4}/√n, the two-point estimate P_{β_c}(x↔y) ≍ ∥x−y∥^{−d+α}, and the three-point estimate with an explicit √(1/log d_min) correction. T

Load-bearing premise

The computed (log n)^{1/4} correction rests on the asymptotic equality of the two vertex factors V_r and ̃V_r which track different cluster interactions; if that equality carries errors that are not logarithmically integrable, the logarithmic-correction exponent would change.

Editorial extensions

If this is right

  • At d=3α<6 the critical volume tail is C (log n)^{1/4}/√n, the same logarithmic correction as hierarchical long-range percolation and not the (log n)^{2/7} conjectured for nearest-neighbour Z^6 percolation.
  • The critical cluster, scaled by ζ(r) ∼ const r^{2α}(log r)^{-1/2}, converges to the integrated symmetric α-stable superprocess excursion measure; the number of typical large clusters on scale r grows like const log r.
  • The two-point function has no logarithmic correction, P_{β_c}(x↔y) ≍ ∥x−y∥^{−d+α}, while the three-point function carries a √(1/log d_min) correction, showing that both the tree-graph and Gladkov bounds are off by a √(log) factor at the critical dimension.
  • Theorem III.1.11 also implies the correlation-length condition for effectively long-range critical behaviour, so the second paper's low-dimensional results, including the pointwise two-point estimate, apply along d=3α<6.
  • The hydrodynamic condition fails in low effective dimension and holds here, giving a sharp geometric distinction: at the critical dimension large clusters interact only very weakly, so mean-field ODEs hold with slowly varying corrections.

Reading between the lines

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

  • A direct numerical test of the paper's core identity is to compute V_r and ̃V_r from the cut-off model at d=3α<6: the proof needs their ratio to tend to 1, and any nonzero limiting deviation would break the second-order RG flow and the (log n)^{1/4} exponent.
  • Because the equality of V_r and ̃V_r is obtained by solving moment recurrences rather than by a model-specific argument, the same mechanism may apply on the other critical lines, such as d=6, α=2 or d=3α>6, once the hydrodynamic condition is available there.
  • The proof of the hydrodynamic condition is ineffective, leaving the rate of Mr open; the paper conjectures Mr ∼ const (log log r)/√(log r) r^{2α}, a prediction that could be checked numerically and would quantify how far the largest cluster sits above the typical large-cluster scale.
  • The explicit constant C = 2∫_B κ^{*4}(y) dy and the interpretation of the logarithmic-correction exponent as a left derivative of the mean-field exponent suggest a diagram-counting rule for logarithmic corrections at upper critical dimensions, which could be tested against the nearest-neighbour Z^6 prediction (log n)^{2/7}.
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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

3 major / 4 minor

Summary. The paper analyzes long-range percolation on Z^d with kernel ~ ||x-y||^{-d-α} at the upper critical dimension d = 3α < 6. It proves the hydrodynamic condition (Theorem III.1.11), then invokes the RG framework of Paper I to obtain superprocess scaling limits with slowly varying corrections (Theorem III.1.6), and finally computes second-order corrections to the RG flow, yielding the volume-tail asymptotics P_{β_c}(|K|≥n) ~ C (log n)^{1/4}/√n and matching lower/upper bounds for the two- and three-point functions (Theorems III.1.2 and III.1.8). The proof of the hydrodynamic condition proceeds by contradiction under a fictitious failure; the computation of logarithmic corrections is based on diagrammatic second-order asymptotics for the error terms D^{(1)}, D^{(2)}, including a claimed asymptotic equality of two vertex factors V_r and \tilde V_r.

Significance. If the second-order computation is correct, this is a major rigorous advance: it gives the first non-perturbative determination of critical exponents and logarithmic corrections at the upper critical dimension for a long-range percolation model, including a superprocess scaling limit with explicitly identified slowly varying factors. The predictions match hierarchical percolation and differ from the conjectured behaviour of nearest-neighbour percolation on Z^6, so the paper has real conceptual content. The manuscript is also commendably explicit about its limitations: Theorem III.1.11 is openly described as ineffective, the dependence on the deferred Lemma III.6.11 is flagged, and the hypotheses of the main theorems are stated precisely. These strengths make the central strategy coherent, but the decisive second-order step is not fully verified in the present version.

major comments (3)
  1. [§III.6.1–III.6.2, Eqs. (III.6.13)–(III.6.20)] This is the load-bearing step. Theorem III.6.6 is stated with unquantified o(1) remainders in (III.6.13)–(III.6.14), and the proof of Proposition III.6.2 then asserts the existence of a logarithmically integrable error function δ_r in (III.6.20). That conversion is essential: Lemma III.6.3 requires δ_r to be logarithmically integrable in order to force the solution f(r) ~ (aγC log r)^{-1/γ} r^a. A remainder that is merely o(1), e.g. δ_r ~ 1/log log r, is not logarithmically integrable and would change the solution to r^{3α}(log r)^{-1/2}(log log r)^c, destroying the constant-prefactor form of Theorem III.1.2. The displayed argument does not establish log-integrability; pointwise o(1) convergence does not imply it. The authors must either prove a quantitative version of Theorem III.6.6 with log-integrable error bounds, or supply a separate argument showing that the errors in (III.6.15)–(I
  2. [§III.6.3, Lemma III.6.11] Lemma III.6.11 is explicitly described as the key extra ingredient for Theorem III.6.6, and its proof is deferred to §III.6.3. In the version supplied for refereeing, that proof is not present, so the central second-order computation is incomplete as written. Moreover, the error term stated in the lemma is only o(|B_r| V^{...} r^{deg(P)} (E_r|K|)^{...}), which by itself does not provide the log-integrability required for Proposition III.6.2. Even if the deferred proof establishes the claimed equality, it may not yield a remainder that is summable against ds/s. This needs to be addressed directly: either the proof of Lemma III.6.11 is included and its remainder is shown to be log-integrable, or a different argument must be given.
  3. [§III.6.2, passage from Theorem III.6.6 to sums over B_r] The inference from polynomial-moment convergence to the ball integrals in (III.6.15)–(III.6.16) is not justified in detail. Theorem III.6.6 gives asymptotics for ∑_y D^{(i)}_r(0,y) P(y/r) for polynomials P; passing to P = 1_{B} via Carleman's criterion requires a tightness or uniform-integrability argument for the normalized signed measures. Lemma III.6.9 gives bounds of the correct order, but the manuscript does not spell out why the o(1) errors in the polynomial moments survive passage to the discontinuous indicator 1_B with any uniformity. Without such an argument, the rate at which E_{1,r} and E_{2,r} approach their limits is uncontrolled, which is exactly the same log-integrability problem raised above.
minor comments (4)
  1. [§III.1.2] The word 'edian' in the definition of M_r appears to be a typo for 'median'.
  2. [§III.3 (overview) and §III.4.1] There are several typos: 'fictirious' should be 'fictitious', 'mininum' should be 'minimum', and 'important important' appears duplicated.
  3. [§III.3.1] In the proof of Lemma III.3.4, the text refers to 'K_λ as in the proof of Lemma III.3.4'; the set is actually defined during the proof of Lemma III.3.11. This cross-reference should be corrected.
  4. [§III.6.2] The proof of Proposition III.6.2 introduces several error quantities (E_{0,r}, E_{1,r}, E_{2,r}, \bar E_{1,r}, \tilde H_r) in quick succession. A short summary table or labels would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hydrodynamic condition is proved rather than assumed, and the logarithmic corrections are derived from a second-order ODE whose input is a first-order scaling limit with undetermined slowly varying function.

full rationale

I walked the derivation chain. Theorem III.1.11 (hydrodynamic condition) is proved in Sections III.3–III.4 by contradiction: the failure of the modified hydrodynamic condition is shown to force simultaneously mean-field and low-dimensional behaviour, and the contradiction is derived from the paper's own estimates. The prior papers' theorems (I.1.15, I.1.17, I.5.21, etc.) are applied only after this hypothesis is established, so citing them is structural, not circular. The logarithmic corrections are not fitted: Proposition III.6.2 is a second-order asymptotic ODE whose error function is asserted to be logarithmically integrable, and Lemma III.6.3 converts that ODE into the (log n)^{1/4} tail. The constant C is computed from the superprocess scaling limit kernel κ via Theorem III.6.6, and the first-order scaling limit itself involves an undetermined slowly varying function A_r, so the second-order computation does not presuppose the logarithmic exponent. The equality of the two vertex factors V_r and Ṽ_r is presented as a theorem (III.6.6) proved by moment ODEs and recurrence (III.6.24), not as an ansatz imported from prior work. The deferred Lemma III.6.11 is an internal proof ingredient, not a self-citation. Possible concerns about whether the o(1) errors in Theorem III.6.6 are logarithmically integrable would be a correctness gap, not circularity: nothing in the paper's argument makes the predicted exponent equal to an input by construction. Self-citations to [45,46,50–52] are load-bearing only in the sense that they supply theorems with explicit hypotheses that are checked here (Hydro) or independent input bounds (e.g., the universal tightness theorem), which is legitimate external evidence rather than circular reduction.

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

No fitted parameters. The kernel normalization (∗) sets the constant C in (III.1.4) to 1, stated as WLOG scaling (changes β_c, not the law). Slowly varying functions A_r / A*_n are initially undetermined outputs (Theorems I.1.15, I.1.17) and are then computed, not fit. The secondary constants (C in Prop. III.6.2, prefactors in (III.6.21)-(III.6.22)) are expressions in the model's own parameters (α, β_c, κ). No new physical entities (particles, forces, dimensions, conserved quantities) are postulated; technical objects such as the edian M_r, the modified measures \tilde{P}_{r,λ}, the vertex factors V_r and \tilde{V}_r, and the kernel κ are definitions used within proofs, not entities requiring independent evidence.

assumptions (4)
  • standard math Standard percolation inequalities: BK, Reimer, Harris-FKG, Russo, OSSS; Aizenman-Newman tree-graph inequality; Gladkov inequality
    Invoked throughout §§III.2-III.6, e.g., Lemmas III.2.1-III.2.5, III.3.6, III.4.3, with citations [4], [30], [34], [49], [61], [64].
  • domain assumption Theorems and lemmas of paper I of the series ([45]) and paper II ([46]), in particular Theorems I.1.15 and I.1.17 (moment relations and superprocess limits conditional on the hydrodynamic condition), Theorem I.5.21 (scaling limit of moments), Lemma I.5.27, Corollary II.1.4, Theorem II.1.7
    The entire log-correction pipeline invokes these results once Theorem III.1.11 is proved. They are cited, not re-proved here. This is structural reliance on the author's prior work, not circularity, but it is a load-bearing external input.
  • domain assumption Kernel regularity: J(x,y)=J(||x-y||) decreasing, differentiable, |J'(r)|=(1+δ_r)r^{-d-α-1} with δ_r logarithmically integrable; normalization (∗)
    Stated in §III.1.1 as the definitional class of kernels; universality claims are relative to this class.
  • domain assumption Universal tightness theorem [50, Thm 2.2] and the spatially averaged two-point upper bound (III.1.10) from [51]
    These give M_r = O(r^{(d+α)/2}) (III.1.15) and underpin Lemma III.4.4 and the fictitious-regime estimates of §III.3.

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Pith. "Pith review of Critical long-range percolation III: The upper critical dimension." pith.science (2026). https://pith.science/paper/FXE7NTAJ

@misc{pith2026250818809,
  author       = {Pith},
  title        = {Pith review of: Critical long-range percolation III: The upper critical dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FXE7NTAJ}},
  note         = {Machine review of arXiv:2508.18809}
}
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. Here, we analyze the model at its upper critical dimension $d=3\alpha<6$. We prove the hydrodynamic condition holds, which allows us to apply our first paper's RG analysis to deduce that the model has the same superprocess scaling limits as in high dimension, after accounting for slowly varying corrections to scaling. We then compute the precise logarithmic corrections to scaling by analyzing the RG flow to second order. Our results yield in particular that for $d=3\alpha < 6$ the critical volume tail is \[ \mathbb{P}_{\beta_c}(|K|\geq n) \sim C \frac{(\log n)^{1/4}}{\sqrt{n}} \] as $n\to \infty$, while the critical two- and three-point functions are \[ \mathbb{P}_{\beta_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+\alpha} \; \text{ and } \; \mathbb{P}_{\beta_c}(x\leftrightarrow y \leftrightarrow z) \asymp \sqrt{\frac{\|x-y\|^{-d+\alpha}\|y-z\|^{-d+\alpha}\|z-x\|^{-d+\alpha}}{\log(1+\min\{\|x-y\|,\|y-z\|,\|z-x\|\})}}. \] These logarithmic corrections match those in hierarchical percolation but differ from those conjectured for nearest-neighbour percolation on $\mathbb{Z}^6$.

Figures

Figures reproduced from arXiv: 2508.18809 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, α ≥ 1 (where there is either no phase transition when α > 1 or a disconti… view at source ↗
Figure 2
Figure 2. Schematic illustrations of the events bounded by Lemmas III.6.15–III.6.17. Each lemma controls one way in which three clusters can “all interact with each other” in the coupling described at the beginning of the subsection. Left: Lemma III.6.15 bounds the probability that the clusters K0 and Kx x both cut the connection from y to a. Each violet box in the diagram denotes a copy of the vertex factor, while the orange… view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.PR 2026-07 accept novelty 8.0 of 10

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