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This paper proves the exact large-distance form of the k-point function for high-dimensional critical percolation, and derives the super-Brownian scaling limit of clusters from it.

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2026-08-01 04:54 UTC pith:TJIRXRLK

load-bearing objection Major result, but the load-bearing two-blob estimate has a genuinely unproven domination step; worth refereeing carefully rather than desk rejecting. the 3 major comments →

arxiv 2607.22387 v1 pith:TJIRXRLK submitted 2026-07-24 math.PR math-phmath.MP

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

Arthur Blanc-Renaudie , Tom Hutchcroft This is my paper
classification math.PR math-phmath.MP MSC 60K3582B43
keywords percolationk-point functionsuper-Brownian excursionscaling limitcontinuum random treeincipient infinite clusterlace expansioncritical phenomena
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Critical percolation sits between independent percolation and branching random walk: at criticality in high dimensions, the self-interaction that distinguishes them should vanish at large scales. This paper proves that the vanishing is exact in a strong sense. It establishes the asymptotic form of the k-point function — the probability that k well-separated points lie in the same cluster — as a sum over binary trees of products of lattice Green's functions, with a single constant factor encoding local interactions. From that formula it derives the full scaling limit: the cluster of the origin, rescaled in its internal metrics by r^2 and in space by r, converges to the continuum random tree embedded in R^d by Brownian motion, simultaneously for chemical, pivotal, and resistance distances. A corollary is the first-order one-arm asymptotics, upgrading previously known up-to-constants estimates.

Core claim

The central claim is that for critical Bernoulli bond percolation on Z^d with d large enough (or d > 6 on a spread-out lattice), the k-point function satisfies T_{p_c}(x_1,...,x_k) ~ V^{k-2} A^{2k-3} G(x_1,...,x_k) as the points separate, where G is the sum over ternary trees of products of lattice Green's functions and A and V are lattice-dependent constants. This is the exact mean-field formula: all large-distance correlations are carried by Green's functions, and the only trace of the model's self-interaction is the constant V at each branching point. The paper then shows this scalar formula upgrades to a full distributional statement: conditional on the cluster being large, the quadruple

What carries the argument

The load-bearing object is the scheme function: the probability that a finite 'scheme' of marked points, prescribed microscopic edge sets, and required or forbidden connections holds. The paper proves an asymptotic factorization theorem: as the marked points separate, a scheme function splits into a product of local 'blob factors' (one for each cluster's microscopic geometry) times products of Green's functions for each required connection. This factorization is derived from just two inputs — the two-point asymptotics (A) and a two-blob mixing estimate (B) asserting that conditioning on two distant points being connected leaves the local geometry around each point asymptotically as an indepe

Load-bearing premise

The load-bearing premise is the two-blob mixing estimate: that conditional on two distant points being connected, the cluster's local geometry near the two points decorrelates to two independent incipient infinite clusters, at the rate given by the two-point function; the central theorems collapse if this fails even when the two-point asymptotics hold, and the paper's proof of this estimate inherits quantitative lace-expansion bounds from prior works.

What would settle it

Find a graph or model satisfying the two-point asymptotics (A) but where the two-blob estimate (B) fails — for instance, where the incipient infinite cluster limit depends on the direction of approach — and check whether the k-point function formula and the super-Brownian scaling limit still hold; the paper's Theorem 1.12 predicts they must fail. Concretely, on a spread-out lattice in dimension 7 with modest L, simulate the ratio of the three-point function to G(x,y)G(y,z)G(z,x) aggregated over directions: convergence to a direction-independent constant V is the paper's prediction, and any sys

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The k-point function of critical percolation has the conjectured tree-graph asymptotics in every dimension that satisfies the two stated perturbative conditions.
  • The counting measure on the cluster, rescaled by r^4 in mass, converges to integrated super-Brownian excursion — the same limit as critical branching random walk.
  • The cluster as an embedded metric-measure space converges to the continuum random tree with Brownian embedding simultaneously for chemical, pivotal, and resistance metrics, and these distances are asymptotically constant multiples of one another.
  • One-arm probabilities have the sharp asymptotics C r^{-2} (extrinsic) and C r^{-1} (chemical), upgrading previously known up-to-constants estimates.
  • The same conclusions hold for any percolation model satisfying the two-point asymptotics and the two-blob mixing estimate, without direct appeal to the lace expansion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the proof is packaged as the pair of assumptions (A) and (B), it gives a template for transferring super-Brownian scaling limits to other high-dimensional models that admit a two-point estimate and a decorrelation principle, even when the lace expansion is unavailable.
  • The identification of the constants C_vol = A^2 V and C_prob = AV / N(μ(R^d)≥1) suggests that the universal super-Brownian object appears with model-dependent prefactors; the ratio C_prob C_vol^{-1/2} could serve as a stable fingerprint of the lattice structure in simulations.
  • The asymptotic proportionality of chemical, pivotal, and resistance distances suggests that on large critical clusters, pivotal edges are distributed along the tree structure in a way that makes all three metrics share the same coarse geometry; this could be probed by computing the constant ratios on smaller lattices.
  • The sharp one-arm asymptotics open the door to refined questions about the outer boundary of large critical clusters, such as fluctuation exponents for the set of points at chemical distance r.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper proves that, under two hypotheses—the two-point asymptotics (A) and a two-blob mixing estimate (B)—the critical percolation k-point function has the tree-graph asymptotics conjectured by Aizenman and Newman, and that the cluster of the origin, suitably rescaled, converges to integrated super-Brownian excursion jointly for the chemical, pivotal, and resistance metrics. A non-perturbative criterion, Theorem 1.12, reduces Theorems 1.4, 1.6, and 1.8 to (A) and (B). The authors then use the lace expansion in Section 2 to establish (B) for high-dimensional and spread-out lattices. The main body from Section 3 onward is a self-contained derivation from (A) and (B), involving new scheme-function factorization and overcount-localization arguments. The paper also derives first-order one-arm asymptotics and answers a question of Heydenreich and van der Hofstad.

Significance. If the proof of (B) is fully valid, this is a landmark result: it resolves long-standing conjectures of Aizenman–Newman, Hara–Slade, and van der Hofstad–Slade, and gives the first super-Brownian scaling limit for unoriented finite-range percolation, with precise constants and simultaneous convergence of several intrinsic metrics. The conditional framework of Theorem 1.12 is itself valuable and likely to be influential, since it cleanly separates the probabilistic decorrelation input from the analytic derivation of the scaling limit. The paper is also commendable for its careful treatment of overcount factors and for the scheme-function formalism, which yields k-arm IIC measures as byproducts. However, the unconditional claims rest on Section 2's proof of the two-blob estimate, and that proof is the weakest point; the delegation to [48,49] is not sufficiently detailed to certify the required estimate as written.

major comments (3)
  1. [Section 2.1, proof of Theorem 2.3] The displayed domination inequality Π^{(i+1)}(x,y) ≥ [min_{w∈V(W)} T(x,w) min_e P(e)] Π^{(i)}_W(x,y) is load-bearing: it is the step that converts the lace-expansion bounds of [48] into the bound Π_W(0,x) ≤ C_W ⟨x⟩^{-2d+6} used in Corollary 2.5 and hence in Theorem 1.11. The inequality is asserted with the phrase “considering the contribution…” and no derivation. It is not obvious that the “only via W” events defining Π_W are dominated by the 2-edge-connected events entering the standard Π diagrams; if the inequality fails, the two-blob estimate does not follow from the cited bounds. The proof must either prove this inequality carefully, or cite a specific result in [48] that directly gives the needed bound, including uniformity for the finitely edge-deleted graphs used later in Theorem 1.11.
  2. [Section 2.1, Remark 2.4] The direction-independence of the IIC limit, and hence the identification of the P_IIC(F) factors in Theorem 1.11, is asserted to require the Liouville property, but this is not proved. The remark itself says “one also expects” directional dependence in other settings, which flags that this is not a vacuous point. Since (2.8) in the proof of Theorem 1.11 uses the IIC measure as a direction-independent limit, a proof or precise reference for the Liouville property of the relevant graph and its implication for the two-point function of the IIC should be supplied.
  3. [Section 2.2, proof of Theorem 1.11] The convergence argument for [(I_{W_1}+Π_{W_1})P − (I_{W_1}+Π_{y,W_1})P_y](0,z) → 0 uses dominated convergence first over oriented edges and then over expansion depth, with the uniform bound provided by Theorem 2.3 and Observation 2.7. This is a legitimate strategy, but it inherits all the gaps of Theorem 2.3. In particular, the claimed uniformity of the bound for Π_{y,W_1} in the edge-deleted graph Z^d_{L,y} is not transparent from the proof of Theorem 2.3, because that proof is written for the full lattice and the displayed domination inequality involves the full-lattice two-point function T. The authors should state and prove explicitly the version of the bound that is valid on the deleted graphs, or give the exact reference from [48] that yields it.
minor comments (3)
  1. [Section 3.1, proof of Lemma 4.3] The phrase “the open triangle condition holds” is used without definition. Since this property is used to control sums of products of Green's functions, a short definition or reference would help the reader.
  2. [Theorem 1.6] In the display, the random array on the right has d(Y_i,Y_j) without a subscript #, while the left has d_#(X_i,X_j); the text explains this is intentional, but the notation is initially confusing and a small parenthetical in the display would improve readability.
  3. [Throughout] The letter T is used both for the two-point function T_{p_c} and for combinatorial trees T ∈ Tr(k). This is standard in the subject but can be ambiguous in displays like Theorem 1.4; a short notation note would be helpful.

Circularity Check

0 steps flagged

No significant circularity: the k-point and scaling-limit theorems are genuine consequences of the two-point and two-blob inputs, with constants characterized rather than fitted.

full rationale

The paper's derivation chain is not circular. Theorem 1.12 reduces the main theorems to two inputs: the two-point asymptotics (A) and the two-blob estimate (B). Section 2 proves (B) from the lace-expansion bounds of Hara, van der Hofstad and Slade [48,49], which are external to this paper and do not presuppose the k-point or scaling-limit conclusions. The constants A and V are characterized as limits of independent two-point and localized blob/scheme quantities (e.g., in the proof of Theorem 1.4 the localized factor C(R1,R2) is shown to converge to V because the left-hand side is independent of R1,R2), not fitted to the predicted k-point function. The overcount-factor localization (Proposition 4.8) compares N(x1,...,xk) with a product of local arm sets via Ψ_T-usual estimates, again an independent combinatorial/probabilistic computation rather than a restatement of the desired asymptotics. The self-citations in Remarks 1.1-1.3 are contextual; the one place where prior work of the second author is used substantively, the moment characterization of integrated super-Brownian excursion from [68], is a parameter-free uniqueness theorem used only to convert independently proved moment convergence into weak convergence, so it is independent support rather than a circular load-bearing citation. The genuine soft spot is the asserted domination inequality in the proof of Theorem 2.3 ('considering the contribution...' yielding Π^{(i+1)}(x,y) ≥ ... Π^{(i)}_W(x,y)); this is a proof gap or correctness risk, not a circular reduction, because no displayed equation in the paper defines a predicted quantity in terms of itself and no fitted parameter is renamed as a prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No fitted or hand-chosen parameters: A is fixed by the known two-point function asymptotics, V is defined as the double limit of blob factors W in §3.1 and proven positive by Lemma 4.13; the scaling constants C_# are determined as geodesic factors G_#A, C_vol=A²V, C_prob=AV/N(ν(T)≥1). No new physical entities are postulated; the new mathematical objects (schemes, blob factors, k-arm IIC measures) are constructed and proven to exist within the paper.

axioms (4)
  • standard math Two-point function asymptotics (A): T_{p_c}(x,y) ~ A G(x,y), proven in [47,48] under d≥d0 or d>6 with L≥L0.
    Primary input to Theorem 1.12; assumed throughout Sections 3–6.
  • standard math Quantitative lace expansion bounds (Theorem 2.3): operators Π_W decay as C_W⟨x⟩^{-2d+6} and remainders R_N→0; stated as 'essentially equivalent to the estimates proven in [48]'.
    Underpins the proof of the two-blob estimate (Theorem 1.11) and the existence of the IIC (Corollary 2.5). This is the least self-contained premise.
  • domain assumption Liouville property of Z^d_L (tail-trivial random walk) so that IIC limits do not depend on direction.
    Remark 2.4: without this, lim_{x→∞} P(·|0↔x) could be direction-dependent (as on regular trees); the two-blob estimate is formulated in terms of the IIC measure.
  • standard math Green's function convolution estimates for d>6 (open triangle condition, finiteness of G*G*G, decay exponents d-2, d-4).
    Used throughout Lemmas 3.5, 4.3, 5.7, and the scheme-function arguments to show that non-local interactions are negligible.

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read the original abstract

We prove that there exist positive constants $A$ and $V$ such that the high-dimensional critical percolation $k$-point function is given by \[ T_{p_c}(x_1,x_2,\ldots,x_{k}) \sim V^{k-2} A^{2k-3} \sum_{T\in \mathsf{Tr}(k)} \sum_{\substack{\Phi:V(T)\to \mathbb{Z}^d \\ \Phi(i)=x_i \forall 1\leq i \leq k}} \prod_{\substack{u,v\in V(T)\\u\sim v}}G(\Phi(u),\Phi(v)) \] as $\min_{i\neq j}\|x_i-x_j\|\to \infty$, where $\mathsf{Tr}(k)$ is a set of isomorphism class representatives of trees with $k$ labelled leaves $\{1,\ldots,k\}$ and unlabelled internal vertices all of which have degree $3$ and $G$ is the lattice Green's function. This verifies a conjecture of Aizenman and Newman (1984) subject to the usual perturbative conditions needed for convergence of the lace expansion. It follows from this theorem that the law of the cluster of the origin, considered as the counting measure on its range, converges under rescaling to the canonical measure of the integrated super-Brownian excursion. By computing the asymptotics of various more complicated variations on the $k$-point function, we also prove the stronger result that the cluster converges as an embedded metric-measure space to the continuum random tree equipped with its Brownian embedding into $\mathbb{R}^d$. This convergence holds simultaneously with respect to the chemical distance, pivotal distance, and resistance distance on the cluster, which we prove are asymptotic to constant multiples of each other. This resolves conjectures of Hara and Slade (1998) and van der Hofstad and Slade (2003). As a corollary of our results we prove that there exists a positive constant $C$ such that $\mathbb{P}_{p_c}(0\leftrightarrow \mathbb{Z}^d \setminus [-r,r]^d)\sim C r^{-2}$, answering a question of Heydenreich and van der Hofstad (2017).

Figures

Figures reproduced from arXiv: 2607.22387 by Arthur Blanc-Renaudie, Tom Hutchcroft.

Figure 1
Figure 1. Figure 1: Aldous’ construction of the continuum random tree [ [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of a cutting bond for the event {x ↔ y only via W}. The elliptical regions represent 2-connected components of the cluster, while straight lines represent bridges. The pink shaded regions represent the set W. The cutting bond is unique when it exists. with percolation on a general graph. (Specifically, we will later apply the lace expansion to the graph formed by deleting some finite set of edges f… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of a scheme: a plan together with assignments of blobs to each vertex. In this example, the scheme is a 3-scheme whose associated plan has one vertex of degree 3 and three vertices of degree 1, so that the degree 3 vertex is assigned a 3-blob and each of the three degree 1 vertices are assigned 1-blobs. of critical percolation on the spread-out lattice Z d L . We define a k-blob to b… view at source ↗

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