Pith. sign in

REVIEW 2 major objections 4 minor 62 references

Range convergence and sharp one-arm asymptotics for the critical percolation cluster in high dimensions

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Critical percolation clusters converge to super-Brownian range in d≥11.

desk verdict Upgrades k-point convergence to law, range, and sharp one-arm asymptotics; main risk is reliance on external preprints. read the letter →

arxiv 2607.24561 v1 pith:VNUZDGQK submitted 2026-07-27 math.PR

M. Cabezas , D. Croydon , A. Fribergh , N. Kawamoto This is my paper · ORCID
classification math.PR MSC 60K3560J6860F1782B43
keywords criticalpercolationhigh-dimensionalsuper-Brownianmotioncanonicalmeasureone-armexponentrangeconvergencek-pointfunctionslowermassbound
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

The paper proves the super-Brownian scaling limit for high-dimensional critical percolation at the level of laws. For critical nearest-neighbour bond percolation on $\mathbb{Z}^d$ with $d\ge 11$, the rescaled empirical measure of the cluster of the origin, conditioned to carry macroscopic mass, converges in a $\sigma$-finite sense to the canonical total occupation measure of super-Brownian motion. A new uniform lower mass bound—any macroscopic region the cluster reaches contains of order $R^4$ points—then upgrades this to convergence of the rescaled cluster as a compact set, in the Hausdorff metric, to the range of super-Brownian motion. The same machinery yields the sharp one-arm asymptotics $r^2\mathbb{P}(0\leftrightarrow \partial B_r)\to\theta_1\in(0,\infty)$, replacing the previously known order-only bound.

What carries the argument

The load-bearing identity is the exact moment dictionary between lattice connection functions and tree integrals. The rescaled $(m+1)$-point functions of percolation converge to sums over binary trees of integrals $I_T$ of Brownian Green kernels, and the same tree integrals are the moment densities of the total occupation measure of super-Brownian motion under its canonical measure. Around this identity, the proof uses size-biasing by $\mu(\varphi_0)$ to turn the infinite $\sigma$-finite limit into finite measures with convergent moments, exponential-moment determinacy to identify limits, and the one-arm bound for tightness; a separate pioneer-point regularity argument over annuli produces the lower mass bound, and a deterministic support-convergence lemma (weak convergence plus uniform lower mass bound implies Hausdorff convergence of supports) closes the geometric step.

What would settle it

Run high-precision simulations of critical bond percolation on $\mathbb{Z}^{11}$ and record $r^2\mathbb{P}(0\leftrightarrow\partial B_r)$ for $r$ a sequence of dyadic scales; if the sequence does not converge to a positive finite constant, or if the constant disagrees with $\theta_1$ evaluated numerically from the radial PDE $\tfrac12 v''+\tfrac{d-1}{2r}v'=\tfrac{\lambda}{2a}v^2$ with $v\to\infty$ at $1$, the central claim fails. Equivalently, simulate the conditioned cluster $\{|\mathcal C|\ge R^4\}$ and test the lower mass bound: if with positive probability some ball of radius $\delta R$ met by the cluster contains $o(R^4)$ sites as $R\to\infty$ at fixed $\delta$, Theorem 1.2 and the range convergence collapse.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that moment-level convergence of connection functions can be promoted to law-level convergence of the cluster and its geometry. With $\mu_n=(an^4)^{-1}\sum_{x\in\mathcal C}\delta_{x/n}$ and $\nu_n=n^2\mathbb{P}(\mu_n\in\cdot)$, Theorem 1.1 states that for every $\varepsilon>0$, $\nu_n(\cdot;\mu_n(\mathbb{R}^d)>\varepsilon)$ converges weakly to $N_{0,\lambda/a}(\cdot;\mu(\mathbb{R}^d)>\varepsilon)$, the canonical measure of a $(\tfrac12\Delta,\lambda/a)$-super-Brownian motion restricted to total mass above $\varepsilon$. Theorem 1.2 states a uniform lower mass bound: conditionally on the cluster reaching scale $R$, every ball of radius $\delta R$ that the cluster meets contains at least order $R^4$ sites, up to an arbitrarily small failure probability as $\delta\to0$. Combining these, Theorem 1.3 gives joint convergence of $(\mu_n,n^{-1}\mathcal C)$ to $(\mu,\operatorname{supp}\mu)$ under the conditioned canonical measure, and Theorem 1.4 identifies $\lim_r r^2\mathbb{P}(0\leftrightarrow\partial B_r)=\theta_1=(a/\lambda)N_{0,1}(\mathcal R\not\subset B_1)\in(0,\infty)$.

Load-bearing premise

The conclusions rest on two load-bearing inputs from outside this paper: the uniform two-point bound (1.2) and the $k$-point convergence (1.3); if either failed at the needed uniformity, the measure convergence, range convergence, and one-arm constant would not follow.

Editorial extensions

If this is right

  • The one-arm exponent is exactly $2$ with an identified constant: $r^2\mathbb{P}(0\leftrightarrow\partial B_r)\to\theta_1$, so the cluster's reach has the same tail as the super-Brownian range.
  • Conditioned on carrying macroscopic mass, the rescaled cluster converges as a compact set, so macroscopic geometric quantities that are continuous functions of the range—such as diameter—converge to the corresponding super-Brownian quantities.
  • The uniform lower mass bound holds under the two-point bound alone for $d>6$, giving a ready-made no-thin-region input for other scaling-limit programs on critical structures.
  • The measure convergence recovers the known cluster-size tail $\mathbb{P}(|\mathcal C|\ge N)\asymp N^{-1/2}$ with the explicit constant forced by the normalization, confirming consistency of the constants $a,\lambda$.
  • Because Theorem 1.1 factors through the two-point bound and the $k$-point convergence, any lattice model verified to satisfy those two hypotheses inherits the same three conclusions.

Reading between the lines

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

  • If the expected extension of the $k$-point convergence to spread-out models in $d>6$ materializes, the same proof would give measure, range, and one-arm convergence there without modification, since the argument is hypothesis-driven.
  • The constant $\theta_1$ is characterized by the radial boundary blow-up problem $\tfrac12 v'' + \tfrac{d-1}{2r}v' = \tfrac{\gamma}{2}v^2$ with $v\to\infty$ at $r=1$; numerically solving this ODE would yield a quantitative prediction that lattice simulations could test.
  • The lower mass bound supplies exactly the volume-growth condition demanded by resistance-form scaling-limit criteria, so a random-walk ('ant in the labyrinth') scaling limit on the high-dimensional critical cluster is a plausible next application.
  • The size-biasing and un-biasing template may transfer to other $\sigma$-finite scaling limits where only moment convergence is known.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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. The paper proves, for critical nearest-neighbour bond percolation on Z^d with d ≥ 11, that the rescaled empirical cluster measure (1.1) converges in a σ-finite sense to the total occupation measure of super-Brownian motion, conditional on the total mass exceeding ε. It also proves a uniform lower mass bound for the critical cluster in the Euclidean metric, derives Hausdorff convergence of the rescaled cluster to the range of super-Brownian motion, and obtains the sharp one-arm asymptotics r^2 P(0 ↔ ∂B_r) → θ_1 ∈ (0,∞). The derivation is carried out under the explicit external hypotheses (1.2) and (1.3), the latter being the k-point convergence result of [21]; the paper is transparent that d ≥ 11 serves only to guarantee these inputs.

Significance. If the external inputs are accepted, this is a major step: it upgrades the moment-level k-point convergence of [21] to law-level convergence of the cluster measure in the high-dimensional nearest-neighbour setting, and it refines the Kozma–Nachmias one-arm bound to an exact asymptotic with an identified constant. The lower mass bound is of independent interest for random-walk and Gromov–Hausdorff-type scaling programmes. The paper is carefully structured: the size-biasing mechanism converts the infinite canonical measure into a finite measure, the moment-convergence step is an exact lattice-to-continuum rewriting rather than an approximation, and the deterministic support-convergence lemma cleanly isolates the role of the lower mass bound. The dependence on the recent preprints [21] and [9] is clearly flagged; there is no sign of circularity or of fitted constants manufacturing the conclusion.

major comments (2)
  1. [§2.4, display after (2.20)] In the proof of Theorem 2.1, the displayed estimate ∑_{i=ε^{-1/2}}^{ε^{-1}} [r^{d+1}e^{-c(log αr^2)^4} + α/(iε)^2 + r^2 e^{-δr^2/2}] ≲_Λ ε^{-1}[r^{d+1}e^{-c(log αr^2)^4} + α + r^2 e^{-δr^2/2}] is not correct for the middle term: ∑_{i=ε^{-1/2}}^{ε^{-1}} (iε)^{-2} ≍ ε^{-3/2}, not ε^{-1}. With the stated choices α ≍ ε^2 and ε ≍ η^{1/4}, the correctly bounded first term is of order η^{1/8}, not η^{1/4}. Thus the proof as written establishes at most limsup ≤ C η^{1/8}, and Theorem 2.1's stated η^{1/4} rate is not justified. The later uses of Theorem 2.1 only require the bound to vanish as η↓0, so the main scaling-limit theorems are not endangered, but the theorem statement and Lemma 4.2 should be adjusted to the weaker rate or the argument revised.
  2. [§2.2, Eq. (2.7)] Lemma 2.6 relies on the exterior-geometry estimate (2.7), which is described only as an analogue of [9, Claim 5.5] with the assertion that 'the same argument remains valid.' This estimate is load-bearing for Lemmas 2.6 and 2.7 and hence for Theorem 2.1. Since [9] is a recent preprint rather than a published theorem, the authors should either prove (2.7) in the present paper or state it as a standalone lemma with full hypotheses and a sufficiently detailed proof sketch, so that the lower mass bound is not contingent on an unverified adaptation.
minor comments (4)
  1. [§2.2–§2.3] The symbol C(K) denotes different quantities in Lemma 2.7 and in the proof of Lemma 2.10; renaming one of them would avoid confusion.
  2. [Theorem 1.3 statement] There is a stray extra parenthesis in the sentence 'with a, λ as in (1.3)), jointly in M_F(R^d) × K(R^d)'; the closing parenthesis after (1.3) should be removed.
  3. [Lemma 2.7] In the definition of C(K) = K^d / p_c^{2dK}, please check whether the exponent should be dK rather than 2dK, to match the binomial domination parameter p_c^{dK} used a few lines later in the proof.
  4. [§2.1 and Proposition 3.8] The one-arm estimate (2.1) is introduced as a known input at the start of Section 2, but it is also proved/later stated as Proposition 3.8; adding a forward cross-reference would clarify the logical dependence of Section 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central derivation is a genuine upgrade of the externally supplied k-point convergence, and no conclusion is equivalent to its inputs by construction.

full rationale

The paper's chain of reasoning is self-contained once its clearly flagged external inputs are accepted, and the main new conclusions require analytic work not present in those inputs. Section 3 starts from the k-point convergence (1.3), which supplies moment densities only, and converts it into law-level convergence through four independent ingredients: the exact lattice-to-continuum rewriting (3.13)-(3.14), dominated convergence using the Aizenman-Newman tree-graph inequality (Lemma 3.9), tightness from the Kozma-Nachmias one-arm bound and a second-moment estimate (Lemma 3.15), and moment determinacy of the size-biased canonical measure via finite exponential moments (Lemmas 3.13-3.14). None of these steps is an identity rewriting of (1.3): the size-biasing and un-biasing procedure is a genuine measure-theoretic device needed because N0 is sigma-finite and the restriction events are invisible to moments. The constants a and lambda are external inputs from [21], and gamma is set to lambda/a so that the SBM canonical measure has the correct moment formula; this is a normalization choice, not a fitted prediction, and it does not force the distributional convergence. Section 2's uniform lower mass bound, Theorem 1.2, is proved by an involved percolation argument using only the uniform two-point bound (1.2), the one-arm bound of [50], and adaptations of [9]; it is not derived from Theorem 1.1 and is stronger than the input bounds it uses. Theorem 1.3 combines the measure convergence with the lower mass bound and the deterministic Athreya-Lohr-Winter support-convergence principle (Lemma 4.4); the support limit is not built into the measure convergence, since weak convergence alone cannot force every support point to carry mass. Theorem 1.4 obtains the sharp one-arm asymptotics by evaluating the range convergence on an exit event after localization by phi0 and removing the localization through a thin-region estimate; the coarse Kozma-Nachmias bound P(0<->dB_r) as r^-2 is used only as an envelope in tightness and in Lemma 4.2, not as the sharp limit, whose constant theta1 comes from the SBM range exit probability. The self-citations present ([15], [16], [24]-[26]) appear in the introduction and related-work discussion as motivation and context and are not load-bearing in any proof.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims inherit a small number of external percolation inputs, all cited and all valid for nearest-neighbour d≥11; the genuinely new analytic work is the lower mass bound (Section 2) and the size-bias/tightness route to measure convergence (Section 3). No free parameter is fitted in this paper: the constants a, λ arise from the prior k-point theorem.

assumptions (6)
  • domain assumption Uniform two-point bound (1.2): K_1^{-1}(1∨|x|)^{-(d-2)} ≤ τ_2(0,x) ≤ K_1(1∨|x|)^{-(d-2)}.
    Used throughout Sections 2 and 3; established for nearest-neighbour percolation in d≥11 by Hara [35], Fitzner–van der Hofstad [33] and Hara–van der Hofstad–Slade [36], and recorded as an input in Section 1.2.
  • domain assumption Convergence of k-point functions (1.3) to tree integrals with constants a, λ.
    Main percolation input from Chatterjee, Chinmay, Hanson and Sosoe [21]; supplies the pointwise limit in the dominated convergence argument for all moments, Section 3.2.3.
  • domain assumption Kozma–Nachmias one-arm estimate (Prop 3.8): P(0↔∂B_r) ≍ r^{-2}.
    Used for diameter tightness in Sections 3.3.2 and 4.3 and for probability comparisons in Theorem 1.1; follows from (1.2), cited to [50] and [9].
  • domain assumption Cluster-size tail (2.21): P(|C|≥R) ≍ R^{-1/2}.
    Used in Section 2.5 for the |C|≥R^4 version of the lower mass bound; the paper states this follows from the triangle condition, citing Barsky–Aizenman [14] and Hara–Slade [37].
  • standard math Fixed-time moment formulas and Brownian snake representation for super-Brownian motion.
    Occupation moment formula (Prop 3.4) and compact connected range (Prop 4.1) rely on standard superprocess theory recalled in Section 1.4; references [30,54,59].
  • standard math Aizenman–Newman tree-graph inequality (Prop 3.7).
    Used for the domination bound in Lemma 3.9; stated with proof reference [3] in Section 3.2.1.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Range convergence and sharp one-arm asymptotics for the critical percolation cluster in high dimensions." pith.science (2026). https://pith.science/paper/VNUZDGQK

@misc{pith2026260724561,
  author       = {Pith},
  title        = {Pith review of: Range convergence and sharp one-arm asymptotics for the critical percolation cluster in high dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNUZDGQK}},
  note         = {Machine review of arXiv:2607.24561}
}
abstract

For critical Bernoulli bond percolation on $\Z^d$ in the high-dimensional regime, we prove that the rescaled empirical measure of the cluster of the origin converges, in a suitable $\sigma$-finite sense, to the total occupation measure of super-Brownian motion. Combined with a uniform lower mass bound for the critical cluster with respect to the extrinsic (Euclidean) metric, which we also prove and which is of independent interest, the measure convergence further yields the convergence of the rescaled cluster as a compact set, in the Hausdorff metric. As a consequence, we are able to obtain the sharp one-arm asymptotics $r^2\,\bP(0\leftrightarrow \partial B_r)\to \theta_1\in(0,\infty)$, hence refining a result of Kozma and Nachmias.

Figures

Figures reproduced from arXiv: 2607.24561 by the authors.

Figure 1
Figure 1. Schematic for the proof of Theorem 2.1. The solid black circles represent pioneer points. and the two-point function estimate τ (x) ≍ |||x|||2−d , ∀x ∈ Z d , (2.2) where we have written |||x||| = ∥x∥ ∨ 1. Here τ (x) = τ2(0, x) is the two-point function of (1.2), and ∥· ∥ denotes the Euclidean norm (any fixed norm gives the same estimates up to constants). In particular, the two-sided estimate (2.2) is the uniform tw… view at source ↗
Figure 2
Figure 2. Left: Schematic for the definition of K-line good points. The line segments represent L(B), and the solid black circles represent the K-line good points of B. Right: Schematic for the definition of Lx, for x lying within distance K of an edge of ∂Qz,i. The segment first extends orthogonally into Qz,i (perpendicular to the face on which x lies), then bends and extends orthogonally again – this time parallel to the fi… view at source ↗
Figure 3
Figure 3. Left: Schematic for the definitions of E1 and A1. The gray small boxes are those in Bz,i(w). Right: Enlarged view of the blue boxed region in the left panel. If the red boxed active box is chosen as q2 at step 2, it is added to E1 to form E2 = E1 ∪ q2, and the box to its right becomes active, since it can be reached from 0 by paths going only through E2. We will show shortly that conditioning on N(w), the quantity N… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The blue box centered at y and Y are Qy,L, and QY,K, respectively. In the illustration, there is another point of Reg′ i (C) in Qy,L such that y is connected, within Qy,2L and off C, to the boundary of the box of side length K centered at that point. The point Y , howe…
Figure 5
Figure 5. Figure 5: The net construction in Lemma 4.2. A point x ∈ C realizing the thin-region event lies within δn/2 of a net point ˆx (grey dots); the cluster then reaches the box Qx,r ˆ n (solid), while the enlarged box Qx, ˆ 2rn (dashed) is contained in the ball B(x, 3 √ d δn) (dotted…
Figure 6
Figure 6. Figure 6: Localization in the proof of Theorem 1.4. On {0 ↔ ∂BKn} the cluster crosses the annulus {K/8 ≤ |y| ≤ K/4}; off the thin-region event, the crossing point x deposits mass ≥ ηδ4/a in the ball B(x/n, 3 √ d δ) (dotted), which is contained in the shaded disc B(K/4)+3√ d δ, o…

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 54 canonical work pages

  1. [21]

    Chatterjee, P

    S. Chatterjee, P. Chinmay, J. Hanson, and P. Sosoe. Convergence ofk-point functions in high dimensional percolation, 2026. arXiv:2604.08462

    arXiv 2026

  2. [9]

    Asselah, B

    A. Asselah, B. Schapira, and P. Sousi. Capacity in high dimensional percolation, 2025. Preprint, arXiv:2509.21253

    arXiv 2025

  3. [18]

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

    A. Blanc-Renaudie and T. Hutchcroft. Super-Brownian limits and thek-point function for high-dimensional percolation, 2026. Preprint, arXiv:2607.22387

    work page Pith review arXiv 2026

  4. [1]

    R. J. Adler. Superprocess local and intersection local times and their corresponding particle pictures. InSeminar on Stochastic Processes 1992, volume 33 ofProgr. Probab., pages 1–42. Birkh¨ auser, Boston, 1993

    work page 1992

  5. [2]

    Aizenman

    M. Aizenman. On the number of incipient spanning clusters.Nuclear Phys. B, 485:551–582, 1997

    work page 1997

  6. [3]

    Aizenman and C

    M. Aizenman and C. M. Newman. Tree graph inequalities and critical behavior in perco- lation models.J. Statist. Phys., 36:107–143, 1984

    work page 1984

  7. [4]

    D. Aldous. Tree-based models for random distribution of mass.J. Statist. Phys., 73:625– 641, 1993

    work page 1993

  8. [5]

    C. D. Aliprantis and K. C. Border.Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin, 3rd edition, 2006

    work page 2006

Show all 62 references
  1. [6]

    Alon and J

    N. Alon and J. H. Spencer.The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ, 2008

    2008

  2. [7]

    Angel, D

    O. Angel, D. A. Croydon, S. Hernandez-Torres, and D. Shiraishi. Scaling limits of the three- dimensional uniform spanning tree and associated random walk.Ann. Probab., 49(6):3032– 3105, 2021

    2021

  3. [8]

    Archer and D

    E. Archer and D. A. Croydon. Scaling limit of critical percolation clusters on hyperbolic random half-planar triangulations and the associated random walks.Random Structures Algorithms, 67(4):Paper No. e70020, 44, 2025

    2025

  4. [10]

    Athreya, W

    S. Athreya, W. L¨ ohr, and A. Winter. The gap between Gromov-vague and Gromov- Hausdorff-vague topology.Stochastic Process. Appl., 126:2527–2553, 2016. 58

    2016

  5. [11]

    M. T. Barlow, D. A. Croydon, and T. Kumagai. Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree.Ann. Probab., 45(1):4–55, 2017

    2017

  6. [12]

    M. T. Barlow, A. A. J´ arai, T. Kumagai, and G. Slade. Random walk on the incipient infinite cluster for oriented percolation in high dimensions.Comm. Math. Phys., 278(2):385–431, 2008

    2008

  7. [13]

    M. T. Barlow and T. Kumagai. Random walk on the incipient infinite cluster on trees. Illinois J. Math., 50(1-4):33–65, 2006

    2006

  8. [14]

    D. J. Barsky and M. Aizenman. Percolation critical exponents under the triangle condition. Ann. Probab., 19:1520–1536, 1991

    1991

  9. [15]

    Ben Arous, M

    G. Ben Arous, M. Cabezas, and A. Fribergh. Scaling limit for the ant in a simple high- dimensional labyrinth.Probab. Theory Related Fields, 174(1-2):553–646, 2019

    2019

  10. [16]

    Ben Arous, M

    G. Ben Arous, M. Cabezas, and A. Fribergh. Scaling limit for the ant in high-dimensional labyrinths.Comm. Pure Appl. Math., 72(4):669–763, 2019

    2019

  11. [17]

    Billingsley.Probability and Measure

    P. Billingsley.Probability and Measure. Wiley, New York, 3rd edition, 1995

    1995

  12. [19]

    Bramson, J

    M. Bramson, J. T. Cox, and J.-F. Le Gall. Super-Brownian limits of voter model clusters. Ann. Probab., 29:1001–1032, 2001

    2001

  13. [20]

    Chatterjee, P

    S. Chatterjee, P. Chinmay, J. Hanson, and P. Sosoe. Robust construction of the incipient infinite cluster in high dimensional critical percolation, 2025. arXiv:2502.10882

    2025 arXiv

  14. [22]

    Chatterjee and J

    S. Chatterjee and J. Hanson. Restricted percolation critical exponents in high dimensions. Comm. Pure Appl. Math., 73(11):2370–2429, 2020

    2020

  15. [23]

    D. A. Croydon. Scaling limit for the random walk on the largest connected component of the critical random graph.Publ. Res. Inst. Math. Sci., 48(2):279–338, 2012

    2012

  16. [24]

    D. A. Croydon. An introduction to stochastic processes associated with resistance forms and their scaling limits.RIMS Kokyuroku, 2030: paper no. 1, 2017

    2017

  17. [25]

    D. A. Croydon. Scaling limits of stochastic processes associated with resistance forms.Ann. Inst. Henri Poincar´ e Probab. Stat., 54(4):1939–1968, 2018

    1939

  18. [26]

    D. A. Croydon, B. M. Hambly, and T. Kumagai. Time-changes of stochastic processes associated with resistance forms.Electron. J. Probab., 22, 2017. paper no. 82, 41 pp

    2017

  19. [27]

    Dankovic, M

    I. Dankovic, M. Markering, J. Miller, and Y. Yuan. The scaling limit of random walk and the intrinsic metric on planar critical percolation. Preprint available at arXiv:2604.14122, 2026. 59

    2026 arXiv

  20. [28]

    de Gennes

    P.-G. de Gennes. La percolation: un concept unificateur.La Recherche, 7:919–927, 1976

    1976

  21. [29]

    Derbez and G

    E. Derbez and G. Slade. The scaling limit of lattice trees in high dimensions.Comm. Math. Phys., 193:69–104, 1998

    1998

  22. [30]

    E. B. Dynkin. Representation for functionals of superprocesses by multiple stochastic inte- grals, with applications to self-intersection local times.Ast´ erisque, 157–158:147–171, 1988

    1988

  23. [31]

    E. B. Dynkin. A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Related Fields, 89:89–115, 1991

    1991

  24. [32]

    A. M. Etheridge.An Introduction to Superprocesses, volume 20 ofUniversity Lecture Series. Amer. Math. Soc., Providence, 2000

    2000

  25. [33]

    Fitzner and R

    R. Fitzner and R. van der Hofstad. Mean-field behavior for nearest-neighbor percolation in d>10.Electron. J. Probab., 22(43), 2017

    2017

  26. [34]

    Grimmett.Percolation, volume 321 ofGrundlehren der mathematischen Wissenschaften

    G. Grimmett.Percolation, volume 321 ofGrundlehren der mathematischen Wissenschaften. Springer, Berlin, 2nd edition, 1999

    1999

  27. [35]

    T. Hara. Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals.Ann. Probab., 36:530–593, 2008

    2008

  28. [36]

    T. Hara, R. van der Hofstad, and G. Slade. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models.Ann. Probab., 31:349–408, 2003

    2003

  29. [37]

    Hara and G

    T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys., 128:333–391, 1990

    1990

  30. [38]

    Hara and G

    T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents.J. Statist. Phys., 99:1075–1168, 2000

    2000

  31. [39]

    Hara and G

    T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion.J. Math. Phys., 41:1244–1293, 2000

    2000

  32. [40]

    Heydenreich and R

    M. Heydenreich and R. van der Hofstad.Progress in High-Dimensional Percolation and Random Graphs. CRM Short Courses. Springer, Cham, 2017

    2017

  33. [41]

    Heydenreich, R

    M. Heydenreich, R. van der Hofstad, and T. Hulshof. Random walk on the high-dimensional IIC.Comm. Math. Phys., 329(1):57–115, 2014

    2014

  34. [42]

    van der Hofstad and G

    R. van der Hofstad and G. Slade. Convergence of critical oriented percolation to super- Brownian motion above 4 + 1 dimensions.Ann. Inst. H. Poincar´ e Probab. Statist., 39:413– 485, 2003

    2003

  35. [43]

    Holmes and E

    M. Holmes and E. Perkins. Weak convergence of measure-valued processes andr-point functions.Ann. Probab., 35:1769–1782, 2007

    2007

  36. [44]

    Holmes and E

    M. Holmes and E. Perkins. On the range of lattice models in high dimensions.Probab. Theory Related Fields, 176(3–4):941–1009, 2020. 60

    2020

  37. [45]

    Hutchcroft

    T. Hutchcroft. Critical long-range percolation I: High effective dimension, 2025. Preprint, arXiv:arXiv:2508.18807

    2025 arXiv

  38. [46]

    Kallenberg.Foundations of modern probability

    O. Kallenberg.Foundations of modern probability. Probability and its Applications. Springer, New York, 2nd edition, 2002

    2002

  39. [47]

    Kallenberg.Random Measures, Theory and Applications, volume 77 ofProbability The- ory and Stochastic Modelling

    O. Kallenberg.Random Measures, Theory and Applications, volume 77 ofProbability The- ory and Stochastic Modelling. Springer, 2017

    2017

  40. [48]

    J. B. Keller. On solutions of ∆u=f(u).Comm. Pure Appl. Math., 10:503–510, 1957

    1957

  41. [49]

    Kozma and A

    G. Kozma and A. Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math., 178(3):635–654, 2009

    2009

  42. [50]

    Kozma and A

    G. Kozma and A. Nachmias. Arm exponents in high dimensional percolation.J. Amer. Math. Soc., 24:375–409, 2011

    2011

  43. [51]

    T. Kumagai. Anomalous random walks and diffusions: from fractals to random media. In Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. IV, pages 75–94. Kyung Moon Sa, Seoul, 2014

    2014

  44. [52]

    Kumagai.Random walks on disordered media and their scaling limits, volume 2101 ofLecture Notes in Mathematics

    T. Kumagai.Random walks on disordered media and their scaling limits, volume 2101 ofLecture Notes in Mathematics. Springer, Cham, 2014. Lecture notes from the 40th Probability Summer School held in Saint-Flour, 2010, ´Ecole d’´Et´ e de Probabilit´ es de Saint- Flour. [Saint-Fl...

    2014

  45. [53]

    Kumagai and J

    T. Kumagai and J. Misumi. Heat kernel estimates for strongly recurrent random walk on random media.J. Theoret. Probab., 21:910–935, 2008

    2008

  46. [54]

    Le Gall.Spatial Branching Processes, Random Snakes and Partial Differential Equa- tions

    J.-F. Le Gall.Spatial Branching Processes, Random Snakes and Partial Differential Equa- tions. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser, Basel, 1999

    1999

  47. [55]

    Liu and G

    Y. Liu and G. Slade. Gaussian deconvolution and the lace expansion.Probab. Theory Related Fields, 195(1–2):3–29, 2026

    2026

  48. [56]

    Liu and G

    Y. Liu and G. Slade. Gaussian deconvolution and the lace expansion for spread-out models. Ann. Inst. Henri Poincar´ e Probab. Stat., 62(1):68–85, 2026

    2026

  49. [57]

    R. Noda. Convergence of local times of stochastic processes associated with resistance forms, 2023. Preprint, arXiv:2305.13224

    2023 arXiv

  50. [58]

    Osserman

    R. Osserman. On the inequality ∆u≥f(u).Pacific J. Math., 7:1641–1647, 1957

    1957

  51. [59]

    E. Perkins. Dawson–Watanabe superprocesses and measure-valued diffusions. InLectures on Probability Theory and Statistics (Saint-Flour, 1999), volume 1781 ofLecture Notes in Math., pages 125–329. Springer, Berlin, 2002

    1999

  52. [60]

    L. C. Petersen. On the relation between the multidimensional moment problem and the one-dimensional moment problem.Math. Scand., 51:361–366, 1982

    1982

  53. [61]

    Schneider and W

    R. Schneider and W. Weil.Stochastic and Integral Geometry. Probability and its Applica- tions. Springer, Berlin, 2008. 61

    2008

  54. [62]

    G. Slade. Lattice trees, percolation and super-Brownian motion. InPerplexing problems in probability: Festschrift in honor of Harry Kesten, volume 44 ofProgr. Probab., pages 35–51. Birkh¨ auser, Boston, 1999. 62

    1999

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.