REVIEW 2 major objections 4 minor 1 cited by
Critical long-range percolation I: High effective dimension
T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read Critical long-range percolation in high effective dimension has exactly computable mean-field cluster volumes and superprocess scaling limits, with a sharp transition at α = 2.
desk verdict The d>3α core is a genuine non-perturbative breakthrough for long-range percolation scaling limits; the high-dimensional claims beyond that are openly conditional on two-point bounds the paper does not prove for its own kernels. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The carrying object is the real-space renormalization group built from the cut-off kernel J_r(x,y) = ∫_{||x−y||}^{r} |J'(s)| ds: one studies critical observables under the subcritical measures P_{β_c,r} as r grows. Russo's formula turns r-derivatives of cluster moments into exact expressions involving two clusters joined by a newly added edge; in high effective dimension these clusters are asymptotically independent, so the derivatives collapse to the triangular mean-field ODE system d/dr E|K|^p ∼ β_c r^{−α−1} Σ_{ℓ=0}^{p−1} C(p,ℓ) E|K|^{ℓ+1} E|K|^{p−ℓ}. The key bootstrap validates these ODEs for d > 3α using only integrability of r^{3α−d} and the tree-graph inequalities. ODE comparison lemma
What would settle it
Measure the size-biased radius of gyration of the critical cluster in the cut-off model for large r in a case with d > 3α, say d = 2, α = 1/2: the paper predicts ξ_2(r) ∼ const r for α < 2; observing any different power, or a cluster-volume tail exponent different from 1/2, would falsify the central claim.
Extended reading notes
Core claim
Under (HD), the paper proves three facts. First, cut-off cluster moments satisfy E_{β_c,r}|K|^p ∼ (2p−3)!! A^{p−1}(α/β_c) r^{(2p−1)α}, and the true cluster tail is P_{β_c}(|K| ≥ n) ∼ (α/β_c)√(2/(πA)) n^{−1/2}. Second, large critical clusters, rescaled by ζ(R), converge in the measure sense to the canonical measure N of the integrated superprocess excursion whose spatial motion is symmetric α-stable Lévy for α < 2 and Brownian for α ≥ 2; ζ(R) is R^{2α} in the long-range regime, R^4(log R)^{-2} at α = 2, and R^4 for α > 2. Third, at α = 2 the model is marginally short-range: the radius of gyration picks up a √log r factor while the volume tail does not. The case d > 3α is non-perturbative; the
Load-bearing premise
The argument leans on asymptotic independence of the clusters at the two ends of a newly added long edge, which the paper proves directly only when d > 3α and otherwise imports as an assumed power-law two-point upper bound (with a log correction at d = 6, α = 2) that is not proved here.
Editorial extensions
If this is right
- The exact moment asymptotics imply that the cluster volume tail decays as a constant times n^{-1/2} throughout the high-effective-dimensional regime, so critical clusters have the same size distribution as near-critical branching processes.
- The scaling limit changes at α = 2: clusters are embedded by symmetric α-stable Lévy motion for α < 2 and by Brownian motion for α ≥ 2, so the long-range/short-range crossover appears in geometry rather than in the volume distribution.
- For d > 3α the theorems are non-perturbative, covering the full power-law kernel class without spread-out or small-parameter assumptions.
- At d = 6, α = 2 there are logarithmic corrections to the radius of gyration and to the cluster normalization ζ(R), but the n^{-1/2} volume tail persists.
- Under the hydrodynamic condition, the same scaling-limit conclusions extend to the critical dimension d = 3α < 6 with slowly varying corrections, a condition verified in the third paper of the series.
Reading between the lines
- Beyond the paper, I would expect the bootstrap argument to transfer to other long-range models such as the Ising model, self-avoiding walk, and lattice trees, giving non-perturbative mean-field exponents and scaling limits there; the paper states the method is intended to apply, but those results are not in this manuscript.
- The α = 2 threshold is likely a general finite-range/long-range crossover signature, so heavy-tailed random graph models might show the same R^4(log R)^{-2} cluster-scale correction at marginality.
- The hydrodynamic condition M_r = o(r^{(d+α)/2}) offers a portable criterion: any model whose largest clusters in adjacent blocks merge only with vanishing probability should fall into the mean-field superprocess class, regardless of the ambient dimension.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper develops a non-perturbative real-space renormalization-group method for long-range percolation on Z^d with kernel J(x,y) ~ ||x-y||^{-d-\alpha}. Under the hypotheses (HD), it proves first-order asymptotics for cluster volume moments E_{\beta_c,r}|K|^p ~ (2p-3)!! A^{p-1} (\alpha/\beta_c) r^{(2p-1)\alpha}, the volume tail P_{\beta_c}(|K| \ge n) ~ (\alpha/\beta_c) \sqrt{2/(\pi A)} n^{-1/2}, and superprocess scaling limits with normalization factors that transition between super-L\'evy (\alpha<2) and super-Brownian (\alpha \ge 2) behaviour. The d>3\alpha part is unconditional and self-contained, based on a bootstrap (Lemma I.4.9) that does not use external two-point estimates. The SR-HD and mSR-HD parts rely on the lace-expansion two-point bounds in hypotheses (HD)2 and 3, and the d=3\alpha<6 scaling limits rely on the hydrodynamic condition, both clearly flagged as external or deferred.
Significance. If taken at face value, the d>3\alpha results are a substantial advance: they give the first non-perturbative computation of mean-field exponents and full scaling limits in a long-range percolation model, without small-parameter assumptions. The proof architecture is coherent: exact derivative identities via Russo's formula and the mass-transport principle, error control via BK, tree-graph and Gladkov inequalities, and a self-contained bootstrap. The use of explicit ODEs rather than matching to conjectured exponents is a particular strength. The conditional parts are honestly labelled, but they are conditional on inputs that are not proved for the paper's own kernel class, which limits the scope of the otherwise sweeping claims in the introduction.
major comments (2)
- [I.1.3] Theorems I.1.6 and I.1.9 cover the SR-HD and mSR-HD regimes only under pointwise two-point bounds P_{\beta_c}(x \leftrightarrow y) \preceq ||x-y||^{-d+2} (with a log correction at d=6, \alpha=2). These bounds are not proved for the kernel class in Definition I.1.3; they are imported from lace-expansion results for spread-out models under numerical assumptions. The paper is explicit about this in the abstract and Figure 1, but the theorem statements in Section I.1.3 could easily be read as applying to the full power-law kernel class. The two-point bounds are load-bearing in Lemma I.4.3 and Lemmas I.5.11--I.5.13 and I.5.29. I recommend adding an explicit 'Conditional on the lace-expansion two-point estimates' label to the relevant theorems and a remark that no proof is supplied for (HD)2 and 3 in this paper.
- [I.5.1] The four-point Gladkov inequality is used in the proof of Proposition I.5.2 (Case 2) and hence in Theorem I.1.9, but its proof is explicitly deferred to the second paper of the series. This is an omitted proof of a load-bearing lemma. Since the current paper is being considered as a standalone contribution, the authors should either include the proof, give a citation to a publicly available version of the proof, or explicitly state that the affected scaling-limit results depend on the forthcoming paper II.
minor comments (4)
- [I.1.4] Theorems I.1.15 and I.1.17 are conditional on the hydrodynamic condition, whose verification is deferred to paper III. This is clearly stated, but it should also be noted in the theorem statements themselves that these results are not self-contained in this paper.
- [I.5.1] In the statement of the lemma, 'x, y, z, wall connected' should presumably be 'x, y, z, w all connected'.
- [I.5.1] The text refers to 'estimate (I.5.13)' before that equation number is defined; the cross-reference should be fixed.
- [Figure 1] Minor typo: 'occuring' should be 'occurring'.
Circularity Check
No significant circularity: central d>3α derivation is self-contained and solved from explicit ODEs; remaining regimes are explicitly conditional on external inputs, not derived from their own conclusions.
full rationale
The paper's main high-dimensional results (Theorems I.1.6 and I.1.9) are obtained by proving asymptotic ODEs such as (I.1.3)–(I.1.5) and solving them analytically; constants such as A and Sigma are limit objects fixed by the kernel and error integrals, not fitted to reproduce the claimed exponents or tail. The d>3alpha core is proven by the self-contained bootstrap of Lemma I.4.9, without importing the two-point bounds used in hypotheses (HD)2–3. Those two-point bounds are stated explicitly as hypotheses and attributed to lace-expansion results for spread-out models [35,36,67]; the corresponding theorems are therefore conditional, not circular. Similarly, the d=3alpha results are stated under the explicitly named hydrodynamic condition, whose verification is deferred to paper III; the theorems do not purport to derive the condition from their own conclusions. Self-citations to [74,76,78] invoke previously proved theorems (e.g., universal tightness, two-point upper bounds) that are independent of the target results and are not used to forbid alternatives or smuggle in an ansatz. Remark I.1.11 notes that Theorem I.1.9 implies the volume-tail estimate (I.1.7) used in its proof; this is not a circular dependency because (I.1.7) is proved independently in Section I.3 from the moment asymptotics of Theorem I.1.6, before the scaling-limit proof in Section I.5. No equation is shown to reduce by construction to an input, and no fitted quantity is renamed as a prediction. The main caveats are external conditions and unverified-for-this-kernel lace-expansion bounds, which are limitations on scope rather than circular reasoning. Honest non-finding with score 0.
Assumptions & free parameters
free parameters (2)
- A (Theorem I.1.6)
- Sigma (covariance matrix, Theorem I.1.9, alpha > 2)
assumptions (6)
- standard math Russo's formula applies to infinite-volume moments of the cutoff model at beta <= beta_c, and the mass-transport principle holds on Z^d (Lemma I.4.1)
- standard math Tree-graph (Aizenman-Newman) inequalities bound k-point functions by products of two-point functions (I.2.1), (I.2.2)
- domain assumption Universal tightness theorem of Hutchcroft [74] controls the maximum cluster size and moment ratios (Corollary I.2.5)
- domain assumption Two-point function bounds P_{beta_c}(x <-> y) <= ||x-y||^{-d+2} (with log correction at d = 6, alpha = 2), from the lace expansion for spread-out models
- ad hoc to paper Hydrodynamic condition M_r = o(r^{(d+alpha)/2}) (Definition I.1.14)
- domain assumption Existing two-point upper bound sum over B_r P_{beta_c}(0 <-> x) <= r^{alpha} [76], and M_r <= r^{(d+alpha)/2} (Corollary I.2.15)
Cite this review
Pith. "Pith review of Critical long-range percolation I: High effective dimension." pith.science (2026). https://pith.science/paper/FVLW644S
@misc{pith2026250818807,
author = {Pith},
title = {Pith review of: Critical long-range percolation I: High effective dimension},
year = {2026},
howpublished = {\url{https://pith.science/paper/FVLW644S}},
note = {Machine review of arXiv:2508.18807}
}
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 first of a series 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. In this paper, we introduce our non-perturbative real-space renormalization group method and apply this method to analyze the HD regime $d>\min\{6,3\alpha\}$. In particular, we compute the tail of the cluster volume and establish the superprocess scaling limits of the model, which transition between super-Levy and super-Brownian behavior when $\alpha=2$. All our results hold unconditionally for $d> 3\alpha$, without any perturbative assumptions on the model; beyond this regime, when $d> 6$ and $\alpha \geq d/3$, they hold under the assumption that appropriate two-point function estimates hold as provided for spread-out models by the lace expansion. Our results on scaling limits also hold (with possible slowly-varying corrections to scaling) in the critical-dimensional regime with $d=3\alpha<6$ subject to a marginal-triviality condition we call the hydrodynamic condition; this condition is verified in the third paper in this series, in which we also compute the precise logarithmic corrections to mean-field scaling when $d=3\alpha<6$.
Figures
Forward citations
Cited by 1 Pith paper
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Super-Brownian limits and the $k$-point function for high-dimensional percolation
High-dimensional critical percolation clusters rescale to super-Brownian excursion, verifying the 1984 Aizenman–Newman k-point conjecture under lace-expansion hypotheses.
Reference graph
Works this paper leans on
-
[1]
A. Abdesselam, A. Chandra, and G. Guadagni. Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions.arXiv preprint arXiv:1302.5971, 2013
arXiv 2013
-
[2]
Addario-Berry, O
L. Addario-Berry, O. Angel, G. Chapuy, É. Fusy, and C. Goldschmidt. Voronoi tessellations in the CRT and continuum random maps of finite excess. InProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 933–946. SIAM, 2018
2018
-
[3]
Aizenman
M. Aizenman. On the number of incipient spanning clusters.Nuclear Phys. B, 485(3):551–582, 1997
1997
-
[4]
Aizenman and C
M. Aizenman and C. M. Newman. Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys., 36(1-2):107–143, 1984
1984
-
[5]
Aizenman and C
M. Aizenman and C. M. Newman. Discontinuity of the percolation density in one-dimensional 1/|x − y|2 percolation models. Comm. Math. Phys., 107(4):611–647, 1986
1986
-
[6]
D. Aldous. The continuum random tree. I.Ann. Probab., 19(1):1–28, 1991
1991
-
[7]
D. Aldous. The continuum random tree. II. an overview.Stochastic analysis (Durham, 1990), 167:23–70, 1991
1990
-
[8]
D. Aldous. The continuum random tree III.The Annals of Probability, pages 248–289, 1993
1993
Show all 113 references
-
[9]
Angel, T
O. Angel, T. Hutchcroft, and A. Járai. On the tail of the branching random walk local time.Probability Theory and Related Fields, 180(1):467–494, 2021
2021
-
[10]
Arratia, S
R. Arratia, S. Garibaldi, and A. W. Hales. The van den berg–kesten–reimer operator and inequality for infinite spaces. arXiv preprint arXiv:1508.05337, 2015
2015 arXiv
-
[11]
Asselah and B
A. Asselah and B. Schapira. On the intersection of critical percolation clusters and other tree-like random graphs. arXiv preprint arXiv:2411.19145, 2024
2024 arXiv
-
[12]
D. J. Barsky and M. Aizenman. Percolation critical exponents under the triangle condition.Ann. Probab., 19(4):1520–1536, 1991
1991
-
[13]
Bateman and A
H. Bateman and A. Erdélyi. Higher transcendental functions, volume ii. Bateman Manuscript Project) Mc Graw-Hill Book Company, 410, 1953
1953
-
[14]
Bauerschmidt, D
R. Bauerschmidt, D. C. Brydges, and G. Slade. Critical two-point function of the 4-dimensional weakly self- avoiding walk.Communications in Mathematical Physics, 338(1):169–193, 2015
2015
-
[15]
Logarithmiccorrectionforthesusceptibilityofthe4-dimensional weakly self-avoiding walk: a renormalisation group analysis.Comm
R.Bauerschmidt, D.C.Brydges, andG.Slade. Logarithmiccorrectionforthesusceptibilityofthe4-dimensional weakly self-avoiding walk: a renormalisation group analysis.Comm. Math. Phys., 337(2):817–877, 2015
2015
-
[16]
Bauerschmidt, D
R. Bauerschmidt, D. C. Brydges, and G. Slade.Introduction to a renormalisation group method, volume 2242 of Lecture Notes in Mathematics. Springer, Singapore, 2019
2019
-
[17]
Bauerschmidt, G
R. Bauerschmidt, G. Slade, A. Tomberg, and B. C. Wallace. Finite-order correlation length for four-dimensional weakly self-avoiding walk andφ4 spins. Annales Henri Poincaré, 18(2):375–402, 2017
2017
-
[18]
Bauerschmidt, G
R. Bauerschmidt, G. Slade, and B. C. Wallace. Four-dimensional weakly self-avoiding walk with contact self- attraction. Journal of Statistical Physics, 167:317–350, 2017
2017
-
[19]
J. Bäumler. Distances in1/∥x − y∥2d percolation models for all dimensions.Communications in Mathematical Physics, 404(3):1495–1570, 2023
2023
-
[20]
Bäumler and N
J. Bäumler and N. Berger. Isoperimetric lower bounds for critical exponents for long-range percolation.Annales de l’Institut Henri Poincare (B) Probabilites et statistiques, 60(1):721–730, 2024
2024
-
[21]
Behan, L
C. Behan, L. Rastelli, S. Rychkov, and B. Zan. A scaling theory for the long-range to short-range crossover and an infrared duality.Journal of Physics A: Mathematical and Theoretical, 50(35):354002, 2017
2017
-
[22]
N. Berger. Transience, recurrence and critical behavior for long-range percolation. Comm. Math. Phys., 226(3):531–558, 2002
2002
-
[23]
Billingsley.Convergence of probability measures
P. Billingsley.Convergence of probability measures. John Wiley & Sons, 2013
2013
-
[24]
N. H. Bingham, C. M. Goldie, and J. L. Teugels.Regular variation, volume 27. Cambridge university press, 1989
1989
-
[25]
Biskup and A
M. Biskup and A. Krieger. Arithmetic oscillations of the chemical distance in long-range percolation onZd. The Annals of Applied Probability, 34(3):2986–3017, 2024
2024
-
[26]
Blanc-Renaudie and T
A. Blanc-Renaudie and T. Hutchcroft. super-Brownian limits and thek-point function for high-dimensional percolation. In final preparation. 100
-
[27]
P. Bleher. Critical phenomena in the Dyson hierarchical model and renormalization group. arXiv preprint arXiv:1010.5855, 2010
2010 arXiv
-
[28]
Borgs, J
C. Borgs, J. Chayes, R. van der Hofstad, and G. Slade. Mean-field lattice trees.Annals of Combinatorics, 3:205–221, 1999
1999
-
[29]
Bramson, J
M. Bramson, J. T. Cox, and J.-F. Le Gall. Super-Brownian limits of voter model clusters.The Annals of Probability, 29(3):1001–1032, 2001
2001
-
[30]
Brezin, G
E. Brezin, G. Parisi, and F. Ricci-Tersenghi. The crossover region between long-range and short-range interac- tions for the critical exponents.Journal of Statistical Physics, 157:855–868, 2014
2014
-
[31]
Brydges and T
D. Brydges and T. Spencer. Self-avoiding walk in5 or more dimensions.Comm. Math. Phys., 97(1-2):125–148, 1985
1985
-
[32]
Cabezas, A
M. Cabezas, A. Fribergh, M. Holmes, and E. Perkins. Random skeletons in high-dimensional lattice trees. arXiv preprint arXiv:2503.19230, 2025
2025 arXiv
-
[33]
D. Callan. A combinatorial survey of identities for the double factorial. 2009. Unpublished. Available at http://https://arxiv.org/abs/0906.1317
2009 arXiv
-
[34]
F. Camia. Conformal covariance of connection probabilities and fields in 2D critical percolation.Communica- tions on Pure and Applied Mathematics, 77(3):2138–2176, 2024
2024
-
[35]
Chen and A
L.-C. Chen and A. Sakai. Critical two-point functions for long-range statistical-mechanical models in high dimensions. Ann. Probab., 43(2):639–681, 2015
2015
-
[36]
Chen and A
L.-C. Chen and A. Sakai. Critical two-point function for long-range models with power-law couplings: the marginal case ford ⩾ dc. Comm. Math. Phys., 372(2):543–572, 2019
2019
-
[37]
T. Cox, R. Durrett, and E. A. Perkins. Rescaled particle systems converging to super-brownian motion. Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten, pages 269–284, 1999
1999
-
[38]
de Alcantara Bonfirm, J
O. de Alcantara Bonfirm, J. Kirkham, and A. McKane. Critical exponents for the percolation problem and the Yang-Lee edge singularity.Journal of Physics A: Mathematical and General, 14(9):2391, 1981
1981
-
[39]
DeMasi and E
A. DeMasi and E. Presutti.Mathematical methods for hydrodynamic limits. Springer, 2006
2006
-
[40]
J. Ding, Z. Fan, and L.-J. Huang. Uniqueness of the critical long-range percolation metrics.arXiv preprint arXiv:2308.00621, 2023
2023 arXiv
-
[41]
Duminil-Copin, C
H. Duminil-Copin, C. Garban, and V. Tassion. Long-range models in 1d revisited. 60(1):232–241, 2024
2024
-
[42]
Duminil-Copin, K
H. Duminil-Copin, K. K. Kozlowski, D. Krachun, I. Manolescu, and M. Oulamara. Rotational invariance in critical planar lattice models.arXiv preprint arXiv:2012.11672, 2020
2012 arXiv
-
[43]
Duminil-Copin and R
H. Duminil-Copin and R. Panis. An alternative approach for the mean-field behaviour of spread-out Bernoulli percolation in dimensionsd >6. arXiv preprint arXiv:2410.03647, 2024
2024 arXiv
-
[44]
E. B. Dynkin.An introduction to branching measure-valued processes. Number 6. American Mathematical Soc., 1994
1994
-
[45]
F. J. Dyson. Existence of a phase-transition in a one-dimensional Ising ferromagnet. Comm. Math. Phys., 12(2):91–107, 1969
1969
-
[46]
Essam, D
I. Essam, D. Gaunt, and A. Guttmann. Percolation theory at the critical dimension.Journal of Physics A: Mathematical and General, 11(10):1983, 1978
1983
-
[47]
W. Feller. An introduction to probability theory and its applications. Vol. II.Second edition. John Wiley & Sons Inc., New York, 1971
1971
-
[48]
M. E. Fisher, S.-k. Ma, and B. Nickel. Critical exponents for long-range interactions.Physical Review Letters, 29(14):917, 1972
1972
-
[49]
Fitzner and R
R. Fitzner and R. van der Hofstad. Mean-field behavior for nearest-neighbor percolation ind >10. Electron. J. Probab., 22:Paper No. 43, 65, 2017
2017
-
[50]
Gawedzki and A
K. Gawedzki and A. Kupiainen. Non-Gaussian fixed points of the block spin transformation. Hierarchical model approximation. Comm. Math. Phys., 89(2):191–220, 1983
1983
-
[51]
Gawedzki and A
K. Gawedzki and A. Kupiainen. Massless latticeφ4 4 theory: rigorous control of a renormalizable asymptotically free model. Communications in mathematical physics, 99:197–252, 1985
1985
-
[52]
N. Gladkov. Percolation inequalities and decision trees.arXiv preprint arXiv:2408.08457, 2024. 101
2024 arXiv
-
[53]
G. Gori, M. Michelangeli, N. Defenu, and A. Trombettoni. One-dimensional long-range percolation: a numerical study. Physical Review E, 96(1):012108, 2017
2017
-
[54]
Gould and J
H. Gould and J. Quaintance. Double fun with double factorials.Math. Mag., 85(3):177–192, 2012
2012
-
[55]
Grimmett
G. Grimmett. Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999
1999
-
[56]
T. Hara. A rigorous control of logarithmic corrections in four-dimensionalϕ4 spin systems: I. Trajectory of effective hamiltonians. Journal of statistical physics, 47:57–98, 1987
1987
-
[57]
T. Hara, T. Hattori, and H. Watanabe. Triviality of hierarchical Ising model in four dimensions.Comm. Math. Phys., 220(1):13–40, 2001
2001
-
[58]
Hara and G
T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions.Comm. Math. Phys., 128(2):333–391, 1990
1990
-
[59]
Hara and G
T. Hara and G. Slade. The incipient infinite cluster in high-dimensional percolation. Electronic Research Announcements of the American Mathematical Society, 4(8):48–55, 1998
1998
-
[60]
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(5-6):1075–1168, 2000
2000
-
[61]
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.Journal of Mathematical Physics, 41(3):1244–1293, 2000
2000
-
[62]
Hara and H
T. Hara and H. Tasaki. A rigorous control of logarithmic corrections in four-dimensionalϕ4 spin systems: II. Critical behavior of susceptibility and correlation length.Journal of statistical physics, 47:99–121, 1987
1987
-
[63]
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(1):349–408, 2003
2003
-
[64]
Heydenreich and R
M. Heydenreich and R. van der Hofstad.Progress in high-dimensional percolation and random graphs. CRM Short Courses. Springer, Cham; Centre de Recherches Mathématiques, Montreal, QC, 2017
2017
-
[65]
Heydenreich, R
M. Heydenreich, R. van der Hofstad, and T. Hulshof. High-dimensional incipient infinite clusters revisited. Journal of Statistical Physics, 155:966–1025, 2014
2014
-
[66]
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
-
[67]
Heydenreich, R
M. Heydenreich, R. van der Hofstad, and A. Sakai. Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk.J. Stat. Phys., 132(6):1001–1049, 2008
2008
-
[68]
M. Holmes. Convergence of lattice trees to super-brownian motion above the critical dimension. 2008
2008
-
[69]
Holmes and E
M. Holmes and E. Perkins. On the range of lattice models in high dimensions.Probability Theory and Related Fields, 176(3):941–1009, 2020
2020
-
[70]
T. Hulshof. The one-arm exponent for mean-field long-range percolation.Electron. J. Probab., 20:no. 115, 26, 2015
2015
-
[71]
Hutchcroft
T. Hutchcroft. Critical long-range percolation II: Low effective dimension
-
[72]
Hutchcroft
T. Hutchcroft. Critical long-range percolation III: The upper critical dimension
-
[73]
Hutchcroft
T. Hutchcroft. Pointwise two-point function estimates and a non-pertubative proof of mean-field critical be- haviour for long-range percolation.Probability Theory and Related Fields. To appear
-
[74]
Hutchcroft
T. Hutchcroft. Power-law bounds for critical long-range percolation below the upper-critical dimension.Probab. Theory Related Fields, 181(1-3):533–570, 2021
2021
-
[75]
Hutchcroft
T. Hutchcroft. On the derivation of mean-field percolation critical exponents from the triangle condition. Journal of Statistical Physics, 189(1):6, 2022
2022
-
[76]
Hutchcroft
T. Hutchcroft. Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on zd. Journal of Mathematical Physics, 63(11), 2022
2022
-
[77]
Hutchcroft
T. Hutchcroft. The critical two-point function for long-range percolation on the hierarchical lattice.The Annals of Applied Probability, 34(1B):986–1002, 2024
2024
-
[78]
Hutchcroft
T. Hutchcroft. Critical cluster volumes in hierarchical percolation.Proceedings of the London Mathematical Society, 130(1):e70023, 2025
2025
-
[79]
I. Iscoe. On the supports of measure-valued critical branching brownian motion.The Annals of Probability, pages 200–221, 1988. 102
1988
-
[80]
S. Ken-Iti. Lévy processes and infinitely divisible distributions, volume 68. Cambridge university press, 1999
1999
-
[81]
H. Kesten. Scaling relations for2D-percolation. Comm. Math. Phys., 109(1):109–156, 1987
1987
-
[82]
S. Kotz, T. Kozubowski, and K. Podgórski.The Laplace distribution and generalizations: a revisit with ap- plications to communications, economics, engineering, and finance. Number 183. Springer Science & Business Media, 2001
2001
-
[83]
G. Kozma. The triangle and the open triangle.Ann. Inst. Henri Poincaré Probab. Stat., 47(1):75–79, 2011
2011
-
[84]
Kozma and A
G. Kozma and A. Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math., 178(3):635–654, 2009
2009
-
[85]
Kozma and A
G. Kozma and A. Nachmias. Arm exponents in high dimensional percolation.J. Amer. Math. Soc., 24(2):375– 409, 2011
2011
-
[86]
Lawler, O
G. Lawler, O. Schramm, W. Werner, et al. One-arm exponent for critical 2d percolation.Electronic Journal of Probability, 7, 2002
2002
-
[87]
J.-F. Le Gall. Spatial branching processes, random snakes and partial differential equations. Springer Science & Business Media, 1999
1999
-
[88]
Y. Liu. High-dimensional long-range statistical mechanical models have random walk correlation functions. arXiv preprint arXiv:2502.12104, 2025
2025
-
[89]
Lohmann, G
M. Lohmann, G. Slade, and B. C. Wallace. Critical two-point function for long-rangeO(n) models below the upper critical dimension.J. Stat. Phys., 169(6):1132–1161, 2017
2017
-
[90]
C. M. Newman and L. S. Schulman. One dimensional1/|j −i|s percolation models: The existence of a transition for s ≤ 2. Communications in Mathematical Physics, 104(4):547–571, 1986
1986
-
[91]
Pauli and F
W. Pauli and F. Villars. On the invariant regularization in relativistic quantum theory.Reviews of Modern Physics, 21(3):434, 1949
1949
-
[92]
E. Perkins. Polar sets and multiple points for super-Brownian motion.The Annals of Probability, pages 453–491, 1990
1990
-
[93]
E. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics, pages 125–329, 2002
2002
-
[94]
D. Reimer. Proof of the van den Berg-Kesten conjecture.Combin. Probab. Comput., 9(1):27–32, 2000
2000
-
[95]
J. J. Ruiz-Lorenzo. Logarithmic corrections for spin glasses, percolation and Lee-Yang singularities in six dimensions. Journal of Physics A: Mathematical and General, 31(44):8773, 1998
1998
-
[96]
J. Sak. Recursion relations and fixed points for ferromagnets with long-range interactions.Physical Review B, 8(1):281, 1973
1973
-
[97]
L. S. Schulman. Long range percolation in one dimension.Journal of Physics A: Mathematical and General, 16(17):L639, 1983
1983
-
[98]
G. Slade. Scaling limits and super-Brownian motion.Notices AMS, 49(9):1056–1067, 2002
2002
-
[99]
G. Slade. The lace expansion and its applications, volume 1879 ofLecture Notes in Mathematics. Springer- Verlag, Berlin, 2006. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, Edited and with a foreword by Jean Picard
2006
-
[100]
G. Slade. Critical exponents for long-range O(n) models below the upper critical dimension.Comm. Math. Phys., 358(1):343–436, 2018
2018
-
[101]
S. Smirnov. Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits.Comptes Rendus de l’Académie des Sciences-Series I-Mathematics, 333(3):239–244, 2001
2001
-
[102]
Smirnov and W
S. Smirnov and W. Werner. Critical exponents for two-dimensional percolation.Math. Res. Lett., 8(5-6):729– 744, 2001
2001
-
[103]
Suzuki, Y
M. Suzuki, Y. Yamazaki, and G. Igarashi. Wilson-type expansions of critical exponents for long-range interac- tions. Physics Letters A, 42(4):313–314, 1972
1972
-
[104]
A. Swan. Superprobability on graphs. PhD thesis, 2021
2021
-
[105]
van den Berg and U
J. van den Berg and U. Fiebig. On a combinatorial conjecture concerning disjoint occurrences of events.Ann. Probab., 15(1):354–374, 1987
1987
-
[106]
van der Hofstad and M
R. van der Hofstad and M. Holmes. The survival probability and r-point functions in high dimensions.Annals of mathematics, pages 665–685, 2013. 103
2013
-
[107]
Van der Hofstad and G
R. Van der Hofstad and G. Slade. Convergence of critical oriented percolation to super-Brownian motion above 4 + 1dimensions. 39(3):413–485, 2003
2003
-
[108]
W. Walter. Differential and integral inequalities, volume 55. Springer Science & Business Media, 2012
2012
-
[109]
Watanabe
S. Watanabe. A limit theorem of branching processes and continuous state branching processes.Journal of Mathematics of Kyoto University, 8(1):141–167, 1968
1968
-
[110]
H. S. Wilf.generatingfunctionology. A K Peters, Ltd., Wellesley, MA, third edition, 2006
2006
-
[111]
K. G. Wilson. The renormalization group: Critical phenomena and the kondo problem.Reviews of modern physics, 47(4):773, 1975
1975
-
[112]
K. G. Wilson and M. E. Fisher. Critical exponents in 3.99 dimensions.Physical Review Letters, 28(4):240, 1972
1972
-
[113]
H. Zessin. The method of moments for random measures. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 62(3):395–409, 1983. 104
1983
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