Upper Bounds on the Percolation Correlation Length

Duminil-Copin, Hugo; Kozma, Gady; Tassion, Vincent

doi:10.1007/978-3-030-60754-8_16

Cited by 6 publications

(4 citation statements)

References 32 publications

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“…Arguments following the supercritical strategy tend to be ineffective in the sense that they give little or no quantitative information about percolation at p c ; see however the recent work of Duminil-Copin, Kozma, and Tassion [27] for some progress towards reversing this trend. The subcritical strategy.…”

Section: About the Proofmentioning

confidence: 99%

Power-law bounds for critical long-range percolation below the upper-critical dimension

Hutchcroft

2021

Probab. Theory Relat. Fields

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We study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

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Section: About the Proofmentioning

confidence: 99%

Power-law bounds for critical long-range percolation below the upper-critical dimension

Hutchcroft

2021

Probab. Theory Relat. Fields

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“…Highlights of the literature regarding the other regimes include [3,21,51] for the subcritical regime, [5,28,44,45] for the critical and slightly subcritical regimes, and [4,16,27,42] for the supercritical regime. See for example, [13,26,31] for overviews of this literature and of open problems in high-dimensional percolation, and [18] for some interesting recent partial progress on slightly supercritical percolation. Let us also mention that a good understanding of slightly supercritical percolation appears to be a prerequisite to the solution of several important open problems regarding invasion percolation and minimal spanning forests, see [31,Section 16.1] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Slightly supercritical percolation on non‐amenable graphs I: The distribution of finite clusters

Hutchcroft

2022

Proceedings of London Math Soc

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We study the distribution of finite clusters in slightly supercritical (𝑝 ↓ 𝑝 𝑐 ) Bernoulli bond percolation on transitive non-amenable graphs, proving in particular that if 𝐺 is a transitive non-amenable graph satisfying the 𝐿 2 boundedness condition (𝑝 𝑐 < 𝑝 2→2 ) and 𝐾 denotes the cluster of the origin and then there exists 𝛿 > 0 such that if 𝑝 ∈ (𝑝 𝑐 − 𝛿, 𝑝 𝑐 + 𝛿), thenandfor every 𝑛, 𝑟 ⩾ 1, where all implicit constants depend only on 𝐺. We deduce in particular that the critical exponents 𝛾 ′ and Δ ′ describing the rate of growth of the moments of a finite cluster as 𝑝 ↓ 𝑝 𝑐 take their meanfield values of 1 and 2, respectively. These results apply in particular to Cayley graphs of non-elementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius 𝜌 < 1∕2. In particular, every finitely generated non-amenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet

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“…This is in contrast to the much studied, but locally more amorphous Bernoulli percolation model, for which celebrated results have been derived on two-dimensional lattices, see [31] and references therein, or on Z d for sufficiently large d (in the mean-field regime), following [18], cf. [19] for an extensive account; see also [8,14] for recent progress on Z 3 , and [9] regarding excursion sets of continuous Gaussian fields with rapid correlation decay.…”

Section: Introductionmentioning

confidence: 99%

Critical exponents for a percolation model on transient graphs

2022

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We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.

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Upper Bounds on the Percolation Correlation Length

Cited by 6 publications

References 32 publications

Power-law bounds for critical long-range percolation below the upper-critical dimension

Power-law bounds for critical long-range percolation below the upper-critical dimension

Slightly supercritical percolation on non‐amenable graphs I: The distribution of finite clusters

Critical exponents for a percolation model on transient graphs

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