On the cluster size distribution for percolation on some general graphs

Bandyopadhyay, Antar; Steif, Jeffrey E.; Timár, Ádám

doi:10.4171/rmi/608

Cited by 10 publications

(19 citation statements)

References 31 publications

(67 reference statements)

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“…As in critical percolation, the corresponding cluster size distribution is heavy-tailed, i.e. integral moments do not exist [4]. The result is a polynomial decay in time.…”

Section: Context and Resultsmentioning

confidence: 99%

“…Let µ be an invariant bond percolation on the 2-dimensional Euclidean lattice G = Z 2 , N.N. with a µ-a.s. finite percolation cluster C o having a size distribution obeying (4).…”

Section: Integrated Density Of States For Zmentioning

confidence: 99%

“…It is well-known that for the homogeneous tree of finite degree, a = b = 1 2 (see [1], [20], and [4]). Just as in the previous proof, the constant C δ can be chosen so large, that the estimate is valid for all t ≥ 1.…”

Section: Corollary 23 Ii); Upper and Lower Boundmentioning

confidence: 99%

“…I am grateful for the discussion with Peter Müller concerning the asymptotics of P t and its comparison with the incipient infinite cluster in the case of homogeneous trees. Thanks also to Adam Timar informing me about [4], to Geoffrey Grimmett and an 'anonymous expert' for remarks leading to the discussion following Corollary 2.5, and to Daniel Lenz and Brian Rider for their helpful comments regarding improvements of the style of the presentation. This work has been written with the support of the Project P18703 of the Austrian Science Foundation (FWF), and during the author's stay at the Mathematisches Institut of the University of Jena.…”

Section: Acknowledgmentmentioning

confidence: 99%

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Bounds for the annealed return probability on large finite percolation graphs

Sobieczky

2012

Electron. J. Probab.

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Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times t in this case is of order t −3/4 .

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“…As in critical percolation, the corresponding cluster size distribution is heavy-tailed, i.e. integral moments do not exist [4]. The result is a polynomial decay in time.…”

Section: Context and Resultsmentioning

confidence: 99%

“…Let µ be an invariant bond percolation on the 2-dimensional Euclidean lattice G = Z 2 , N.N. with a µ-a.s. finite percolation cluster C o having a size distribution obeying (4).…”

Section: Integrated Density Of States For Zmentioning

confidence: 99%

Section: Corollary 23 Ii); Upper and Lower Boundmentioning

confidence: 99%

Section: Acknowledgmentmentioning

confidence: 99%

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Bounds for the annealed return probability on large finite percolation graphs

Sobieczky

2012

Electron. J. Probab.

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“…in [BanST08], for such questions it is important that there should exist some absolute constant k such that the vertex boundary of F n is connected in the distance k Rips complex for all n. This is the case for any Følner sequence when is the Cayley graph of a finitely presented group. On the other hand, the usual Følner sequence of the lamplighter group Z 2 o Z is not such, but it is not very difficult to augment each F n with some paths such that the resulting sequence F n already has this property [BanST08].…”

Section: Proof First Assume For Simplicity Thatmentioning

confidence: 99%

Scale-invariant groups

Nekrashevych¹,

Pete²

2011

Groups Geom. Dyn.

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Abstract. Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F o Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS.1; m/; the affine groups A Ë Z d , for any A Ä GL.Z; d /. However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant.Mathematics Subject Classification (2010). 20E08, 20F18, 05B45, 60B99, 60K35.

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Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

Hermon

Hutchcroft

2020

Invent. math.

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Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and $$p > p_c(G)$$ p > p c ( G ) then there exists a positive constant $$c_p$$ c p such that $$\begin{aligned} \mathbf {P}_p(n \le |K| < \infty ) \le e^{-c_p n} \end{aligned}$$ P p ( n ≤ | K | < ∞ ) ≤ e - c p n for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We deduce the following two corollaries: Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56–84, 1999). For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.

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On the cluster size distribution for percolation on some general graphs

Cited by 10 publications

References 31 publications

Bounds for the annealed return probability on large finite percolation graphs

Bounds for the annealed return probability on large finite percolation graphs

Scale-invariant groups

Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

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