Percolation on Transitive Graphs as a Coalescent Process: Relentless Merging Followed by Simultaneous Uniqueness

Häggström, Olle; Peres, Yuval; Schonmann, Roberto H.

doi:10.1007/978-1-4612-2168-5_4

Cited by 55 publications

(71 citation statements)

References 27 publications

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“…The result for all quasi-transitive graphs was conjectured by Benjamini and Schramm (1996) and first established by in the unimodular case, using a more complicated argument. Very recently, this result on ends was extended to the nonunimodular case by Haggstrom, Peres and Schonmann (1998). Their argument is based on different principles; in particular, it is not known in the nonunimodular case whether for p E (Pc, p), every infinite cluster must contain infinitely many encounter points.…”

Section: --->0mentioning

confidence: 99%

Critical Percolation on any Nonamenable Group has no Infinite Clusters

Benjamini

Lyons

Peres

et al. 2011

Selected Works of Oded Schramm

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We show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by the present authors earlier as a eorollary ofa general study of group-invariant percolation. The goal here is to present a simpler self-contained proof that easily extends to quasi-transitive graphs with a unimodular automorphism group. The key tool is II "mass-transport" method, which is a technique of averaging in nonamenable settings.

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Section: --->0mentioning

confidence: 99%

Critical Percolation on any Nonamenable Group has no Infinite Clusters

Benjamini

Lyons

Peres

et al. 2011

Selected Works of Oded Schramm

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“…28. Note that the condition that G is a Cayley graph can actually be weakened to the condition that G is transitive [HPS99]. However, the proof we present here uses the mass transport method, which we only proved for groups.…”

Section: Uniqueness Monotonicitymentioning

confidence: 99%

Coarse Geometry and Randomness

Benjamini¹,

Saint-Flour²

2013

Lecture Notes in Mathematics

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“…For.9i' c 2 E (Gl, we write I1 e .9i':= {ileA: A E.9i'}. A bond percolation process (P, w) , Haggstrom, Peres and Schonmann (1999), Schonmann (1999a) and Alexander (1995) for examples where this process is studied.…”

Section: R Lyons and O Schrammmentioning

confidence: 99%

“…However, Haggstrom, Peres and Schonmann (1999) have recently shown that even without the unimodularity assumption, when p > Pc so-called "robust" properties do not distinguish between the infinite clusters of Bernoulli( p) percolation.…”

Section: R Lyons and O Schrammmentioning

confidence: 99%

Indistinguishability of Percolation Clusters

Lyons

Schramm

2011

Selected Works of Oded Schramm

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We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for Pu'

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Percolation on Transitive Graphs as a Coalescent Process: Relentless Merging Followed by Simultaneous Uniqueness

Cited by 55 publications

References 27 publications

Critical Percolation on any Nonamenable Group has no Infinite Clusters

Critical Percolation on any Nonamenable Group has no Infinite Clusters

Coarse Geometry and Randomness

Indistinguishability of Percolation Clusters

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