Indistinguishability of Percolation Clusters

Lyons, Russell; Schramm, Oded

doi:10.1007/978-1-4419-9675-6_23

Cited by 25 publications

(17 citation statements)

References 34 publications

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“…For the Li.d. bond percolation case we can then invoke the cluster indistinguishability result of Lyons and Schramm [56] (Theorem 6.1 below) to deduce that all infinite clusters have the same SRW speed.…”

Section: Random Walks On Percolation Clustersmentioning

confidence: 96%

“…This strongly suggests that DSRW on such an infinite cluster should be transient. Lyons and Schramm [56] converts this intuition into a proof by combining the result mentioned in the final paragraph of Section 3 about such infinite clusters C having Pc,bond( C) < 1, with a basic comparison of random walk and percolation on trees due to Lyons [50]. (Alternatively, we could quote Theorem 5.1 here, but that would be a detour.)…”

Section: Lp'(-yznpx E B) = Lp'(x E B)mentioning

confidence: 99%

“…Lyons and Schramm decided to put their most general version of the stationarity result not in [56] but in a separate paper [57]. Without unimodularity, however, stationarity fails, as is easily seen by considering DSRW in Example 3.4: here, Zo has probability ~ of sitting in the "topmost" (closest to ~) component of its open cluster, while the probability that Zn does so tends to ° as n ---+ 00.)…”

Section: Lp'(-yznpx E B) = Lp'(x E B)mentioning

confidence: 99%

“…For further reactions to Oded's untimely death, and memories of his life and work, see for instance Lyons [54], Haggstrom [33] and Werner [79], as well as the blog [59].…”

Section: Introductionmentioning

confidence: 99%

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Percolation beyond ℤ d : the contributions of Oded Schramm*

Häggström¹

2011

Selected Works of Oded Schramm

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Oded Schramm (1961Schramm ( -2008 influenced greatly the development of percolation theory beyond the usual 7!.,d setting, in particular the case of nonamenable lattices. Here we review some of his work in this field. Introd uctionOded Schramm was born in 1961 and died in a hiking accident in 2008, in what otherwise seemed to be the middle of an extraordinary mathematical career. Although he made seminal contributions to many areas of mathematics in general and probability in particular, I will here restrict attention to his work on percolation processes taking place on graph structures more exotic than the usual 7J.,d setting. The title I've chosen alludes to the short but highly influential paper Percolation beyond 7J.,d, many questions and a few answers from 1996 by Itai Benjamini and Oded Schramm [13].I need to point out, however, that there are at least two respects in which I will fail to deliver on what my chosen title suggests. Firstly, I will not come anywhere near an exhaustive exposition of Oded's contributions to the field. All I can offer is a personal and highly subjective selection of highlights. Secondly, Oded was a very collaborative mathematician, and I will make no attempt (if it even makes sense) at identifying his individual contributions as opposed to his coauthors'. Suffice it to say that everyone who worked with him knew him as a very generous person and as ' someone who would not put his name on a paper unless he had contributed at least his fair share. I'll just quote one recollection from Oded's long-time collaborator and friend Russ Lyons:To me, Oded's most distinctive mathematical talent was his extraordinary clarity of thought, which led to dazzling proofs and results. Technical difficulties did not obscure his vision. Indeed, they often melted away under his gaze. At one point when the four of us [Oded, Russ, Itai Benjamini and Yuval Peres] were working on uniform spanning forests, Oded came up with a brilliant new application of the Mass-Transport Principle. We weren't sure it was kosher, and I still recall Yuval asking me if I believed it, saying that it seemed to be "smoke and mirrors". However, when Oded explained it again, the smoke vanished. [59] Following the spirit of the aformentioned paper [13], I will take "percolation beyond Zd" to mean percolation process that are not naturally thought of as embedded in d How it beganIt will be assumed throughout that G = (V, E) is an infinite but locally finite connected graph. In i.i.d. site percolation on G with retention parameter P E [0, 1], each vertex v E V is declared open (retained, value 1) with probability p and closed (deleted, value 0) with the remaining probability 1-p, and this is done independently for different vertices. Alternatively, one may consider LLd. bond percolation, which is similar except that it is the edges rather than the vertices that are declared open or closed. Write lP' p,site and lP' p,bond for the resulting probability measures on {O, I} v and {O,l}E, respectively. The choice whether t...

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Section: Random Walks On Percolation Clustersmentioning

confidence: 96%

Section: Lp'(-yznpx E B) = Lp'(x E B)mentioning

confidence: 99%

Section: Lp'(-yznpx E B) = Lp'(x E B)mentioning

confidence: 99%

“…For further reactions to Oded's untimely death, and memories of his life and work, see for instance Lyons [54], Haggstrom [33] and Werner [79], as well as the blog [59].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Percolation beyond ℤ d : the contributions of Oded Schramm*

Häggström¹

2011

Selected Works of Oded Schramm

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“…The finite energy property was considered in [NS81]. In the sense of Lyons and Schramm [LS99], it corresponds to insertion and deletion tolerance: given an edge e, the conditional probability that e is present (resp. absent) given the status of all the other edges is positive.…”

Section: Introductionmentioning

confidence: 99%

Homogenization via sprinkling

Benjamini¹,

Tassion²

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. We show that a superposition of an ε-Bernoulli bond percolation and any everywhere percolating subgraph of Z d , d ≥ 2, results in a connected subgraph, which after a renormalization dominates supercritical Bernoulli percolation. This result, which confirms a conjecture from [BHS00], is mainly motivated by obtaining finite volume characterizations of uniqueness for general percolation processes.

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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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Indistinguishability of Percolation Clusters

Cited by 25 publications

References 34 publications

Percolation beyond ℤ d : the contributions of Oded Schramm*

Percolation beyond ℤ d : the contributions of Oded Schramm*

Homogenization via sprinkling

Critical Percolation on any Nonamenable Group has no Infinite Clusters

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