Percolation Perturbations in Potential Theory and Random Walks

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dc.creatorBenjamini, Itai
dc.creatorLyons, Russell
dc.creatorSchramm, Oded
dc.date1998-04-02
dc.date.accessioned2026-07-07T05:24:17Z
dc.date.available2026-07-07T05:24:17Z
dc.descriptionWe show that on a Cayley graph of a nonamenable group, almost surely the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoperimetric constant. We also show that percolation clusters in any amenable Cayley graph almost surely admit no nonconstant harmonic Dirichlet functions. Conversely, on a Cayley graph admitting nonconstant harmonic Dirichlet functions, almost surely the infinite clusters of $p$-Bernoulli percolation also have nonconstant harmonic Dirichlet functions when $p$ is sufficiently close to 1. Many conjectures and questions are presented.
dc.description29 pages. For related papers, see the authors' WWW pages. Benjamini: http://www.wisdom.weizmann.ac.il/~itai Lyons: http://php.indiana.edu/~rdlyons Schramm: http://www.wisdom.weizmann.ac.il/~schramm
dc.identifierhttps://arxiv.org/abs/math/9804010
dc.identifierhttp://arxiv.org/abs/math/9804010
dc.identifierRandom walks and discrete potential theory (Cortona, 1997), 56--84, Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge, 1999
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/76781
dc.subjectProbability
dc.subjectMathematical Physics
dc.subject60B99; 60D05; 31C20; 60J15; 31B05; 20F32
dc.titlePercolation Perturbations in Potential Theory and Random Walks
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