Percolation Perturbations in Potential Theory and Random Walks

Details Percolation Perturbations in Potential Theory and Random Walks Percolation Perturbations in Potential Theory and Random Walks

1998-04-02

arxiv.org/abs/math/9804010v1

Random walks and discrete potential theory (Cortona, 1997), 56--84, Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge, 19

Mathematics

Probability

29 pages. For related papers, see the authors' WWW pages. Benjamini: http://www.wisdom.weizmann.ac.il/~itai Lyons: http://php.

Scientific paper

We 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.

Affiliated with

Also associated with

No associations

Rating

Percolation Perturbations in Potential Theory and Random Walks does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Percolation Perturbations in Potential Theory and Random Walks, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Percolation Perturbations in Potential Theory and Random Walks will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-121755