Oriented Random Walk on the Heisenberg Group and Percolation

Abstract

It is shown that oriented random walk on the Heisenberg group admits exponential intersection tail. As a corollary we get that on any transitive graph of polynomial volume growth, which is not a finite extension of $\mathbb{Z},{\mathbb{Z}}^2$, the infinite cluster of percolation with retention parameter $p$, close enough to $1$, is transient.

Written in 1998, 6 pages. arXiv admin note: text overlap with arXiv:math/9701227